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Mary Pyo

Mary Pyo

Measuring Angles in Triangles

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Table of Contents

Section 1: Tools of Geometry
Coordinate Plane

16m 41s

Intro
0:00
The Coordinate System
0:12
Coordinate Plane: X-axis and Y-axis
0:15
Quadrants
1:02
Origin
2:00
Ordered Pair
2:17
Coordinate Plane
2:59
Example: Writing Coordinates
3:01
Coordinate Plane, cont.
4:15
Example: Graphing & Coordinate Plane
4:17
Collinear
5:58
Extra Example 1: Writing Coordinates & Quadrants
7:34
Extra Example 2: Quadrants
8:52
Extra Example 3: Graphing & Coordinate Plane
10:58
Extra Example 4: Collinear
12:50
Points, Lines and Planes

17m 17s

Intro
0:00
Points
0:07
Definition and Example of Points
0:09
Lines
0:50
Definition and Example of Lines
0:51
Planes
2:59
Definition and Example of Planes
3:00
Drawing and Labeling
4:40
Example 1: Drawing and Labeling
4:41
Example 2: Drawing and Labeling
5:54
Example 3: Drawing and Labeling
6:41
Example 4: Drawing and Labeling
8:23
Extra Example 1: Points, Lines and Planes
10:19
Extra Example 2: Naming Figures
11:16
Extra Example 3: Points, Lines and Planes
12:35
Extra Example 4: Draw and Label
14:44
Measuring Segments

31m 31s

Intro
0:00
Segments
0:06
Examples of Segments
0:08
Ruler Postulate
1:30
Ruler Postulate
1:31
Segment Addition Postulate
5:02
Example and Definition of Segment Addition Postulate
5:03
Segment Addition Postulate
8:01
Example 1: Segment Addition Postulate
8:04
Example 2: Segment Addition Postulate
11:15
Pythagorean Theorem
12:36
Definition of Pythagorean Theorem
12:37
Pythagorean Theorem, cont.
15:49
Example: Pythagorean Theorem
15:50
Distance Formula
16:48
Example and Definition of Distance Formula
16:49
Extra Example 1: Find Each Measure
20:32
Extra Example 2: Find the Missing Measure
22:11
Extra Example 3: Find the Distance Between the Two Points
25:36
Extra Example 4: Pythagorean Theorem
29:33
Midpoints and Segment Congruence

42m 26s

Intro
0:00
Definition of Midpoint
0:07
Midpoint
0:10
Midpoint Formulas
1:30
Midpoint Formula: On a Number Line
1:45
Midpoint Formula: In a Coordinate Plane
2:50
Midpoint
4:40
Example: Midpoint on a Number Line
4:43
Midpoint
6:05
Example: Midpoint in a Coordinate Plane
6:06
Midpoint
8:28
Example 1
8:30
Example 2
13:01
Segment Bisector
15:14
Definition and Example of Segment Bisector
15:15
Proofs
17:27
Theorem
17:53
Proof
18:21
Midpoint Theorem
19:37
Example: Proof & Midpoint Theorem
19:38
Extra Example 1: Midpoint on a Number Line
23:44
Extra Example 2: Drawing Diagrams
26:25
Extra Example 3: Midpoint
29:14
Extra Example 4: Segment Bisector
33:21
Angles

42m 34s

Intro
0:00
Angles
0:05
Angle
0:07
Ray
0:23
Opposite Rays
2:09
Angles
3:22
Example: Naming Angle
3:23
Angles
6:39
Interior, Exterior, Angle
6:40
Measure and Degrees
7:38
Protractor Postulate
8:37
Example: Protractor Postulate
8:38
Angle Addition Postulate
11:41
Example: Angle addition Postulate
11:42
Classifying Angles
14:10
Acute Angle
14:16
Right Angles
14:30
Obtuse Angle
14:41
Angle Bisector
15:02
Example: Angle Bisector
15:04
Angle Relationships
16:43
Adjacent Angles
16:47
Vertical Angles
17:49
Linear Pair
19:40
Angle Relationships
20:31
Right Angles
20:32
Supplementary Angles
21:15
Complementary Angles
21:33
Extra Example 1: Angles
24:08
Extra Example 2: Angles
29:06
Extra Example 3: Angles
32:05
Extra Example 4 Angles
35:44
Section 2: Reasoning & Proof
Inductive Reasoning

19m

Intro
0:00
Inductive Reasoning
0:05
Conjecture
0:06
Inductive Reasoning
0:15
Examples
0:55
Example: Sequence
0:56
More Example: Sequence
2:00
Using Inductive Reasoning
2:50
Example: Conjecture
2:51
More Example: Conjecture
3:48
Counterexamples
4:56
Counterexample
4:58
Extra Example 1: Conjecture
6:59
Extra Example 2: Sequence and Pattern
10:20
Extra Example 3: Inductive Reasoning
12:46
Extra Example 4: Conjecture and Counterexample
15:17
Conditional Statements

42m 47s

Intro
0:00
If Then Statements
0:05
If Then Statements
0:06
Other Forms
2:29
Example: Without Then
2:40
Example: Using When
3:03
Example: Hypothesis
3:24
Identify the Hypothesis and Conclusion
3:52
Example 1: Hypothesis and Conclusion
3:58
Example 2: Hypothesis and Conclusion
4:31
Example 3: Hypothesis and Conclusion
5:38
Write in If Then Form
6:16
Example 1: Write in If Then Form
6:23
Example 2: Write in If Then Form
6:57
Example 3: Write in If Then Form
7:39
Other Statements
8:40
Other Statements
8:41
Converse Statements
9:18
Converse Statements
9:20
Converses and Counterexamples
11:04
Converses and Counterexamples
11:05
Example 1: Converses and Counterexamples
12:02
Example 2: Converses and Counterexamples
15:10
Example 3: Converses and Counterexamples
17:08
Inverse Statement
19:58
Definition and Example
19:59
Inverse Statement
21:46
Example 1: Inverse and Counterexample
21:47
Example 2: Inverse and Counterexample
23:34
Contrapositive Statement
25:20
Definition and Example
25:21
Contrapositive Statement
26:58
Example: Contrapositive Statement
27:00
Summary
29:03
Summary of Lesson
29:04
Extra Example 1: Hypothesis and Conclusion
32:20
Extra Example 2: If-Then Form
33:23
Extra Example 3: Converse, Inverse, and Contrapositive
34:54
Extra Example 4: Converse, Inverse, and Contrapositive
37:56
Point, Line, and Plane Postulates

17m 24s

Intro
0:00
What are Postulates?
0:09
Definition of Postulates
0:10
Postulates
1:22
Postulate 1: Two Points
1:23
Postulate 2: Three Points
2:02
Postulate 3: Line
2:45
Postulates, cont..
3:08
Postulate 4: Plane
3:09
Postulate 5: Two Points in a Plane
3:53
Postulates, cont..
4:46
Postulate 6: Two Lines Intersect
4:47
Postulate 7: Two Plane Intersect
5:28
Using the Postulates
6:34
Examples: True or False
6:35
Using the Postulates
10:18
Examples: True or False
10:19
Extra Example 1: Always, Sometimes, or Never
12:22
Extra Example 2: Always, Sometimes, or Never
13:15
Extra Example 3: Always, Sometimes, or Never
14:16
Extra Example 4: Always, Sometimes, or Never
15:03
Deductive Reasoning

36m 3s

Intro
0:00
Deductive Reasoning
0:06
Definition of Deductive Reasoning
0:07
Inductive vs. Deductive
2:51
Inductive Reasoning
2:52
Deductive reasoning
3:19
Law of Detachment
3:47
Law of Detachment
3:48
Examples of Law of Detachment
4:31
Law of Syllogism
7:32
Law of Syllogism
7:33
Example 1: Making a Conclusion
9:02
Example 2: Making a Conclusion
12:54
Using Laws of Logic
14:12
Example 1: Determine the Logic
14:42
Example 2: Determine the Logic
17:02
Using Laws of Logic, cont.
18:47
Example 3: Determine the Logic
19:03
Example 4: Determine the Logic
20:56
Extra Example 1: Determine the Conclusion and Law
22:12
Extra Example 2: Determine the Conclusion and Law
25:39
Extra Example 3: Determine the Logic and Law
29:50
Extra Example 4: Determine the Logic and Law
31:27
Proofs in Algebra: Properties of Equality

44m 31s

Intro
0:00
Properties of Equality
0:10
Addition Property of Equality
0:28
Subtraction Property of Equality
1:10
Multiplication Property of Equality
1:41
Division Property of Equality
1:55
Addition Property of Equality Using Angles
2:46
Properties of Equality, cont.
4:10
Reflexive Property of Equality
4:11
Symmetric Property of Equality
5:24
Transitive Property of Equality
6:10
Properties of Equality, cont.
7:04
Substitution Property of Equality
7:05
Distributive Property of Equality
8:34
Two Column Proof
9:40
Example: Two Column Proof
9:46
Proof Example 1
16:13
Proof Example 2
23:49
Proof Example 3
30:33
Extra Example 1: Name the Property of Equality
38:07
Extra Example 2: Name the Property of Equality
40:16
Extra Example 3: Name the Property of Equality
41:35
Extra Example 4: Name the Property of Equality
43:02
Proving Segment Relationship

41m 2s

Intro
0:00
Good Proofs
0:12
Five Essential Parts
0:13
Proof Reasons
1:38
Undefined
1:40
Definitions
2:06
Postulates
2:42
Previously Proven Theorems
3:24
Congruence of Segments
4:10
Theorem: Congruence of Segments
4:12
Proof Example
10:16
Proof: Congruence of Segments
10:17
Setting Up Proofs
19:13
Example: Two Segments with Equal Measures
19:15
Setting Up Proofs
21:48
Example: Vertical Angles are Congruent
21:50
Setting Up Proofs
23:59
Example: Segment of a Triangle
24:00
Extra Example 1: Congruence of Segments
27:03
Extra Example 2: Setting Up Proofs
28:50
Extra Example 3: Setting Up Proofs
30:55
Extra Example 4: Two-Column Proof
33:11
Proving Angle Relationships

