Home » Mathematics » Geometry » Angles

Mary Pyo

Mary Pyo

Angles

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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 (18)

1 answer

Last reply by: Tamara Orujzade
Wed Jul 15, 2015 7:54 AM

Post by Cristal Barrios on June 15, 2015

I am having problems with the videos they load but they get stuck...and our internet is working fine and I am paying for this program

0 answers

Post by Aravinda Fernando on November 18, 2013

An obtuse angle is:

More than 90° and less than 180°.

Great Lecture, Thanks

1 answer

Last reply by: Professor Pyo
Fri Aug 2, 2013 1:44 AM

Post by Shahram Ahmadi N. Emran on July 14, 2013

The pair of vertical angles you drew at 18:38 are sharing both a vertex and a side. The only difference seems to be that they are on opposite sides of the intersecting line. Could you clarify this a bit?

1 answer

Last reply by: Professor Pyo
Fri Aug 2, 2013 1:42 AM

Post by Manfred Berger on May 27, 2013

The pair of vertical angles you drew at 18:38 are sharing both a vertex and a side. The only difference seems to be that they are on opposite sides of the intersecting line. Could you clarify this a bit?

1 answer

Last reply by: Ruby Zhang
Wed Jun 14, 2017 11:10 PM

Post by Abdihakim Ibrahim on November 20, 2011

oh no the popo

0 answers

Post by Abdihakim Ibrahim on November 20, 2011

c.c

1 answer

Last reply by: Ruby Zhang
Wed Jun 14, 2017 11:10 PM

Post by Abdihakim Ibrahim on November 20, 2011

>:D hahahah im a kid!

1 answer

Last reply by: Mary Pyo
Fri Feb 3, 2012 11:32 PM

Post by Abdihakim Ibrahim on November 19, 2011

?

1 answer

Last reply by: Mary Pyo
Sat Oct 29, 2011 11:20 PM

Post by Teresa Thumbi on October 19, 2011

Amazing , but i still dont get how she got 2n-5 ?

1 answer

Last reply by: Carol Corrigan
Fri Jul 8, 2011 9:34 AM

Post by Ahmed Shiran on June 5, 2011

Good Lesson,

Angles

  • Angle: Figure formed by two non-collinear rays with a common endpoint
  • Ray: Segment with one endpoint and one end extending indefinitely
  • Opposite rays: Two rays that extend in opposite directions to form a line
  • An angle separates a plane into 3 parts: interior, exterior, and the angle itself
  • Angles are measured in units called degrees
  • The number of degrees in the angle is the measure
  • Protractor Postulate: Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is r
  • Acute angle: Angles with a measure less than 90 degrees
  • Right angle: Angles that measure 90 degrees
  • Obtuse angle: Angles that measure more than 90 degrees
  • Angle bisector: A segment, ray, or line that divides an angle into two congruent angles
  • Adjacent angles: Angles with common vertex and common side
  • Vertical angles: Two nonadjacent angles formed by intersecting lines
  • Linear pair: Adjacent angles formed by opposite rays
  • Right angles are formed by perpendicular lines
  • Supplementary angles: Two angles whose measures have a sum of 180 degrees
  • Complementary angles: Two angles whose measures have a sum of 90 degrees

Angles

Decide which ones are the right forms of the ray in the following figure.

(A) M→N (B) N→M (C) N←M (D) ↔MN
Only (A) is right.
Decide which is the same angle as ∠2.

∠BEC, ∠AEC , ∠E , ∠CED , ∠CEB .
∠BEC and ∠CEB are the same as ∠2.
D is interior of ∠ABC, m∠CBD = 15o, m∠ABD = 30o, find m∠ABC.
m∠ABC = m∠CBD + m∠ABD = 15o + 30o = 45o.
OP is bisector of ∠MON, m∠MON is 30o, find m∠MOP.
m∠MOP = [1/2](m∠MON) = [1/2] ×30o = 15o.
∠AOB and ∠BOC are linear pair angles, find m∠AOC.
m∠AOC = m∠AOB + m∠BOC = 180o.
m∠1 = 125o, ∠2 and ∠1 are supplementary angles, ∠3 and ∠2 are complementary angles, find m∠2 and m∠3.
  • m∠1 + m∠2 = 180o
  • m∠2 = 180o − m∠1 = 180o − 125o = 55o
  • m∠2 + m∠3 = 90o
m∠3 = 90o − 55o = 35o.
and↔CD intersect at O,

