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Mary Pyo

Mary Pyo

Ratios in Right 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 (5)

0 answers

Post by Kevin Yuan on February 16, 2014

I always forget which one to use even though there is the sohcahtoa. Any advice?

0 answers

Post by Dania Aljilani on November 4, 2013

Thank you soooo much! My math teacher spent three periods trying to explain this and I understood it in one lecture. You are a great teacher!

2 answers

Last reply by: Denise Bermudez
Wed Mar 11, 2015 5:50 PM

Post by Matthew Johnston on August 17, 2013

I am putting everything you are and I am getting wrong answers everytime??

Related Articles:

Ratios in Right Triangles

  • Trigonometry: The study of involving triangle measurement
  • sine (sin) = opposite/hypotenuse
  • cosine (cos) = adjacent/hypotenuse
  • tangent (tan) = opposite/adjacent
  • SOHCAHTOA

Ratios in Right Triangles

Write the inverse trig function of sin.
sin − 1
Find the value of sin70o.
0.940.
Find the value of cos30o
0.866.
Find the value of tan54o.
1.376.
Find the measure of ∠A, tan A = 0.8.
  • m∠ A = tan − 10.8
m∠ A = 38.7o.

Write sin M, cos P, and tan P.
  • sin M = [NP/MP]
  • cos P = [NP/MP]
tan P = [MN/NP].

AB = 2, BC = 4, AC = 5, find cos A.
cos A = [AB/AC] = [2/5].

m∠ M = 60o, find tan P.
  • m∠ P = 30o
tan P = tan30o = 0.577.
Determine whether the following statement is true or false.
In a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle.
True.

cos A = 0.6, find m∠ A.
m∠ A = cos − 1 0.6 = 53.1o.

*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

Ratios in Right 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
  • Trigonometric Ratios 0:08
    • Definition of Trigonometry
    • Sine (sin), Cosine (cos), & Tangent (tan)
  • Trigonometric Ratios 3:04
    • Trig Functions
    • Inverse Trig Functions
  • SOHCAHTOA 8:16
    • sin x
    • cos x
    • tan x
    • Example: SOHCAHTOA & Triangle
  • 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

Transcription: Ratios in Right Triangles

Welcome back to Educator.com.0000

For this lesson, we are going to continue on right triangles.0003

We are going to go over ratios and right triangles.0006

Trigonometric ratios: first of all, trigonometry is the study involving triangle measurement.0010

Because we are going to go over trigonometric ratios, this all has to do with trigonometry.0024

Now, I know that you are probably thinking that this is geometry, not trigonometry.0030

But it involves a lot of trigonometry, because of the triangles.0034

And we are going over right triangles, so for this section, and for the next couple of sections, we are going to be using a lot of trigonometric ratios.0041

The most common trigonometric ratios are these three right here.0051

The first one is sine; it is pronounced "sign"; this is how you spell it out: S-I-N-E...but we always write it as "sin," but it is pronounced "sign."0056

The next one is cosine, but we shorten it as "cos"; but we call it "cosine."0071

The next one is tangent, but we only write "tan."0082

Now, one thing to remember about trigonometric functions: there are more than three, but these are the main ones that we are going to go over.0089

These are the most common; and for now, we are only going to use these three.0100

But these, we can only use with angle measures; it is very, very important to remember that we can only use0106

the sine of an angle measure, the cosine of an angle measure, and the tangent of an angle measure.0114

They never stand alone; they always have to go with an angle measure.0121

So, if I say "the sine of 90 degrees," that is one way that I would say it; I am looking for the sine of 90 degrees.0124

Or I can say "cosine of 45 degrees," "tangent of 60 degrees..."0138

Always remember that sine, cosine, and tangent must be with an angle measure; you can only find the sine of an angle measure.0149

These trigonometric functions don't stand alone; they are with angle measures, and only angle measures.0159

You can't find the sine of just any random number; it has to be the sine of angle measure; that number must be an angle measure.0167