33m 37s

Intro
0:00
Supplement Theorem
0:05
Supplementary Angles
0:06
Congruence of Angles
2:37
Proof: Congruence of Angles
2:38
Angle Theorems
6:54
Angle Theorem 1: Supplementary Angles
6:55
Angle Theorem 2: Complementary Angles
10:25
Angle Theorems
11:32
Angle Theorem 3: Right Angles
11:35
Angle Theorem 4: Vertical Angles
12:09
Angle Theorem 5: Perpendicular Lines
12:57
Using Angle Theorems
13:45
Example 1: Always, Sometimes, or Never
13:50
Example 2: Always, Sometimes, or Never
14:28
Example 3: Always, Sometimes, or Never
16:21
Extra Example 1: Always, Sometimes, or Never
16:53
Extra Example 2: Find the Measure of Each Angle
18:55
Extra Example 3: Find the Measure of Each Angle
25:03
Extra Example 4: Two-Column Proof
27:08
Section 3: Perpendicular & Parallel Lines
Parallel Lines and Transversals

37m 35s

Intro
0:00
Lines
0:06
Parallel Lines
0:09
Skew Lines
2:02
Transversal
3:42
Angles Formed by a Transversal
4:28
Interior Angles
5:53
Exterior Angles
6:09
Consecutive Interior Angles
7:04
Alternate Exterior Angles
9:47
Alternate Interior Angles
11:22
Corresponding Angles
12:27
Angles Formed by a Transversal
15:29
Relationship Between Angles
15:30
Extra Example 1: Intersecting, Parallel, or Skew
19:26
Extra Example 2: Draw a Diagram
21:37
Extra Example 3: Name the Figures
24:12
Extra Example 4: Angles Formed by a Transversal
28:38
Angles and Parallel Lines

41m 53s

Intro
0:00
Corresponding Angles Postulate
0:05
Corresponding Angles Postulate
0:06
Alternate Interior Angles Theorem
3:05
Alternate Interior Angles Theorem
3:07
Consecutive Interior Angles Theorem
5:16
Consecutive Interior Angles Theorem
5:17
Alternate Exterior Angles Theorem
6:42
Alternate Exterior Angles Theorem
6:43
Parallel Lines Cut by a Transversal
7:18
Example: Parallel Lines Cut by a Transversal
7:19
Perpendicular Transversal Theorem
14:54
Perpendicular Transversal Theorem
14:55
Extra Example 1: State the Postulate or Theorem
16:37
Extra Example 2: Find the Measure of the Numbered Angle
18:53
Extra Example 3: Find the Measure of Each Angle
25:13
Extra Example 4: Find the Values of x, y, and z
36:26
Slope of Lines

44m 6s

Intro
0:00
Definition of Slope
0:06
Slope Equation
0:13
Slope of a Line
3:45
Example: Find the Slope of a Line
3:47
Slope of a Line
8:38
More Example: Find the Slope of a Line
8:40
Slope Postulates
12:32
Proving Slope Postulates
12:33
Parallel or Perpendicular Lines
17:23
Example: Parallel or Perpendicular Lines
17:24
Using Slope Formula
20:02
Example: Using Slope Formula
20:03
Extra Example 1: Slope of a Line
25:10
Extra Example 2: Slope of a Line
26:31
Extra Example 3: Graph the Line
34:11
Extra Example 4: Using the Slope Formula
38:50
Proving Lines Parallel

25m 55s

Intro
0:00
Postulates
0:06
Postulate 1: Parallel Lines
0:21
Postulate 2: Parallel Lines
2:16
Parallel Postulate
3:28
Definition and Example of Parallel Postulate
3:29
Theorems
4:29
Theorem 1: Parallel Lines
4:40
Theorem 2: Parallel Lines
5:37
Theorems, cont.
6:10
Theorem 3: Parallel Lines
6:11
Extra Example 1: Determine Parallel Lines
6:56
Extra Example 2: Find the Value of x
11:42
Extra Example 3: Opposite Sides are Parallel
14:48
Extra Example 4: Proving Parallel Lines
20:42
Parallels and Distance

19m 48s

Intro
0:00
Distance Between a Points and Line
0:07
Definition and Example
0:08
Distance Between Parallel Lines
1:51
Definition and Example
1:52
Extra Example 1: Drawing a Segment to Represent Distance
3:02
Extra Example 2: Drawing a Segment to Represent Distance
4:27
Extra Example 3: Graph, Plot, and Construct a Perpendicular Segment
5:13
Extra Example 4: Distance Between Two Parallel Lines
15:37
Section 4: Congruent Triangles
Classifying Triangles

28m 43s

Intro
0:00
Triangles
0:09
Triangle: A Three-Sided Polygon
0:10
Sides
1:00
Vertices
1:22
Angles
1:56
Classifying Triangles by Angles
2:59
Acute Triangle
3:19
Obtuse Triangle
4:08
Right Triangle
4:44
Equiangular Triangle
5:38
Definition and Example of an Equiangular Triangle
5:39
Classifying Triangles by Sides
6:57
Scalene Triangle
7:17
Isosceles Triangle
7:57
Equilateral Triangle
8:12
Isosceles Triangle
8:58
Labeling Isosceles Triangle
9:00
Labeling Right Triangle
10:44
Isosceles Triangle
11:10
Example: Find x, AB, BC, and AC
11:11
Extra Example 1: Classify Each Triangle
13:45
Extra Example 2: Always, Sometimes, or Never
16:28
Extra Example 3: Find All the Sides of the Isosceles Triangle
20:29
Extra Example 4: Distance Formula and Triangle
22:29
Measuring Angles in Triangles

44m 43s

Intro
0:00
Angle Sum Theorem
0:09
Angle Sum Theorem for Triangle
0:11
Using Angle Sum Theorem
4:06
Find the Measure of the Missing Angle
4:07
Third Angle Theorem
4:58
Example: Third Angle Theorem
4:59
Exterior Angle Theorem
7:58
Example: Exterior Angle Theorem
8:00
Flow Proof of Exterior Angle Theorem
15:14
Flow Proof of Exterior Angle Theorem
15:17
Triangle Corollaries
27:21
Triangle Corollary 1
27:50
Triangle Corollary 2
30:42
Extra Example 1: Find the Value of x
32:55
Extra Example 2: Find the Value of x
34:20
Extra Example 3: Find the Measure of the Angle
35:38
Extra Example 4: Find the Measure of Each Numbered Angle
39:00
Exploring Congruent Triangles

26m 46s

Intro
0:00
Congruent Triangles
0:15
Example of Congruent Triangles
0:17
Corresponding Parts
3:39
Corresponding Angles and Sides of Triangles
3:40
Definition of Congruent Triangles
11:24
Definition of Congruent Triangles
11:25
Triangle Congruence
16:37
Congruence of Triangles
16:38
Extra Example 1: Congruence Statement
18:24
Extra Example 2: Congruence Statement
21:26
Extra Example 3: Draw and Label the Figure
23:09
Extra Example 4: Drawing Triangles
24:04
Proving Triangles Congruent

47m 51s

Intro
0:00
SSS Postulate
0:18
Side-Side-Side Postulate
0:27
SAS Postulate
2:26
Side-Angle-Side Postulate
2:29
SAS Postulate
3:57
Proof Example
3:58
ASA Postulate
11:47
Angle-Side-Angle Postulate
11:53
AAS Theorem
14:13
Angle-Angle-Side Theorem
14:14
Methods Overview
16:16
Methods Overview
16:17
SSS
16:33
SAS
17:06
ASA
17:50
AAS
18:17
CPCTC
19:14
Extra Example 1:Proving Triangles are Congruent
21:29
Extra Example 2: Proof
25:40
Extra Example 3: Proof
30:41
Extra Example 4: Proof
38:41
Isosceles and Equilateral Triangles

27m 53s

Intro
0:00
Isosceles Triangle Theorem
0:07
Isosceles Triangle Theorem
0:09
Isosceles Triangle Theorem
2:26
Example: Using the Isosceles Triangle Theorem
2:27
Isosceles Triangle Theorem Converse
3:29
Isosceles Triangle Theorem Converse
3:30
Equilateral Triangle Theorem Corollaries
4:30
Equilateral Triangle Theorem Corollary 1
4:59
Equilateral Triangle Theorem Corollary 2
5:55
Extra Example 1: Find the Value of x
7:08
Extra Example 2: Find the Value of x
10:04
Extra Example 3: Proof
14:04
Extra Example 4: Proof
22:41
Section 5: Triangle Inequalities
Special Segments in Triangles

43m 44s

Intro
0:00
Perpendicular Bisector
0:06
Perpendicular Bisector
0:07
Perpendicular Bisector
4:07
Perpendicular Bisector Theorems
4:08
Median
6:30
Definition of Median
6:31
Median
9:41
Example: Median
9:42
Altitude
12:22
Definition of Altitude
12:23
Angle Bisector
14:33
Definition of Angle Bisector
14:34
Angle Bisector
16:41
Angle Bisector Theorems
16:42
Special Segments Overview
18:57
Perpendicular Bisector
19:04
Median
19:32
Altitude
19:49
Angle Bisector
20:02
Examples: Special Segments
20:18
Extra Example 1: Draw and Label
22:36
Extra Example 2: Draw the Altitudes for Each Triangle
24:37
Extra Example 3: Perpendicular Bisector
27:57
Extra Example 4: Draw, Label, and Write Proof
34:33
Right Triangles