Name:
A pair of adjacent angles
A pair of vertical angles
linear pairs
A pair of supplementary angles.
A pair of adjacent angles: ∠AOD and ∠BOD
A pair of vertical angles: ∠AOD and ∠BOC
linear pairs: ∠AOD and ∠BOD
A pair of supplementary angles: ∠BOC and ∠BOD.
Draw a pair of adjacent angles.
and↔CE intersect at O, B is interior of ∠AOC, m∠BOC = 40o, m∠DOE = 110o, find m∠AOB.
  • ∠AOC and ∠DOE are vertical angles, so m∠AOC = m∠DOE = 110o
m∠AOB = m∠AOC − m∠BOC = 110o − 40o = 70o.
∠AOB and ∠BOC are supplementary angles, bisects ∠BOC, m∠COD = 2x + 1, m∠AOB = 4x + 2, find m∠BOD.
  • m∠BOC = 2m∠COD = 2(2x + 1)
  • m∠AOB + m∠BOC = 180
  • 4x + 2 + 2(2x + 1) = 180
  • x = 22
m∠BOD = m∠COD = 2x + 1 = 45o.

*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

Angles

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
  • Angles 0:05
    • Angle
    • Ray
    • Opposite Rays
  • Angles 3:22
    • Example: Naming Angle
  • Angles 6:39
    • Interior, Exterior, Angle
    • Measure and Degrees
  • Protractor Postulate 8:37
    • Example: Protractor Postulate
  • Angle Addition Postulate 11:41
    • Example: Angle addition Postulate
  • Classifying Angles 14:10
    • Acute Angle
    • Right Angles
    • Obtuse Angle
  • Angle Bisector 15:02
    • Example: Angle Bisector
  • Angle Relationships 16:43
    • Adjacent Angles
    • Vertical Angles
    • Linear Pair
  • Angle Relationships 20:31
    • Right Angles
    • Supplementary Angles
    • Complementary Angles
  • Extra Example 1: Angles 24:08
  • Extra Example 2: Angles 29:06
  • Extra Example 3: Angles 32:05
  • Extra Example 4 Angles 35:44

Transcription: Angles

Welcome back to Educator.com.0000

This next lesson is on angles.0002

OK, first, let's go over what an angle is: an angle is a figure formed by two non-collinear rays with a common endpoint.0007

Let's go over what a ray is: a ray is a segment like this, with one endpoint and one end extending indefinitely.0021

It is like part of it is a line, and part of it is a segment--one endpoint and one end going continuously.0035

An angle is a figure when we have two rays together, like this here, with a common endpoint.0048

Here is one ray, and here is another ray.0059

Now, just to go over rays a little bit: if you have a ray like this, then to write it...0063

now, we know that, if we have a line, we write it like this, using symbols...0072

if we have a ray, then you are going to write it like this, with one arrow.0078

Now, be careful: you cannot write it like this--you can't do that, because the endpoint is at A.0082

So then, the arrow has to be pointing to the point that is closer to the arrow.0095

So, if it is AB, it is going this way; so then you have to point the arrow that way: AB.0101

It can't be BA, because that is showing you that that is going the other way.0106

Now, also, don't write it like that, either--the direction has to be going to the right.0110

This is the right way; not like that, and not like that.0120

Opposite rays are when you have two rays that extend in opposite directions.0130

If they go in opposite directions...they have a common endpoint; one ray is going this way; the other ray is going this way.0137

Those are opposite rays: you have two rays: one is going to the right, and one is going to the left.0147

They form a line, because they are going in exactly opposite directions.0153

So then, opposite rays are two rays that extend in opposite directions to form a line.0159

Here is a diagram of an angle; now, this angle right here, where the two rays meet, the common endpoint is called the vertex.0166

And each of these rays in the angle are called sides.0187

This is a side, and then this is a side; this is a vertex, and these are sides, of an angle.0193

List all of the possible names for the angles: here, this is actually made up of three angles.0204

We have this angle right here, and I can just label that angle 1; this is another angle right here, and then this big angle.0214

To list all of the names, you are going to have to look at the different angles.0229

Depending on what angle you are looking at, you are going to call it by a different name.0236

This first one right here is going to be angle ABD; make sure that the middle letter is the vertex.0240

This has to be angle ABD; if you do angle ADB, that is going to look like that; ADB is like this, and that is not the angle, so it has to be angle ABD.0253