So, if you have a scientific calculator, or if you have a calculator like this, you are going to need it.0175

And we are going to practice finding values on the calculator.0186

Here, trigonometric functions, again, are sine, cosine, and tangent.0194

If you look on your calculator, you should have buttons that say "sin," "cos," and "tan."0200

Now, these right here, inverse trigonometric functions: if you look at the same three buttons,0209

then above it, it should say "sin-1"; above the "cos," it says "cos-1"; and above "tan," it says "tan-1."0217

Those are very, very important; we are going to practice using those key functions here.0230

We have sine, cosine, and tangent; and then above it, the second key is inverse sine, cosine, and tangent.0240

Now, when do we use each of these?--again, for these, we are going to use angle measures.0250

So, if I wanted to find the sine of 60 degrees, then I would punch in "sin(60)"; and you are going to get a number.0256

And that number says 0.866; so the sine of this angle measure equals this.0276

That is when you use trigonometric functions: the sine, cosine, or tangent (depending on what you have to find)0291

of the angle measure equals...and that is what your calculator is going to give you, the answer.0298

Now, when do you use these? Well, if you punch in "sin" and any number right after sine--0302

if you punch in sin, cos, or tan, and you punch in a number after sin, cos, or tan--0311

the calculator is going to think--it is going to assume--that that number that you punched in is an angle measure,0318

because again, you can only find those functions of angle measures.0324

If I have, let's say, the sine of x equals 0.866--so I have the answer, and I am missing the angle measure--0333

that is what I am missing, so that is where it has to go, here, because angle measure always has to go there--0345

always, always, always, it is the sin of an angle measure--because I don't have the angle measure0351

(I only have what it equals), I can't plug in this number here, because this is not the angle measure.0357

So, if I punch in the sine of 0.866, the calculator is going to think that .866 is the angle measure, which it is not.0365

This is the answer; I am looking for the angle measure.0378

So, depending on what you have, you would have to use different things.0382

Now again, if you punch in sin(60), the calculator will know that 60 is the angle measure,0385

and therefore, the calculator is going to give you the answer to that, the sine of that angle measure.0393

If you want the calculator to give you the angle measure (you are not giving the calculator the angle measure--0400

you want the calculator to give you the angle measure), then that is when you have to use the inverse sine.0407

You are telling the calculator, "Well, I have this--I have the answer 0.866;0414

I don't have the angle measure to give you; given the answer, I want the angle measure."0421

Then, you would, second, press the sine (on your screen, is should say that it is the inverse sine); and then you punch in 0.866.0427

By doing that, the calculator now knows that that number that you punched in is not the angle measure.0442

Then, the answer is 59.997, which is 60 degrees; so let's just write 60.0448

Now, that is the angle measure; it is really important to remember which one you have to use.0468

Whatever number you punch in after sine, cosine, or tangent needs to be the angle measure, and the calculator will assume that.0474

So, if it is not the angle measure (you want the calculator to give you the angle measure), then you have to do inverse sine,0482

so that the calculator will know that that number is not the angle measure (it needs to give you the angle measure).0488

Let's just do a few of those...oh, before that, we are going to go over this right here, "Soh-cah-toa."0495

Soh-cah-toa is just an easy way for you to remember three formulas.0507

Here we have "soh"; "cah" is another formula, and so is "toa."0518

Now, I know that it sounds funny; but just say it to yourself a few times, so that you get used to this word, "Soh-cah-toa."0530

And that is going to help you remember three of the formulas, which are also known as the three ratios.0537

Each of these stands for something: S is sine (we are going to write down this formula here).0549

Sine of...again, it has to be an angle measure, so let's write...x, equals...o is for opposite, the side opposite;0558

the opposite side, over...h is for hypotenuse; so all of this "soh" is this right here: "Sine of x equals opposite over the hypotenuse."0570