26m 34s

Intro
0:00
LL Theorem
0:21
Leg-Leg Theorem
0:25
HA Theorem
2:23
Hypotenuse-Angle Theorem
2:24
LA Theorem
4:49
Leg-Angle Theorem
4:50
LA Theorem
6:18
Example: Find x and y
6:19
HL Postulate
8:22
Hypotenuse-Leg Postulate
8:23
Extra Example 1: LA Theorem & HL Postulate
10:57
Extra Example 2: Find x So That Each Pair of Triangles is Congruent
14:15
Extra Example 3: Two-column Proof
17:02
Extra Example 4: Two-column Proof
21:01
Indirect Proofs and Inequalities

33m 30s

Intro
0:00
Writing an Indirect Proof
0:09
Step 1
0:49
Step 2
2:32
Step 3
3:00
Indirect Proof
4:30
Example: 2 + 6 = 8
5:00
Example: The Suspect is Guilty
5:40
Example: Measure of Angle A < Measure of Angle B
6:06
Definition of Inequality
7:47
Definition of Inequality & Example
7:48
Properties of Inequality
9:55
Comparison Property
9:58
Transitive Property
10:33
Addition and Subtraction Properties
12:01
Multiplication and Division Properties
13:07
Exterior Angle Inequality Theorem
14:12
Example: Exterior Angle Inequality Theorem
14:13
Extra Example 1: Draw a Diagram for the Statement
18:32
Extra Example 2: Name the Property for Each Statement
19:56
Extra Example 3: State the Assumption
21:22
Extra Example 4: Write an Indirect Proof
25:39
Inequalities for Sides and Angles of a Triangle

17m 26s

Intro
0:00
Side to Angles
0:10
If One Side of a Triangle is Longer Than Another Side
0:11
Converse: Angles to Sides
1:57
If One Angle of a Triangle Has a Greater Measure Than Another Angle
1:58
Extra Example 1: Name the Angles in the Triangle From Least to Greatest
2:38
Extra Example 2: Find the Longest and Shortest Segment in the Triangle
3:47
Extra Example 3: Angles and Sides of a Triangle
4:51
Extra Example 4: Two-column Proof
9:08
Triangle Inequality

28m 11s

Intro
0:00
Triangle Inequality Theorem
0:05
Triangle Inequality Theorem
0:06
Triangle Inequality Theorem
4:22
Example 1: Triangle Inequality Theorem
4:23
Example 2: Triangle Inequality Theorem
9:40
Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle
12:00
Extra Example 2: Finding the Third Side of a Triangle
13:34
Extra Example 3: Always True, Sometimes True, or Never True
18:18
Extra Example 4: Triangle and Vertices
22:36
Inequalities Involving Two Triangles

29m 36s

Intro
0:00
SAS Inequality Theorem
0:06
SAS Inequality Theorem & Example
0:25
SSS Inequality Theorem
4:33
SSS Inequality Theorem & Example
4:34
Extra Example 1: Write an Inequality Comparing the Segments
6:08
Extra Example 2: Determine if the Statement is True
9:52
Extra Example 3: Write an Inequality for x
14:20
Extra Example 4: Two-column Proof
17:44
Section 6: Quadrilaterals
Parallelograms

29m 11s

Intro
0:00
Quadrilaterals
0:06
Four-sided Polygons
0:08
Non Examples of Quadrilaterals
0:47
Parallelograms
1:35
Parallelograms
1:36
Properties of Parallelograms
4:28
Opposite Sides of a Parallelogram are Congruent
4:29
Opposite Angles of a Parallelogram are Congruent
5:49
Angles and Diagonals
6:24
Consecutive Angles in a Parallelogram are Supplementary
6:25
The Diagonals of a Parallelogram Bisect Each Other
8:42
Extra Example 1: Complete Each Statement About the Parallelogram
10:26
Extra Example 2: Find the Values of x, y, and z of the Parallelogram
13:21
Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram
16:35
Extra Example 4: Slope of Parallelogram
23:15
Proving Parallelograms

42m 43s

Intro
0:00
Parallelogram Theorems
0:09
Theorem 1
0:20
Theorem 2
1:50
Parallelogram Theorems, Cont.
3:10
Theorem 3
3:11
Theorem 4
4:15
Proving Parallelogram
6:21
Example: Determine if Quadrilateral ABCD is a Parallelogram
6:22
Summary
14:01
Both Pairs of Opposite Sides are Parallel
14:14
Both Pairs of Opposite Sides are Congruent
15:09
Both Pairs of Opposite Angles are Congruent
15:24
Diagonals Bisect Each Other
15:44
A Pair of Opposite Sides is Both Parallel and Congruent
16:13
Extra Example 1: Determine if Each Quadrilateral is a Parallelogram
16:54
Extra Example 2: Find the Value of x and y
20:23
Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram
24:05
Extra Example 4: Two-column Proof
30:28
Rectangles

29m 47s

Intro
0:00
Rectangles
0:03
Definition of Rectangles
0:04
Diagonals of Rectangles
2:52
Rectangles: Diagonals Property 1
2:53
Rectangles: Diagonals Property 2
3:30
Proving a Rectangle
4:40
Example: Determine Whether Parallelogram ABCD is a Rectangle
4:41
Rectangles Summary
9:22
Opposite Sides are Congruent and Parallel
9:40
Opposite Angles are Congruent
9:51
Consecutive Angles are Supplementary
9:58
Diagonals are Congruent and Bisect Each Other
10:05
All Four Angles are Right Angles
10:40
Extra Example 1: Find the Value of x
11:03
Extra Example 2: Name All Congruent Sides and Angles
13:52
Extra Example 3: Always, Sometimes, or Never True
19:39
Extra Example 4: Determine if ABCD is a Rectangle
26:45
Squares and Rhombi

39m 14s

Intro
0:00
Rhombus
0:09
Definition of a Rhombus
0:10
Diagonals of a Rhombus
2:03
Rhombus: Diagonals Property 1
2:21
Rhombus: Diagonals Property 2
3:49
Rhombus: Diagonals Property 3
4:36
Rhombus
6:17
Example: Use the Rhombus to Find the Missing Value
6:18
Square
8:17
Definition of a Square
8:20
Summary Chart
11:06
Parallelogram
11:07
Rectangle
12:56
Rhombus
13:54
Square
14:44
Extra Example 1: Diagonal Property
15:44
Extra Example 2: Use Rhombus ABCD to Find the Missing Value
19:39
Extra Example 3: Always, Sometimes, or Never True
23:06
Extra Example 4: Determine the Quadrilateral
28:02
Trapezoids and Kites

30m 48s

Intro
0:00
Trapezoid
0:10
Definition of Trapezoid
0:12
Isosceles Trapezoid
2:57
Base Angles of an Isosceles Trapezoid
2:58
Diagonals of an Isosceles Trapezoid
4:05
Median of a Trapezoid
4:26
Median of a Trapezoid
4:27
Median of a Trapezoid
6:41
Median Formula
7:00
Kite
8:28
Definition of a Kite
8:29
Quadrilaterals Summary
11:19
A Quadrilateral with Two Pairs of Adjacent Congruent Sides
11:20
Extra Example 1: Isosceles Trapezoid
14:50
Extra Example 2: Median of Trapezoid
18:28
Extra Example 3: Always, Sometimes, or Never
24:13
Extra Example 4: Determine if the Figure is a Trapezoid
26:49
Section 7: Proportions and Similarity
Using Proportions and Ratios

20m 10s

Intro
0:00
Ratio
0:05
Definition and Examples of Writing Ratio
0:06
Proportion
2:05
Definition of Proportion
2:06
Examples of Proportion
2:29
Using Ratio
5:53
Example: Ratio
5:54
Extra Example 1: Find Three Ratios Equivalent to 2/5
9:28
Extra Example 2: Proportion and Cross Products
10:32
Extra Example 3: Express Each Ratio as a Fraction
13:18
Extra Example 4: Fin the Measure of a 3:4:5 Triangle
17:26
Similar Polygons

27m 53s

Intro
0:00
Similar Polygons
0:05
Definition of Similar Polygons
0:06
Example of Similar Polygons
2:32
Scale Factor
4:26
Scale Factor: Definition and Example
4:27
Extra Example 1: Determine if Each Pair of Figures is Similar
7:03
Extra Example 2: Find the Values of x and y
11:33
Extra Example 3: Similar Triangles
19:57
Extra Example 4: Draw Two Similar Figures
23:36
Similar Triangles

34m 10s

Intro
0:00
AA Similarity
0:10
Definition of AA Similarity
0:20
Example of AA Similarity
2:32
SSS Similarity
4:46
Definition of SSS Similarity
4:47
Example of SSS Similarity
6:00
SAS Similarity
8:04
Definition of SAS Similarity
8:05
Example of SAS Similarity
9:12
Extra Example 1: Determine Whether Each Pair of Triangles is Similar
10:59
Extra Example 2: Determine Which Triangles are Similar
16:08
Extra Example 3: Determine if the Statement is True or False
23:11
Extra Example 4: Write Two-Column Proof
26:25
Parallel Lines and Proportional Parts

24m 7s

Intro
0:00
Triangle Proportionality
0:07
Definition of Triangle Proportionality
0:08
Example of Triangle Proportionality
0:51
Triangle Proportionality Converse
2:19
Triangle Proportionality Converse
2:20
Triangle Mid-segment
3:42
Triangle Mid-segment: Definition and Example
3:43
Parallel Lines and Transversal
6:51
Parallel Lines and Transversal
6:52
Extra Example 1: Complete Each Statement
8:59
Extra Example 2: Determine if the Statement is True or False
12:28
Extra Example 3: Find the Value of x and y
15:35
Extra Example 4: Find Midpoints of a Triangle
20:43
Parts of Similar Triangles