You can also say angle DBA, as long as the vertex is in the middle.0267

This can also be angle 1, because this whole angle is labeled as angle 1.0277

That is the first one; and then the next one can be angle DBC (again, with the vertex in the middle), angle CBD, or angle 2.0290

And then, the big one...now, if I just had an angle like this, just a single angle, and the vertex...0307

let's say this was different...EFG: if I had an angle like that, this can be angle angle EFG, angle GFE, or angle F.0314

It could be angle F, even though F is just the vertex; as long as you have only one angle--0333

this vertex is only for a single angle--then you can name the angle by its vertex, so this can be angle F.0341

In this case, I have three different angles here, so I can't label this whole...0351

even if I am talking about the big one, the whole thing, I can't label it as angle B, because angle B...this is a vertex for three different angles.0358

So, I cannot label it angle B; so instead, you have to just list it all out.0369

You are going to just say angle...the big one is label ABC; angle CBA; and that is it--those are the only two names for that big one.0375

It is not by 1 and 2; the 1 is for this angle, and then the 2 is for this angle.0390

An angle separates a plane into three parts; if I have an angle (actually, let me just draw it out--0402

there is my angle), then it separates it into its interior (the interior is the inside--that is all of this right here, on the inside),0414

and then the exterior (which is all of this, the outside of the angle), and then on the angle--the angle itself.0428

So, when you have an angle, there are three different parts: the inside, on the angle, and outside the angle.0441

And just so you are more familiar with these words: interior/exterior is inside/outside.0451

Angles are measured in units called degrees: so if you have this angle right here, this angle could be 110 degrees.0458

The number of degrees in an angle is the measure.0475

If I want to say that this angle is angle ABC, then I can say the measure (and m is for the measure) of angle ABC is 110.0480

That is how you would write it: the measure...and you write the m in front of the angle, angle ABC (what you name it) only when you are giving the measure.0499

The Protractor Postulate: from the last lesson (the last lesson was on segments), remember: we went over the Ruler Postulate.0519

This one is the Protractor Postulate; it is the same concept, but then, because it is an angle, you are not using a ruler; you are using a protractor.0529

And again, a postulate is any statement that is assumed to be true.0537

This is the protractor postulate--this whole thing right here, the statement.0542

Once we go over it, we can assume that it is true, because it is a postulate.0546

Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A extending on either side of ray AB0552

such that the measure of the angle formed is r; all that that is saying is that,0567

if you have this ray right here (this is A; this is B), now, if you have a ray AB, there is only one ray0575

or a single ray that you can draw to get a certain angle measure.0597

If I want an angle measure of 80 from AB, then there is only one ray that I can draw that is going to give you an angle measure of 80.0607

You can't draw two different angles; but it is going to be on both sides--so it can be this side,0619

or I can have an angle on...this is AB...then it can go on the other side, too; so this also can be AB.0624

That is all that they are saying: just for this, if you draw a ray like this to make it 80, you can't draw a different type of ray to also make it 80.0634

It is going to be something else.0645

So, for a number r between 0 and 180, there is only one ray that you can extend from this ray right here to give you that angle measure.0647

And the Protractor Postulate is as if you put this at 0 on your protractor, if you have your protractor like that;0661

and then here is your protractor, and then you have all of your angle measures; that is how you would read it.0677

And then, whatever this says right here--this number--that is going to be your angle measure of this.0688

Make sure that this endpoint is at the 0 of the protractor.0694

Angle Addition Postulate: from the last lesson, the segments lesson, we went over the Segment Addition Postulate.0702

The Segment Addition Postulate was when we had a segment that was broken down into its parts.0711

For the Segment Addition Postulate, remember, you had...this is AB...B is just anywhere in between A and C;0720

then we can say that AB + BC equals AC; so this plus this equals AC.0730

In the same way, we have the Angle Addition Postulate.0741

And all that it is saying is that, if R is in the interior (remember, interior is inside) of angle PQS, the measure of angle PQR0748

plus the measure of angle RQS equals the measure of angle PQS.0764

Measure means that you are talking about degrees, the number of degrees in the angle.0776

If the measure of angle PQR, let's say, is 30 (let's say this angle measure is 30),0785

and the measure of angle RQS is, say, 50; then the measure of angle PQS:0794

you just add them up, and then it is going to be...the whole thing is 80 degrees.0805

That is just the Angle Addition Postulate; in the same way, remember, if this was 3, and this was 5, then AC...you add them up, and you get 8.0812

It is the same thing--the Angle Addition Postulate.0824

You can also say the other way around: if the measure of this angle, plus the measure of this angle,0828

equals the measure of the whole thing, then R is in the interior of angle PQS.0837