27m 6s

Intro
0:00
Proportional Perimeters
0:09
Proportional Perimeters: Definition and Example
0:10
Similar Altitudes
2:23
Similar Altitudes: Definition and Example
2:24
Similar Angle Bisectors
4:50
Similar Angle Bisectors: Definition and Example
4:51
Similar Medians
6:05
Similar Medians: Definition and Example
6:06
Angle Bisector Theorem
7:33
Angle Bisector Theorem
7:34
Extra Example 1: Parts of Similar Triangles
10:52
Extra Example 2: Parts of Similar Triangles
14:57
Extra Example 3: Parts of Similar Triangles
19:27
Extra Example 4: Find the Perimeter of Triangle ABC
23:14
Section 8: Applying Right Triangles & Trigonometry
Pythagorean Theorem

21m 14s

Intro
0:00
Pythagorean Theorem
0:05
Pythagorean Theorem & Example
0:06
Pythagorean Converse
1:20
Pythagorean Converse & Example
1:21
Pythagorean Triple
2:42
Pythagorean Triple
2:43
Extra Example 1: Find the Missing Side
4:59
Extra Example 2: Determine Right Triangle
7:40
Extra Example 3: Determine Pythagorean Triple
11:30
Extra Example 4: Vertices and Right Triangle
14:29
Geometric Mean

40m 59s

Intro
0:00
Geometric Mean
0:04
Geometric Mean & Example
0:05
Similar Triangles
4:32
Similar Triangles
4:33
Geometric Mean-Altitude
11:10
Geometric Mean-Altitude & Example
11:11
Geometric Mean-Leg
14:47
Geometric Mean-Leg & Example
14:18
Extra Example 1: Geometric Mean Between Each Pair of Numbers
20:10
Extra Example 2: Similar Triangles
23:46
Extra Example 3: Geometric Mean of Triangles
28:30
Extra Example 4: Geometric Mean of Triangles
36:58
Special Right Triangles

37m 57s

Intro
0:00
45-45-90 Triangles
0:06
Definition of 45-45-90 Triangles
0:25
45-45-90 Triangles
5:51
Example: Find n
5:52
30-60-90 Triangles
8:59
Definition of 30-60-90 Triangles
9:00
30-60-90 Triangles
12:25
Example: Find n
12:26
Extra Example 1: Special Right Triangles
15:08
Extra Example 2: Special Right Triangles
18:22
Extra Example 3: Word Problems & Special Triangles
27:40
Extra Example 4: Hexagon & Special Triangles
33:51
Ratios in Right Triangles

40m 37s

Intro
0:00
Trigonometric Ratios
0:08
Definition of Trigonometry
0:13
Sine (sin), Cosine (cos), & Tangent (tan)
0:50
Trigonometric Ratios
3:04
Trig Functions
3:05
Inverse Trig Functions
5:02
SOHCAHTOA
8:16
sin x
9:07
cos x
10:00
tan x
10:32
Example: SOHCAHTOA & Triangle
12:10
Extra Example 1: Find the Value of Each Ratio or Angle Measure
14:36
Extra Example 2: Find Sin, Cos, and Tan
18:51
Extra Example 3: Find the Value of x Using SOHCAHTOA
22:55
Extra Example 4: Trigonometric Ratios in Right Triangles
32:13
Angles of Elevation and Depression

21m 4s

Intro
0:00
Angle of Elevation
0:10
Definition of Angle of Elevation & Example
0:11
Angle of Depression
1:19
Definition of Angle of Depression & Example
1:20
Extra Example 1: Name the Angle of Elevation and Depression
2:22
Extra Example 2: Word Problem & Angle of Depression
4:41
Extra Example 3: Word Problem & Angle of Elevation
14:02
Extra Example 4: Find the Missing Measure
18:10
Law of Sines

35m 25s

Intro
0:00
Law of Sines
0:20
Law of Sines
0:21
Law of Sines
3:34
Example: Find b
3:35
Solving the Triangle
9:19
Example: Using the Law of Sines to Solve Triangle
9:20
Extra Example 1: Law of Sines and Triangle
17:43
Extra Example 2: Law of Sines and Triangle
20:06
Extra Example 3: Law of Sines and Triangle
23:54
Extra Example 4: Law of Sines and Triangle
28:59
Law of Cosines

52m 43s

Intro
0:00
Law of Cosines
0:35
Law of Cosines
0:36
Law of Cosines
6:22
Use the Law of Cosines When Both are True
6:23
Law of Cosines
8:35
Example: Law of Cosines
8:36
Extra Example 1: Law of Sines or Law of Cosines?
13:35
Extra Example 2: Use the Law of Cosines to Find the Missing Measure
17:02
Extra Example 3: Solve the Triangle
30:49
Extra Example 4: Find the Measure of Each Diagonal of the Parallelogram
41:39
Section 9: Circles
Segments in a Circle

22m 43s

Intro
0:00
Segments in a Circle
0:10
Circle
0:11
Chord
0:59
Diameter
1:32
Radius
2:07
Secant
2:17
Tangent
3:10
Circumference
3:56
Introduction to Circumference
3:57
Example: Find the Circumference of the Circle
5:09
Circumference
6:40
Example: Find the Circumference of the Circle
6:41
Extra Example 1: Use the Circle to Answer the Following
9:10
Extra Example 2: Find the Missing Measure
12:53
Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle
15:51
Extra Example 4: Find the Circumference of Each Circle
19:24
Angles and Arc

35m 24s

Intro
0:00
Central Angle
0:06
Definition of Central Angle
0:07
Sum of Central Angles
1:17
Sum of Central Angles
1:18
Arcs
2:27
Minor Arc
2:30
Major Arc
3:47
Arc Measure
5:24
Measure of Minor Arc
5:24
Measure of Major Arc
6:53
Measure of a Semicircle
7:11
Arc Addition Postulate
8:25
Arc Addition Postulate
8:26
Arc Length
9:43
Arc Length and Example
9:44
Concentric Circles
16:05
Concentric Circles
16:06
Congruent Circles and Arcs
17:50
Congruent Circles
17:51
Congruent Arcs
18:47
Extra Example 1: Minor Arc, Major Arc, and Semicircle
20:14
Extra Example 2: Measure and Length of Arc
22:52
Extra Example 3: Congruent Arcs
25:48
Extra Example 4: Angles and Arcs
30:33
Arcs and Chords

21m 51s

Intro
0:00
Arcs and Chords
0:07
Arc of the Chord
0:08
Theorem 1: Congruent Minor Arcs
1:01
Inscribed Polygon
2:10
Inscribed Polygon
2:11
Arcs and Chords
3:18
Theorem 2: When a Diameter is Perpendicular to a Chord
3:19
Arcs and Chords
5:05
Theorem 3: Congruent Chords
5:06
Extra Example 1: Congruent Arcs
10:35
Extra Example 2: Length of Arc
13:50
Extra Example 3: Arcs and Chords
17:09
Extra Example 4: Arcs and Chords
19:45
Inscribed Angles

27m 53s

Intro
0:00
Inscribed Angles
0:07
Definition of Inscribed Angles
0:08
Inscribed Angles
0:58
Inscribed Angle Theorem 1
0:59
Inscribed Angles
3:29
Inscribed Angle Theorem 2
3:30
Inscribed Angles
4:38
Inscribed Angle Theorem 3
4:39
Inscribed Quadrilateral
5:50
Inscribed Quadrilateral
5:51
Extra Example 1: Central Angle, Inscribed Angle, and Intercepted Arc
7:02
Extra Example 2: Inscribed Angles
9:24
Extra Example 3: Inscribed Angles
14:00
Extra Example 4: Complete the Proof
17:58
Tangents

26m 16s

Intro
0:00
Tangent Theorems
0:04
Tangent Theorem 1
0:05
Tangent Theorem 1 Converse
0:55
Common Tangents
1:34
Common External Tangent
2:12
Common Internal Tangent
2:30
Tangent Segments
3:08
Tangent Segments
3:09
Circumscribed Polygons
4:11
Circumscribed Polygons
4:12
Extra Example 1: Tangents & Circumscribed Polygons
5:50
Extra Example 2: Tangents & Circumscribed Polygons
8:35
Extra Example 3: Tangents & Circumscribed Polygons
11:50
Extra Example 4: Tangents & Circumscribed Polygons
15:43
Secants, Tangents, & Angle Measures

27m 50s

Intro
0:00
Secant
0:08
Secant
0:09
Secant and Tangent
0:49
Secant and Tangent
0:50
Interior Angles
2:56
Secants & Interior Angles
2:57
Exterior Angles
7:21
Secants & Exterior Angles
7:22
Extra Example 1: Secants, Tangents, & Angle Measures
10:53
Extra Example 2: Secants, Tangents, & Angle Measures
13:31
Extra Example 3: Secants, Tangents, & Angle Measures
19:54
Extra Example 4: Secants, Tangents, & Angle Measures
22:29
Special Segments in a Circle

23m 8s

Intro
0:00
Chord Segments
0:05
Chord Segments
0:06
Secant Segments
1:36
Secant Segments
1:37
Tangent and Secant Segments
4:10
Tangent and Secant Segments
4:11
Extra Example 1: Special Segments in a Circle
5:53
Extra Example 2: Special Segments in a Circle
7:58
Extra Example 3: Special Segments in a Circle
11:24
Extra Example 4: Special Segments in a Circle
18:09
Equations of Circles

27m 1s

Intro
0:00
Equation of a Circle
0:06
Standard Equation of a Circle
0:07
Example 1: Equation of a Circle
0:57
Example 2: Equation of a Circle
1:36
Extra Example 1: Determine the Coordinates of the Center and the Radius
4:56
Extra Example 2: Write an Equation Based on the Given Information
7:53
Extra Example 3: Graph Each Circle
16:48
Extra Example 4: Write the Equation of Each Circle
19:17
Section 10: Polygons & Area
Polygons

27m 24s

Intro
0:00
Polygons
0:10
Polygon vs. Not Polygon
0:18
Convex and Concave
1:46
Convex vs. Concave Polygon
1:52
Regular Polygon
4:04
Regular Polygon
4:05
Interior Angle Sum Theorem
4:53
Triangle
5:03
Quadrilateral
6:05
Pentagon
6:38
Hexagon
7:59
20-Gon
9:36
Exterior Angle Sum Theorem
12:04
Exterior Angle Sum Theorem
12:05
Extra Example 1: Drawing Polygons
13:51
Extra Example 2: Convex Polygon
15:16
Extra Example 3: Exterior Angle Sum Theorem
18:21
Extra Example 4: Interior Angle Sum Theorem
22:20
Area of Parallelograms

17m 46s

Intro
0:00
Parallelograms
0:06
Definition and Area Formula
0:07
Area of Figure
2:00
Area of Figure
2:01
Extra Example 1:Find the Area of the Shaded Area
3:14
Extra Example 2: Find the Height and Area of the Parallelogram
6:00
Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices
10:11
Extra Example 4: Find the Area of the Figure
14:31
Area of Triangles Rhombi, & Trapezoids

20m 31s

Intro
0:00
Area of a Triangle
0:06
Area of a Triangle: Formula and Example
0:07
Area of a Trapezoid
2:31
Area of a Trapezoid: Formula
2:32
Area of a Trapezoid: Example
6:55
Area of a Rhombus
8:05
Area of a Rhombus: Formula and Example
8:06
Extra Example 1: Find the Area of the Polygon
9:51
Extra Example 2: Find the Area of the Figure
11:19
Extra Example 3: Find the Area of the Figure
14:16
Extra Example 4: Find the Height of the Trapezoid
18:10
Area of Regular Polygons & Circles

36m 43s

Intro
0:00
Regular Polygon
0:08
SOHCAHTOA
0:54
30-60-90 Triangle
1:52
45-45-90 Triangle
2:40
Area of a Regular Polygon
3:39
Area of a Regular Polygon
3:40
Are of a Circle
7:55
Are of a Circle
7:56
Extra Example 1: Find the Area of the Regular Polygon
8:22
Extra Example 2: Find the Area of the Regular Polygon
16:48
Extra Example 3: Find the Area of the Shaded Region
24:11
Extra Example 4: Find the Area of the Shaded Region
32:24
Perimeter & Area of Similar Figures

18m 17s

Intro
0:00
Perimeter of Similar Figures
0:08
Example: Scale Factor & Perimeter of Similar Figures
0:09
Area of Similar Figures
2:44
Example:Scale Factor & Area of Similar Figures
2:55
Extra Example 1: Complete the Table
6:09
Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures
8:56
Extra Example 3: Find the Unknown Area
12:04
Extra Example 4: Use the Given Area to Find AB
14:26
Geometric Probability

38m 40s

Intro
0:00
Length Probability Postulate
0:05
Length Probability Postulate
0:06
Are Probability Postulate
2:34
Are Probability Postulate
2:35
Are of a Sector of a Circle
4:11
Are of a Sector of a Circle Formula
4:12
Are of a Sector of a Circle Example
7:51
Extra Example 1: Length Probability
11:07
Extra Example 2: Area Probability
12:14
Extra Example 3: Area Probability
17:17
Extra Example 4: Area of a Sector of a Circle
26:23
Section 11: Solids
Three-Dimensional Figures

23m 39s

Intro
0:00
Polyhedrons
0:05
Polyhedrons: Definition and Examples
0:06
Faces
1:08
Edges
1:55
Vertices
2:23
Solids
2:51
Pyramid
2:54
Cylinder
3:45
Cone
4:09
Sphere
4:23
Prisms
5:00
Rectangular, Regular, and Cube Prisms
5:02
Platonic Solids
9:48
Five Types of Regular Polyhedra
9:49
Slices and Cross Sections
12:07
Slices
12:08
Cross Sections
12:47
Extra Example 1: Name the Edges, Faces, and Vertices of the Polyhedron
14:23
Extra Example 2: Determine if the Figure is a Polyhedron and Explain Why
17:37
Extra Example 3: Describe the Slice Resulting from the Cut
19:12
Extra Example 4: Describe the Shape of the Intersection
21:25
Surface Area of Prisms and Cylinders

38m 50s

Intro
0:00
Prisms
0:06
Bases
0:07
Lateral Faces
0:52
Lateral Edges
1:19
Altitude
1:58
Prisms
2:24
Right Prism
2:25
Oblique Prism
2:56
Classifying Prisms
3:27
Right Rectangular Prism
3:28
4:55
Oblique Pentagonal Prism
6:26
Right Hexagonal Prism
7:14
Lateral Area of a Prism
7:42
Lateral Area of a Prism
7:43
Surface Area of a Prism
13:44
Surface Area of a Prism
13:45
Cylinder
16:18
Cylinder: Right and Oblique
16:19
Lateral Area of a Cylinder
18:02
Lateral Area of a Cylinder
18:03
Surface Area of a Cylinder
20:54
Surface Area of a Cylinder
20:55
Extra Example 1: Find the Lateral Area and Surface Are of the Prism
21:51
Extra Example 2: Find the Lateral Area of the Prism
28:15
Extra Example 3: Find the Surface Area of the Prism
31:57
Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder
34:17
Surface Area of Pyramids and Cones

26m 10s

Intro
0:00
Pyramids
0:07
Pyramids
0:08
Regular Pyramids
1:52
Regular Pyramids
1:53
Lateral Area of a Pyramid
4:33
Lateral Area of a Pyramid
4:34
Surface Area of a Pyramid
9:19
Surface Area of a Pyramid
9:20
Cone
10:09
Right and Oblique Cone
10:10
Lateral Area and Surface Area of a Right Cone
11:20
Lateral Area and Surface Are of a Right Cone
11:21
Extra Example 1: Pyramid and Prism
13:11
Extra Example 2: Find the Lateral Area of the Regular Pyramid
15:00
Extra Example 3: Find the Surface Area of the Pyramid
18:29
Extra Example 4: Find the Lateral Area and Surface Area of the Cone
22:08
Volume of Prisms and Cylinders

21m 59s

Intro
0:00
Volume of Prism
0:08
Volume of Prism
0:10
Volume of Cylinder
3:38
Volume of Cylinder
3:39
Extra Example 1: Find the Volume of the Prism
5:10
Extra Example 2: Find the Volume of the Cylinder
8:03
Extra Example 3: Find the Volume of the Prism
9:35
Extra Example 4: Find the Volume of the Solid
19:06
Volume of Pyramids and Cones

22m 2s

Intro
0:00
Volume of a Cone
0:08
Volume of a Cone: Example
0:10
Volume of a Pyramid
3:02
Volume of a Pyramid: Example
3:03
Extra Example 1: Find the Volume of the Pyramid
4:56
Extra Example 2: Find the Volume of the Solid
6:01
Extra Example 3: Find the Volume of the Pyramid
10:28
Extra Example 4: Find the Volume of the Octahedron
16:23
Surface Area and Volume of Spheres

14m 46s

Intro
0:00
Special Segments
0:06
Radius
0:07
Chord
0:31
Diameter
0:55
Tangent
1:20
Sphere
1:43
Plane & Sphere
1:44
Hemisphere
2:56
Surface Area of a Sphere
3:25
Surface Area of a Sphere
3:26
Volume of a Sphere
4:08
Volume of a Sphere
4:09
Extra Example 1: Determine Whether Each Statement is True or False
4:24
Extra Example 2: Find the Surface Area of the Sphere
6:17
Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters
7:25
Extra Example 4: Find the Surface Area and Volume of the Solid
9:17
Congruent and Similar Solids

16m 6s

Intro
0:00
Scale Factor
0:06
Scale Factor: Definition and Example
0:08
Congruent Solids
1:09
Congruent Solids
1:10
Similar Solids
2:17
Similar Solids
2:18
Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither
3:35
Extra Example 2: Determine if Each Statement is True or False
7:47
Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume
10:14
Extra Example 4: Find the Volume of the Larger Prism
12:14
Section 12: Transformational Geometry
Mapping

14m 12s

Intro
0:00
Transformation
0:04
Rotation
0:32
Translation
1:03
Reflection
1:17
Dilation
1:24
Transformations
1:45
Examples
1:46
Congruence Transformation
2:51
Congruence Transformation
2:52
Extra Example 1: Describe the Transformation that Occurred in the Mappings
3:37
Extra Example 2: Determine if the Transformation is an Isometry
5:16
Extra Example 3: Isometry
8:16
Reflections

23m 17s

Intro
0:00
Reflection
0:05
Definition of Reflection
0:06
Line of Reflection
0:35
Point of Reflection
1:22
Symmetry
1:59
Line of Symmetry
2:00
Point of Symmetry
2:48
Extra Example 1: Draw the Image over the Line of Reflection and the Point of Reflection
3:45
Extra Example 2: Determine Lines and Point of Symmetry
6:59
Extra Example 3: Graph the Reflection of the Polygon
11:15
Extra Example 4: Graph the Coordinates
16:07
Translations

18m 43s

Intro
0:00
Translation
0:05
Translation: Preimage & Image
0:06
Example
0:56
Composite of Reflections
6:28
Composite of Reflections
6:29
Extra Example 1: Translation
7:48
Extra Example 2: Image, Preimage, and Translation
12:38
Extra Example 3: Find the Translation Image Using a Composite of Reflections
15:08
Extra Example 4: Find the Value of Each Variable in the Translation
17:18
Rotations

21m 26s

Intro
0:00
Rotations
0:04
Rotations
0:05
Performing Rotations
2:13
Composite of Two Successive Reflections over Two Intersecting Lines
2:14
Angle of Rotation: Angle Formed by Intersecting Lines
4:29
Angle of Rotation
5:30
Rotation Postulate
5:31
Extra Example 1: Find the Rotated Image
7:32
Extra Example 2: Rotations and Coordinate Plane
10:33
Extra Example 3: Find the Value of Each Variable in the Rotation
14:29
Extra Example 4: Draw the Polygon Rotated 90 Degree Clockwise about P
16:13
Dilation

37m 6s

Intro
0:00
Dilations
0:06
Dilations
0:07
Scale Factor
1:36
Scale Factor
1:37
Example 1
2:06
Example 2
6:22
Scale Factor
8:20
Positive Scale Factor
8:21
Negative Scale Factor
9:25
Enlargement
12:43
Reduction
13:52
Extra Example 1: Find the Scale Factor
16:39
Extra Example 2: Find the Measure of the Dilation Image
19:32
Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation
26:18
Extra Example 4: Graphing Polygon, Dilation, and Scale Factor
32:08
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Lecture Comments (8)

0 answers

Post by Merinrose on August 2, 2020

this is really helpful thank you

0 answers

Post by Anthony Salamanca on January 13, 2016

In this example, if there was an auxiliary line above vertex which makes two parallel lines then would angle 105 be equal to x?   Would this make 105 and x alternate congruent angles?