Classifying angles: let's go over the different types of angles.0852

An acute angle is any angle that is less than 90 degrees; less than 90 looks like that--an acute angle.0857

A right angle is any angle that measures perfectly 90 degrees, like that.0871

And then, an obtuse angle is any angle that is greater than 90, like that.0882

This is a small angle, a right angle, and a big angle--an obtuse angle.0893

Angle bisector: we also went over a segment bisector.0905

A segment bisector was when you have a segment (or...it could be a segment; it could be a line;0909

it could be a ray; it could be a plane)...anything that cuts the segment in half; it intersects the segment.0919

Let's draw a line: this line intersects this segment at its midpoint.0929

That means that this line is called the segment bisector, because it is bisecting this segment; it is cutting it in half.0936

The same thing happens with an angle bisector: it could be a segment, ray, or line that divides the angle ABC into two congruent angles.0944

This ray, ray BD, is an angle bisector; BD, that ray, is the angle bisector of angle ABC.0955

Again, the bisector is whatever is doing the cutting, the dividing; it has to divide it into two congruent parts; and the same here.0971

This is a segment bisector; this would be the angle bisector, because it is intersecting the angle at exactly its middle.0985

Then, if this is 30, then this has to be 30.0996

Angle relationships: adjacent angles are angles that are next to each other with a common vertex and a side.1006

Angles 1 and 2 are adjacent angles, because they share a side, which is this right here, and a vertex.1028

If I have an angle like this, and then an angle like that, 1 and 2, even though they share a common side,1037

these angles would not be adjacent, because they don't share a vertex.1049

It has to be a side and a vertex; so this is not adjacent angles; these would be adjacent angles.1056

Again, two angles that share a side and a vertex are adjacent angles.1063

Vertical angles: if you have two lines that are intersecting each other1070

("intersecting," meaning that they cross each other--they meet, right here), then they form vertical angles.1077

Vertical angles, when they cross, would be the opposite angle; so this angle, right here, and this angle are vertical angles.1087

Non-adjacent means that they are not next to each other; they are not sharing a side and a vertex.1096

Even though vertical angles share a vertex, they are not sharing a side.1102

The two sides of this angle are this and this; the two sides of this angle are this and this.1106

They are not adjacent; they are non-adjacent angles formed by intersecting lines.1117

In this one right here, 1 and 2 would be vertical angles.1124

Now, this and this are also vertical angles; so then, in this one, 3 and 4 are also vertical angles.1127

There are two pairs of vertical angles when two lines are intersected; always, there are always vertical angles involved.1140

If I have an angle that looks like that, these are not considered...1150

because this is not a line, these are not vertical angles, and these are not vertical angles.1161

They have to be straight lines that are intersecting.1175

Linear pair: a linear pair are adjacent angles that form a line by opposite rays.1181

Remember: opposite rays form a line, so adjacent angles are two angles, like this, angles 1 and 2.1191

Here is angle 1, and here is angle 2; see how they form a line?1205

They are also adjacent, because they are sharing a side and a vertex.1213

A linear pair is two angles that just form a line; a linear pair is the pair of angles that forms a line.1218

Right angles formed by perpendicular lines have perpendicular lines; they are perpendicular;1233

then we know that this is 90 degrees; then that means that this is 90 degrees,1245

because we know that a straight line measures 180 degrees.1253

This is also 90, and this is 90; we are going to go over this again later, but perpendicular lines form right angles.1262

Supplementary angles are two angles that add up to 180, two angles whose measures have a sum of 180.1276

Supplementary is 180; complementary angles are two angles that add up to 90.1290

It has to be two angles that add up to 90 and two angles that add up to 180; this is 90.1302

Now, if I have an angle like this (let's say that this is 120 degrees), then I can say...1310

OK, if I ask you for the angle that makes it add up to 180, that is called the supplement.1325

This is 60 degrees, because these two angles together...if they are supplementary angles, then they have to add up to 180.1338

Two angles (1,2) that add up to 180...1347

If I want to compare them to each other, then I can say, "This is the supplement of this, and this is the supplement of that."1357

So, if I ask you, "What is the supplement of 120?" the answer will be 60.1364

"What is the supplement of 60?" 120; but together, they are supplementary angles.1373

The same thing for complementary: I have two angles; let's say this is 50, and this is 40; they are complementary angles; together they add up to 90.1380