1 answer

Last reply by: Professor Pyo
Sat Jan 18, 2014 2:55 PM

Post by Yuval Guetta on January 13, 2014

for the flow proof can you write the second statement (that BCD and ACB form a linear pair) before writing the given instead of writing it underneath?

0 answers

Post by Werner Dietrich on May 5, 2011

The class topics/content is located on the left. You can select a topic by scrolling down that list, or you can simply "fast forward" by moving the progress bar below the video.

Hope this helps.

1 answer

Last reply by: Humayun Ali
Tue Dec 22, 2020 12:40 PM

Post by Sonia Oglesby on February 23, 2011

How do you fast forward to what you need. I don't need to watch the entire class.

Measuring Angles in Triangles

  • Angle Sum Theorem: The sum of the measure of the angles of a triangle is 180 degrees
  • Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent
  • Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
  • Triangle Corollaries:
    • The acute angles of a right triangle are complementary
    • There can be at most one right or obtuse angle in a triangle

Measuring Angles in Triangles

m∠MON = 60o, m∠MNO = 75o, find m∠OMN.
  • m∠OMN = 180o - m∠MON - m∠MNO
  • m∠OMN = 180o − 60o − 75o = 45o
m∠OMN = 45o
Fill in the blank in the statement with always, sometimes or never.
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are ____ congruent.
Always
m∠OMN = 60o, m∠MON = 45o, find m∠ONA.
  • m∠ONA = m∠OMN + m∠MON
m∠ONA = 60o + 45o = 105o
m∠ABC = 14 + x, m∠ACB = 8 + 2x, m∠BAC = 48, find x.
  • m∠ABC + m∠ACB + m∠BAC = 180
  • 14 + x + 8 + 2x + 48 = 180o
  • 3x + 70 = 180
x = [110/3]y
Right triangle ABC, m∠BAC = 40o, find m∠ACB.
  • m∠ACB = 90o - m∠BAC
m∠ACB = 90o − 40o = 50o
Determine whether the following statement is true or false.
There can be a right angle and an obtuse angle in the same triangle.
False
Determine whether the following statement is true or false.
There must be at least two acute angles in a triangle.
True
Fill in the blank with always, sometimes, or never.

Triangle ABC and triangle DEF, if ∠1 ≅ ∠5, ∠2 ≅ ∠4, then 3 and 8 are ____ congruent.
Always
m∠ABC = 3x + 5, m∠BAC = 2x + 16, m∠ACD = 6x + 9, find x.
  • m∠ACD = m∠ABC + m∠BAC
  • 6x + 9 = 3x + 5 + 2x + 16
x = 12
m∠DAE = 5x + 4, m∠4 = 2x + 3, AD||BC, find x.
  • m∠1 + m∠2 + m∠3 = 180
  • m∠2 + m∠3 = 180 - m∠1
  • 4 ≅ 1
  • m∠4 = m∠1
  • m∠2 + m∠3 = 180 - m∠4 = 180 - (2x + 3)
  • m∠2 + m∠3 = m∠DAE = 5x + 4
  • 180 − (2x + 3) = 5x + 4
x = [173/7]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Measuring Angles in Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Angle Sum Theorem 0:09
    • Angle Sum Theorem for Triangle
  • Using Angle Sum Theorem 4:06
    • Find the Measure of the Missing Angle
  • Third Angle Theorem 4:58
    • Example: Third Angle Theorem
  • Exterior Angle Theorem 7:58
    • Example: Exterior Angle Theorem
  • Flow Proof of Exterior Angle Theorem 15:14
    • Flow Proof of Exterior Angle Theorem
  • Triangle Corollaries 27:21
    • Triangle Corollary 1
    • Triangle Corollary 2
  • Extra Example 1: Find the Value of x 32:55
  • Extra Example 2: Find the Value of x 34:20
  • Extra Example 3: Find the Measure of the Angle 35:38
  • Extra Example 4: Find the Measure of Each Numbered Angle 39:00

Transcription: Measuring Angles in Triangles

Welcome back to Educator.com.0000

This next lesson is on angles of triangles, so we are going to be measuring unknown angles within triangles.0002

First, the angle sum theorem: this is a very important theorem when it comes to triangles, because you will use it really often.0013

And what it says is that the sum of the measures of all of the angles of a triangle adds up to 180.0026

So, in this triangle right here, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.0033

Now, this line right here is just to help with this diagram; and that is called an auxiliary line.0043

Whenever you draw a line--you just add a line or a line segment to a diagram to help you visualize something--then that is an auxiliary line.0050

So, this is an auxiliary line, because it is just to show that the three angles of a triangle will add up to 180.0062

Now, if this line right here is parallel to this line, this side, and if I extend this line out right here,0071

then from the last chapter we know that, if we have parallel lines, then certain angles have certain relationships.0085

And this right here is going to be our transversal; so then, we have two lines that are parallel, and our transversal.0094

Parallel line, parallel line, transversal...then angle 3, with this angle right here, are alternate interior angles.0104

If we have that, then those angles are alternate interior angles.0117

And remember: as long as the lines are parallel, then alternate interior angles are congruent.0127

So, I can say that this angle right here and this angle right here are the same--they are congruent--because the lines are parallel.0133

Now, the same thing happens here: angle 1 with this angle right here--again, if these two lines are the parallel lines,0141

and this is my transversal (so again, it is going like this), then this is one, and this is one.0151

They are alternate interior angles; and if the lines are parallel, then they are congruent.0160

So, whatever the angle measure is for this, it is going to be the measure of the angle for that.0170

Now, looking at this right here: the measure of angle 1, the measure of angle 2, and the measure of angle 3 are going to equal 180.0177

Why? Because they form a line; 1, 2, and 3 together form a line.0187

We know that a line equals 180; so if it forms 180 this way, then the same thing; angle 1, angle 2, and angle 3 here form 180.0195

So, the three angles of a triangle are going to add up to 180; and we know that because of this--0211

because, if I look at this angle, the same thing as the measure of angle 1 here, and this angle,0216

the same thing as the measure of angle 3 here, all three of them are going to be supplementary,0223

which means that these are supplementary--they are the same angles.0228

So, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.0231

That is the angle sum theorem: All three angles of a triangle add up to 180.0237

So, using the angle sum theorem, let's find the measure of the missing side.0247

I am given two angles out of the three; one is missing; I need to find the measure of angle A.0251

I can say that x + 35 + 85 is going to add up to 180.0259

It is x +...now, if I add this up, it is going to be 120...equals 180; so if I subtract the 120, then I get 60.0272

Right here, this is x = 60 degrees.0292

The third angle theorem: what we just used right now, that theorem, is called the angle sum theorem,0300

where all three angles of a triangle add up to 180.0307

This is the third angle theorem, and this is saying that, if you have two triangles, one of these angles is congruent0311

to an angle on the other triangle (see how this one and this one are congruent), and a second angle is congruent0322

to another angle on the other triangle, then automatically,0332

the third angle for this triangle is going to be congruent to the third angle for this triangle.0338

So, if two angles of one triangle (which are these two) are congruent to two angles of a second triangle0345

(these two), then automatically, the third angles of the triangles are going to be congruent.0353

And that is the third angle theorem; and it is just saying that, if I know this angle measure and this angle measure,0361

then I have to subtract those two from 180, and I get this angle measure, because all three add up to 180.0367

So, if I say that this is, let's say, 80, and this is 60, well, then, those together are going to be 140.0375

That means that I have to subtract it from 180, and that is going to give me this angle measure right here.0390

Well, this angle and this angle are the same; this angle and this angle are the same;0395

again, I have to subtract it from 180; so that means that the sum of these two is going to be the same thing as the sum of those two.0402

So, automatically, the third angle has to be the same as this angle here, because it is the same number,0411

and I have to subtract the number from 180; so that is the same thing right here,0416

and then again...so it is like you are doing the same problem.0427

And that is a 4...0436

If you have an angle congruent to an angle on the other triangle,0445

and a second angle also congruent to another angle on the other triangle,0448

then automatically, the third angles will be congruent; and that is the third angle theorem.0452

We have the angle sum theorem, and we have the third angle theorem.0458

Now, these theorems: again, remember, the theorems are supposed to be proven before we can actually use them.0462

But in your book, it will show you, or it will have you do the proof on these theorems; so just take a look at that.0470

The exterior angle theorem: now, we know that "exterior angle" sounds familiar.0480

An exterior angle is an angle outside; so when it comes to a triangle, what we are dealing with in this lesson here,0486

or this chapter, "exterior angle" is referring to an angle outside the triangle.0493

So, any time anything is exterior, it is always on the outside.0501

So, in this case, it is on the outside of the triangle.0506

Now, this is my triangle right here; I have an angle, angle 4; that is my exterior angle.0510

Now, I could have more than one exterior angle.0524

If I draw this out, then I will have an exterior angle here; if I draw this out, I am going to have an exterior angle here, and so on.0527

So, I could have a few different exterior angles, but this is the one that I am looking at right now.0538

The measure of an exterior angle (and of course, this theorem applies to any of the exterior angles0546

that you see beyond the triangle) is equal to the sum of the measures of the two remote interior angles.0551

This is very important to note, because otherwise you don't know what the exterior angle equals.0567

There are three angles of a triangle: one of them is adjacent to the exterior angle, which is angle 3.0581

So, angle 4 and angle 3 are actually a linear pair.0592

Not including that angle, the other two angles that are not next to the exterior angle,0601

which are angles 1 and 2, are the remote interior angles.0610

Now, if I drew an angle out here, and made this angle 5, the two remote interior angles would be 1 and 3, not 2.0615

If I drew an exterior angle out right here, and made this angle 6, then I wouldn't be talking about angle 1; I would be talking about angles 2 and 3.0625

The two remote interior angles would be the two angles inside the triangle that are far away from it,0634

that are not adjacent, that are not next to the exterior angle.0641

That is the remote interior angles; now, think of "remote" as in the TV remote.0646

Whenever you use the remote, if you are right next to the TV, you are not going to be using the remote;0652

you can touch the buttons on the TV; but the remote is used when you have distance away from the TV.0657

In that same way, remote interior angles would be the angles that are some distance away,0666

that are not right next to it, but further away--the other two angles.0672

The measure of angle 4--that is the exterior angle that we see here in this diagram--0682

is going to be equal to the sum of the measures of the two remote interior angles.0689

Since we know that that is 1 and 2, this angle measure is going to be the same as the measure of angle 1, plus the measure of angle 2.0695

Just to show you the proof of this: the measure of angle 1, plus the measure of angle 20717

(let me do this, actually, in a different color, so you know that this is actually not part of this),0722

plus the measure of angle 3--those are the three angles of a triangle--are going to add up to 180; that is the angle sum theorem.0733

I know that the measure of angle 4, plus the measure of angle 3, adds up to 180.0742

Why?--because it is a linear pair; they make a straight line, so they add up to 180, the measure of angle 4 and angle 3.0755

If you look at this, well, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals 180.0768

The measure of angle 4, plus the measure of angle 3, equals 180.0776

Doesn't that mean that these two have to equal the same?--because all of this, plus whatever this is, is 180;0779

and then, that plus the measure of angle 3, equals 180; so these have to automatically equal each other.0786

If I use the substitution property, and I say, "OK, well, then, if all of this equals 180, and all of this equals 180..."0793

then I can just make all of this equal to all of this, because they both equal 180.0801

So, the measure of angle 1, plus the measure of angle 2, plus the measure of angle 3, equals the measure of angle 4 plus the measure of angle 3.0807

And all I did was just...since all of this equals 180, and all of this equals 180, then I can just make these equal each other.0817

Then, I just made all of this equal to all of that.0828

And that would be the substitution property of equality.0831

Then, from here, if I subtract the measure of angle 3 from both, that would be the subtraction property.0836

Then, the measure of angle 1, plus the measure of angle 2, is equal to the measure of angle 4.0849

That is what this exterior angle theorem is: the measure of angle 1, plus the measure of angle 2, equals the measure of angle 4.0859

So, again, just to explain this proof to you: if all of these three angles of a triangle add up to 180,0866

and these two add up to 180, well, then, this plus 3 equals 180, and this plus 3 equals 180;0875

then these two will equal the measure of angle 4.0884

Just take a look at this again, if you are still a little confused.0893

Just remember the exterior angle theorem: the measure of angle 4, the exterior angle,0897

is equal to the two remote interior angles, the measure of angle 1 and the measure of angle 2.0901

OK, so we just went over the proof, explaining the exterior angle theorem.0916

But I want to actually go over a flow proof.0923

Now, a flow proof is a little different in the format of proofs than what we have done so far.0929

We have done two-column proofs, and we have done paragraph proofs.0940

A flow proof is another type of setup of a proof; you are just showing it in a different format.0945

But it is still the same thing, where you have to show the statement and the reason behind it.0957

This one, the proof that we just did, is just a brief proof; it wasn't the actual full step-by-step proof.0963

So, we are just going to do the full proof here right now.0973

And so, what I am going to do is start with...0977

Now, with a flow proof, whenever one step leads to another step, then you are going to draw an arrow.0982

It is going to consist of boxes, and each of these boxes is going to flow--0989

you are going to draw an arrow to show the next statement and reason, then the next statement and reason,0999

and so on and so on, the next steps, until you get to what you are trying to prove.1003

Now, remember how we talked about the proofs--how it is from step A to step B.1010

If you are driving from point A to point B, then there is a series of steps that you have to take to get there, a series of turns and whatever.1014

In the same way, with a proof, you are going to start from point A; and through a series of steps, you are going to arrive at point B.1025

That is a proof; this is just another format, so I am just going to show you the flow proof of this.1035

The first thing, right here, is going to be: I am going to give you this statement, which is triangle ABC, and that is "Given."1042

I am going to write the statement, and then I am going to write the reason below it.1052

And then, I am going to draw that as a box.1056

And then, like I said, you are going to draw an arrow that leads to the next step, and then to the next step, and so on.1062

If it doesn't--let's say you have a step that wasn't from the step before--then you can just write it below here.1068

So, I am actually going to do that right now: I am going to say that angle BCD and angle ACB (these two angles) form a linear pair.1080

Now, that is not from this step; that is not the next step of this, so I am going to write it below here.1095

I am going to say, "Angle BCD and angle ACB form a linear pair."1102

And then, my reason is going to be...well, that is just the definition of a linear pair.1125

They form a linear pair, so they are a linear pair; this is "definition of linear pair."1132

And that is my first box; then, from here, if we form a linear pair, then what does that mean?1145

That means that angle BCD and angle ACB are supplementary.1156

Whenever two angles form a linear pair, then they are supplementary; and so, my reason behind that is,1174

"If two angles form a linear pair, then they are supplementary."1180

And I have to write that whole thing out as my reason, because there is no name for it.1197

Remember: if there is no name for a theorem or a postulate, then you have to write out the whole thing.1201

You can abbreviate stuff, but you have to actually write out the whole sentence.1205

That is my second box; then, from here, if they are supplementary, then what?1212

I said that they are supplementary; then what is true?1221

That means that the measure of angle BCD plus the measure of angle ACB equals 180.1226

If they are supplementary, then don't they add up to 180?1239

And that is just the definition of supplementary angles.1242

Whenever you go from saying that they are supplementary to then making them equal to 180,1252

that is the definition of supplementary angles, because that is what the definition says.1258

If they are supplementary, then they equal 180.1262

Now, see how our flow proof is going this way; it is going to the right.1269

A flow proof can also go from top down; you can draw arrows going downwards, too; you can go in that direction.1275

OK, now, I want to continue on from this right here.1285

Now, look at what I am trying to prove: I am trying to prove that, again, this exterior angle is going to be equal to the measure of the two remote interior angles.1296

So, this is the measure of angle BCD; this is equal to the measure of angle A, plus the measure of angle B.1305

Like how I showed you before in the last slide, if I have a triangle ABC, then I know that the measure of angle A,1313

plus the measure of angle A, plus the measure of angle C, is going to equal 180; what is my reason?--"angle sum theorem."1323

We just went over the angle sum theorem.1338

Now, in order for you to use the angle sum theorem, you have to have something state that it is a triangle first,1342

because the angle sum theorem says "the three angles of a triangle."1348

So then, see how this is given: "triangle ABC" is saying the statement that you have a triangle.1352

Then, from there, you can say that the three angles of that triangle are going to be 180.1358

Again, you have a triangle, and then you can say that the three angles of that triangle are going to add up to 180.1365

There is that; and then, since you know that all of this is equal to 180, and all of this is equal to 180--1373

this one says that the three angles add up to 180, and this one says that these two add up to 180--1385

then, what I can do for my next step is make all of this (since all of this equals 180, and all of this equals 180) equal to all of this,1393

since they have the same measure; so I can say that the measure of angle A, plus the measure of angle B,1407

plus the measure of angle C, equals the measure of angle BCD, plus the measure of angle ACB.1420

See how all of this equals 180, and all of this equals 180; so I just make them equal to each other.1435

And what is the reason for that?--"substitution property."1440

Now, see how this step right here was derived from this step and this step.1457

See how I drew an arrow from that one; I can also draw an arrow from this to that, because both of these led to this one.1464

Then, if I look at this right here, let's see: there is one thing that I am going to correct here.1474

I don't want to say the measure of angle C here; now, I know this looks like angle C, because it is part of the triangle--1488

it is one of the vertices of the triangle; but see how this angle C can refer to three different angles.1495

It can refer to this angle right here, this angle right here, or this whole angle right here.1507

So, I want to specify this more to "the measure of angle BCA," and the same thing here.1511

Or actually, let's change it to ACB, because that is the name that I called it for this one.1524

I called this angle ACB, so I will just call this the same angle.1534

So then, here also, it is the measure of angle ACB.1539

That way, I know, because this one and this one are supposed to be the same,1546

because this angle is part of this triangle, and this angle is also part of this.1551

So then, the next step...now, I don't have room, so I am going to draw it like that.1558

I can say that the measure of...now, I am going to write this part first;1567

I am going to use the subtraction property to get rid of this.1574

And then, I can say that the measure of angle A, plus the measure of angle B, equals the measure of angle BCD.1578