But I can say that this is the complement of 40, and 40 is the complement of 50; together, they are complementary angles.1400

So again, supplementary is 180, and complementary is 90.1411

Sometimes, just be careful not to get confused between complementary and supplementary--which one is 90 and which one is 180.1418

Complementary starts with a C; it comes before S; C is before S, and then 90 is before 180; that is one way you can remember it.1428

Complementary is 90; supplementary is 180; C before S, 90 before 180.1440

Let's do a few examples: the first one: State all possible names for angle 1.1450

Here is angle 1; all of the possible names are...(now, I am not going to write measure of angle,1457

because I am not dealing with its degree measure)...just naming them, you are going to write1467

angle AFB, angle BFA (make sure that the vertex is in the middle); it is also angle 1.1474

Is that all? That is all; I cannot say angle F, because angle F...1493

that is the vertex of so many different angles that you can't use it to name any of those angles.1499

So then, that would be all--just those three.1508

Number 2: Name a pair of adjacent angles and vertical angles.1512

Adjacent angles would be...we can say that there are a lot; as long as they share a vertex and a side...1518

I can say that angles 1 and 2 are adjacent angles; angles 2 and 3; angles 3 and 4;1530

I can also say angle 4 and angle EFA, since this one doesn't have a number; I am going to say angle...let's see, 2...and angle 3.1540

Now, remember: angle 2 and angle 4 are not adjacent, because, even though they share a vertex, they don't share a side.1556

And for vertical angles, I can say angle AFE and angle...what is vertical to AFE?1566

It has to be angle BFC, or angle 2.1591

Be careful: right here, angle 4 and angle 1 are not vertical, because it has to be straight intersecting lines that form the two vertical angles.1599

So, you can say that 4 and 3 together are vertical to angle 1, because it is formed by the same sides, the lines; but not 4 and 1, and not 4 and 2.1613

The third one: Make a statement for angle EFC, using the Angle Addition Postulate.1628

The Angle Addition Postulate: remember, that means that this angle plus the measure of this angle is going to equal the measure of the full angle.1640

An the Angle Addition Postulate is different than an angle bisector; you can still apply the Angle Addition Postulate,1653

but for an angle bisector, it has to be cut in half; the angle has to be cut into two equal parts.1665

The Angle Addition Postulate could be like this, and you can say this, plus this small part, equals the whole thing, the whole angle.1672

That is the Angle Addition Postulate; I am going to say that the measure of angle EFD,1686

plus the measure of angle DFC, equals the measure of angle EFC; and that is my statement.1699

Number 4: Name a point in the exterior of angle AFE.1717

Angle AFE is right there; a point on the exterior...this is the interior, so it is anything outside of that.1724

I can say point B; I can say point C; or I can say point D.1734

I am just going to write point C; that is on the exterior.1737

The next example: Name a pair of opposite rays.1747

Opposite rays would be two rays with a common endpoint going in opposite directions.1754

I can say rays CF and CA...now, be careful; I can't say AC.1761

I can't say AC, because the ray is going this way, so I have to label it as CA, going towards that.1777

If the measure of angle ACB is 50, and measure of angle ACE, the whole thing, is 110, find the measure of angle BCE.1787

This is what I am looking for, x; that is the angle measure that I am looking for.1808

This is using the Angle Addition Postulate; so this, the measure of this angle, plus the measure of this angle, equals the whole thing.1813

So, the measure of angle ACB, plus the measure of angle ECB, equals the measure of angle ACE.1822

This will be 50; 50 plus the measure of angle ECB equals 110.1843

Subtract the 50; so the measure of angle ECB (or BCE--the same thing) equals 60 degrees.1855

Number 3: Name two angles that form a linear pair.1870

This one was to name a pair of opposite rays, two rays that form a line; this is two angles that form a linear pair.1878

That is a pair of angles that form a line; so I can say this angle right here and this angle right here,1889

angle FCD and angle DCA: angle ACD and angle DCF, or FCD.1899

Draw angles that satisfy the following conditions: Number 1: Two angles that intersect in one point.1928

We just need two angles that intersect in exactly one point.1935

I can draw an angle like this, and I can draw an angle like this; it doesn't matter, as long as they intersect at one point.1942

If it asks for two angles that intersect in four points, then that would be like this...1952

or this actually is two points; and then, if you went the other way, then you can just do four: so 1, 2...I meant 2.1963

The next one, measure of angle DEB plus the measure of angle BEF, equals the measure of angle DEF.1974