But here in my "prove" statement, I have the measure of BCD first.1583

It is the same thing; it is just stating this one first: the measure of angle BCD equals the measure of angle A plus the measure of angle B.1590

I just switched it: and the reason for that is the subtraction property of equality.1601

And that is it--that is the proof.1611

Now, you can do this as a two-column proof; then you would just split up the statements and the reasons, and just number them off.1615

You can also write this as a paragraph proof, where you would just write it in sentences; and then, this is the flow proof,1622

where you are grouping up the statement with each reason,1630

and then you are just drawing the arrows to see how it leads from point A to point B.1634

OK, moving on: triangle corollaries: a corollary is kind of like a theorem, where you have to prove it.1642

But you can use theorems to prove corollaries; it is not as big of a deal as a theorem, but still necessary.1654

The first one is "the acute angles of a right triangle are complementary."1671

If I have a right triangle (this is going to be #1, so I am going to write that right here)...this is kind of slippery...1675

then since I know that all three angles, this angle plus this angle plus this angle, have to add up to 180;1695

but this right here is going to be 90 degrees; well, if this is 90, this right here is just saying that this has to be an acute angle,1705

and this has to be an acute angle, because if this is 90, then these two are going to be 90 together.1722

Together it has to be 90; and in order for them to be anything but acute, it has to be 90 or greater.1731

So, if this angle wants to be not acute, if it wants to be a right or an obtuse angle, then it has to be 90 or greater.1739

The same thing here: if this doesn't want to be an acute angle, then it has to be either 90 degrees1749

(to be a right angle) or greater than 90 (to be an obtuse angle).1755

So, it has no choice but to be acute angles, because they are going to add up to 180.1758

And so, since they do add up to 180, if this is, let's say, A, and this is B; then the measure of angle A,1765

plus the measure of angle B, is going to equal 90 degrees.1773

So then, not only are they acute, but they have to add up to 90, because all together, they are going to add up to 180.1778

So, if the measure of angle A, plus (this is C--let's say that that is C) the measure of angle B,1789

plus the measure of angle C, is equal to 180--all three angles--well, this right here is equal to 90;1795

that means that all of this has to be 90.1808

So, if this is 90, then the other two are left to add up to 90.1818

This corollary--you don't have to use it, because you can just state it like this.1825

But it just makes it easier, because if you have a right triangle, then you don't have to even worry about the right angle.1833

You can just say that the other two angles are going to add up to 90.1839

The next one, #2: There can be at most one right or obtuse angle in a triangle.1844

I kind of already explained #2 when I was explaining #1.1850

You can only have one right angle or one obtuse angle in a triangle, again, because if this is 90...1857

all three angles have to add up to 180; that means if this is 90, then this plus this plus this has to be 90.1867

That way, all three can add up to 180.1879

So, let's say that this wants to be a right angle, along with this one.1883

Well, then, there is my right angle; that is this one.1887

If this wants to be a right angle, too, then it has to go that way, in order for that to be a right angle.1891

There is no way that this could be a triangle; so then, you know that there can only be one right angle in a triangle.1896

And then again, let's say I have an obtuse angle.1902

Let me try to draw more than one obtuse angle in a triangle; there is my obtuse angle there.1912

But see, this angle right here wants to be an obtuse angle.1917

Then, it has to be greater than 90, so let's say I draw it like that.1922

Is there any way that that could be a triangle right there?1927

There is no way that this side and this side are going to meet, unless I draw another side, a fourth side.1932

And if I have four sides, then that is not a triangle.1938

So then, if I have one obtuse angle (this one is greater than 90), then this one has to be less than 90.1940

It can't be greater than 90 also, because then this is going to be, let's say, 100, and let's say this is 100.1949

That is already 200 degrees right there, and you know that all three angles have to add up to 180.1957

So, that is not going to work; that is not going to work.1964

At most, there is one right angle, or at most one obtuse angle, in a triangle.1967

OK, let's go over our examples: Find the value of x.1977

The angle sum theorem: I know that all three angles are going to add up to 180.1983

So, 79 + 36 + x = 180; so this is going to be 115 + x = 180; subtract the 115; x is going to be 65.1988

The next one: 32 + x + x = 180; 32 +...x + x is...2x = 180; subtract the 32; 2x is going to be 148, and then x is going to be 74.2022

So, each one of these is 74.2051

Find the value of x, again: here is my exterior angle; the measure of this angle is going to be equal to2062

the sum of the two remote interior angles, which are the two angles away from the angle,2073

not the one that is next to it (because there are three angles).2079

So, the other two: let me just write out the angle measure: 50 + x...those are my two remote interior angles...equals 105.2082

Add up these two; it is going to be the same thing as that measure right there.2100

So, I subtract the 50, and x will equal 55 degrees.2103

Again, the exterior angle is going to be equal to the sum of the two remote interior angles.2112

That is 25 + x = 128; I subtract the 25; x = 103, so this angle right here is 103.2120

OK, so then, we are going to find the measure of angle A, which is this right here.2140

Now, I have a right triangle for this first one; all three angles are going to add up to 180,2145

but because I have a right angle, this already uses up 90--it uses up half of it.2157

That means that I don't even have to worry about that; I can just say that that means the other two,2164

the remaining two angles, are going to add up to the other 90.2168

So, if this takes up half already, if this takes up 90 degrees, then these two are going to add up to 90,2172

because all three of them together have to add up to 180.2181

So, without worrying about that...you could do that; you could just say that this angle,2183

plus this angle, plus this angle, is going to add up to 180.2190

But I am just going to go ahead and say that these two (forget this angle B, the right angle) are going to add up to 90.2193

So, 2x + 15 + 3x - 5 is going to equal 90; again, that is because this one is 90--that means that the other two have to add up to 90,2202

because they all have to add up to 180; so this is 5x + 10 = 90; I subtract the 10, so 5x = 80; then x = 16.2220

And then, that wouldn't be my answer, because they want us to find the measure of angle A.2241

That means that I have to plug x back into angle A.2247

So, it would be 2 times 16, plus 15; that is 32 + 15 equals 47, so the measure of angle A is 47.2252

And then, the next one: there is my exterior angle; it is equal to the sum of the two remote interior angles.2273

So, it would be these two and not this one: 6x - 10 = 45 + 3x + 5.2281

So, 6x - 10 =...I can just add these up, so this would be 3x (I can write that better) + 50.2294

Then, I am going to subtract 3x, so it is 3x =...add the 10...60, so x = 20.2310

And then, here is angle A; so the measure of angle A equals 3(20) + 5; 60 + 5 is 65, so the measure of angle A equals 65 degrees.2318

My next example: Find the measure of each numbered angle, which means that I need to find the measure of angle 1,2341

the measure of angle 2, and the measure of angle 3.2347

Let's see here: I have two triangles; I have a triangle right here, and I have a triangle right here.2354

Now, if you get this type of problem--you get a problem that looks a little complicated, a little confusing,2361

with kind of a big diagram, then just take a second and just observe it--take a closer look; see what you have.2370

I see here that I have two triangles, because we know that this lesson is on triangles.2380

I am going to look for triangles; I also have an exterior angle.2390

I have two parallel lines; this means that you have parallel lines, right there.2396

Now, I want to find all three angles--the measure of all three of them.2404

Since I have an exterior angle, I am thinking that I might have to use the exterior angle theorem,2409

which I do, because I have two remote interior angles, which is a 42, and then the measure of angle 3.2415

To find the measure of angle 3 (the measure of angle 3, plus 42, equals 110--that is the exterior angle theorem--the exterior angle2424

equals the sum of the remote interior angles), if I subtract the 42, the measure of angle 3 equals 68.2439

Then, if this is 68, let's see here--how am I going to find the measures of angles 1 and 2?2454

Well, maybe I can use the triangle sum theorem for 1 and 2.2470

But look: I am missing two angles, this angle and this angle; I only have the measure of this angle, so I can't use the angle sum theorem yet.2477

But what I can do is look at my parallel lines; if it helps, just extend out the lines so that it will be easier to see your two parallel lines.2489

And then, the line that is crossing both is your transversal, so you could extend that, if you want, too.2501

And then, you can also move your book, or maybe move your paper, if it is on paper--and try to look at the parallel lines2508

so that it would be horizontal or vertical--whichever is easiest for you to see.2521

And then, you can see that this angle right here and this angle right here are alternate interior angles.2527

Now, 48 and 68--are those alternate interior angles?2536

Yes, they are; but because the two lines that are used to form those two angles are not parallel,2542

that is why we have two different angle measures--48 here and 68 here.2553

If these two lines were parallel, then this one and this one would have to be congruent.2557

But those lines are not parallel; so, yes, they are still alternate interior angles;2564

but since the lines are not parallel, they are not congruent.2570

Their relationship would just be alternate interior angles.2574

They are only congruent if the lines are parallel.2579

In this case, with these lines and those transversal (because those are the lines that are used to form those angles),2581

since the lines are parallel, these two would be congruent; so the measure of angle 2, I know, is going to be 42.2592

This one is going to be 42, also.2603

Then, how do I find the measure of angle 1?2607

Now that I have the two angles of this triangle, I can use the angle sum theorem to find the measure of angle 1.2613

So, the measure of angle 1 is going to be added to 48 and 42, and that is going to give you 180 (the angle sum theorem).2621

So, the measure of angle 1, plus...this is 90...equals 180; I subtract the 90, so the measure of angle 1 is 90 degrees.2638

This one right here is 90 degrees.2654

Again, just take a look at your diagram and see what you have; look to see what you don't have and what you can use.2659

They are always going to give you what you need, like here: they gave me parallel lines and an exterior angle.2670

That is it for this lesson; we will see you next time--thank you for watching Educator.com.2677

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