This is the Angle Addition Postulate; I know that this is going to be the whole thing.1987

And then, E has to be the vertex; then D and F...the measure of angle DEB...DE, and then, that means that B is going to be in the interior of the angle.1996

And then, that, plus the measure of angle BEF, equals the measure of angle DEF.2011

Your diagrams might be a little bit different than mine; but as long as you know that this plus this...2015

as long as B is in the middle of this angle, and E is the vertex, then that is fine.2022

Ray AB and ray BC as opposite rays: opposite rays means two rays that go in opposite directions to form a line.2029

There is A, B...I know that this has to be A, because that is how it is written.2041

A has to be at the endpoint; that and BC as opposite rays...well, I can do B, and then C like that, somewhere here.2049

Or, if you want to just draw it longer...it can be like that, or this will probably be AC; and then in that case, it would just be like...2066

A is a common endpoint; AB is going that way, and then AC is going this way.2089

And then, the last one: two angles that are complementary:2096

you can draw any two angles, as long they add up to 90 degrees (complementary, remember, is 90).2099

I can draw it like that; this could be 45 and 45; they are complementary angles.2110

I can draw two angles separately, maybe like that, and then, say, if this is 60, then I have to draw a 30-degree angle.2123

As long as they add up to 90...it is any two angles that add up to 90.2138

OK, the fourth example: BA and BE are opposite rays, and BC bisects the measure of angle ABD.2145

That means that this is a line, because they are opposite rays; so they are saying that it forms a line.2165

BC bisects this angle ABD; that means, since it bisects it, they are the same measure--they are equal.2171

If the measure of angle ABC equals 4x + 1, and the measure of angle CBD equals 6x - 15, then find the measure of angle CBD.2183

We have these two angles; now, when you draw a little line like that, that just means that they congruent to each other--they are equal.2197

And then, if you have two other angles that are not the same as that, then...2211

let's say these two angles are the same; then you can just draw these two, because you did one for each of these,2219

to show that all the angles that you drew one for are congruent.2226

Then, the next pair of congruent angles--you can just draw two.2230

The measure of angle ABC plus the measure of angle CBD is going to be the measure of angle ABD.2239

But then, they are actually wanting you to find the measure of angle CBD; and this is the angle bisector.2247

I can just make them equal to each other; let me just solve it down here.2254

4x + 1 is equal to 6x - 15; I need to solve for x.2258

So, if I subtract the 6x over, I get (let's write out the answer)...-2x; subtract the 1; and I get -16, so x = 8.2267

But they want you to find the measure of angle CBD; that means that, once you find x, you have to plug it back into to the measure of angle CBD.2286

That is 6(8) - 15; this is 48 - 15; that is 33; so the measure of angle CBD equals 33.2299

Number 2: the measure of angle DBC is 12n - 8; the measure of angle ABD, the whole thing, is 22n - 11; find the measure of angle ABC.2324

They give you this whole thing right here, and they give you DBC, and they give you ABD; and they want you to find this angle right here.2346

Since we know that this and this are the same--they have the same measure--this whole thing would be two times one of these.2359

So, if this is 10, then this has to be 10; then the whole thing is 20.2370

I can do 12n - 8, plus 12n - 8 (because this is also 12n - 8--the same thing), equals 22n - 11, the whole thing.2378

Or you can just do 2 times 12n - 8, because it is just this angle, times 2, equals ABD.2390

So, I am just going to do that; number 2 is 2(12n - 8) = 22n - 11.2400

This is going to be...I will use the distributive property...this is 24n - 16 = 22n - 11.2414

If I subtract this, I get 2n; add it over; I get 5; so n = 5/2.2425

And then, n is 5/2, and then we have to find the measure of angle ABC.2438

Now, they don't give me something for ABC; but as long as I find what DBC is, then that is the same measure.2450

I just do 12...substitute in that n...minus 8; this becomes 6; 6 times 5 is 30, so 30 - 8 is 22.2461

That means that the measure of angle ABC is 22.2480

The last one: If the measure of angle EBD, this one right here, is 115, find the measure of the angle supplementary to angle EBD.2493

Supplementary: it is asking you to find the supplement of this angle.2508

Remember: supplementary is 180, so find two angles that add up to 180; it is 115 + something (which is x) is going to add up to 180.2513

You are going to subtract it: x is equal to 65 degrees.2530

This will be the supplement of angle EBD.2539

All right, that is it for this lesson; we will see you next time.2549

Thank you for watching Educator.com!2552

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