That is the "Soh," and that is the ratio for sine.0592

Now, for cosine, it is right here: "cah" is going to be "Cosine of x is equal to"...the a stands for adjacent side, over the hypotenuse for the h.0600

Cosine of x equals the adjacent side over the hypotenuse side.0627

And the last one, "toa" is for tangent: Tangent of x equals...o is for opposite side, over...a is for adjacent side.0634

Then, these three right here are the actual trigonometric functions; and then the rest,0659

the "oh," "ah," "oa," are all for sides: o is for opposite side; a is for adjacent; and h is for hypotenuse.0666

So then, again, "soh" is "Sine of x equals opposite over hypotenuse."0678

This one is "cosine of x equals adjacent over hypotenuse," and then, this one right here is "tangent of x equals opposite over adjacent."0687

This will help you remember these three formulas; and that is what this was for.0697

Sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; and then, tangent of x equals opposite over adjacent.0703

Again, x is going to be the angle measure--only an angle measure can go there.0731

So, let's say C: we are going to find sine of C.0735

That is C, so now we are talking about from this angle's point of view.0748

From this angle's point of view, what is the side opposite?--because we are looking for the side opposite.0752

The opposite side would be side AB; and "over hypotenuse"--what is the hypotenuse? BC, so it is over BC.0762

So again, the sine of C (now, it doesn't always have to be the sine of C; they will either name the angle,0777

or they will tell you from what angle's point of view), for this one, we are going to do angle C's point of view: what is the side opposite?0785

It is that right there, over the hypotenuse, which is BC.0796

And then, for the cosine of C, again, from angle C's point of view, the adjacent side is the side that is next to it--0802

not that one that is opposite, but it is the one that is next to it; it is the other leg, the AC.0813

The side adjacent to angle C is AC, over the hypotenuse, which is BC.0821

And then, the tangent of C is going to be opposite (which is AB), over the adjacent, which is AC.0830

So again, the sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; tangent of x equals opposite over adjacent.0845

This is really, really important to know; again, just say this a few times to yourself: "Soh-cah-toa."0860

And that will definitely help you, because you do need to know these three ratios.0867

Now, let's do a few practices on the calculator, finding trigonometric functions.0879

We are going to find the value of each ratio, or the measure of each angle.0888

The first one, the sine of 15 (now, this is an angle measure, so let me just do that--these are angle measures), equals...0893

and that is what we are looking for: so you go on your calculator, and just punch in "sin(15)".0905

And then, because you are writing "15" right after you punch "sin," the calculator knows that 15 is the angle measure.0916

Then, the answer becomes .2588.0923

And then, the next one: it will be the tangent of 72, and that becomes 3.0777.0936

I am just rounding it to four decimal places: write that over here...3.0777.0953

Now, here, we don't have the angle measure; the angle measure is e--that is what we want to find.0965

We want the angle measure; so when we punch it in, we want the calculator to give us the angle measure.0974

But if you punch in "tan(0.9201)," the calculator will think that this is the angle measure, which is not true.0979

This is not the angle measure; e is the angle measure.0987

So, you tell the calculator, "I am going to give you the answer, and I want you to give me the angle measure."0990

Inverse tangent, remember, is 2nd and then tan: .9201...equals...1001

and then, the calculator knows that you want the angle measure, and that is 42.6 degrees.1011

The same thing here: we don't have the angle measure; we want the angle measure; that is what we are looking for.1027

When you punch it in, you can't punch in cosine of this number, because then the calculator...1033

now, let's just try it: just try clearing your screen, and then just punch in "cos(.2821)".1039

Now, it is going to give you .999987 and so on; that rounds to 1.1057

Now, this number is not the angle measure, because the calculator, because you punched in cosine of this number,1067

would assume that this is the angle measure; and so, it is going to give you the answer if this were to be the angle measure.1078

But what we have to do is tell the calculator that that number is not the angle measure.1088

Convert it to inverse cosine, and then the calculator will give you the angle measure: 73.6 degrees.1101

Just be very, very careful with that: if you have a number here, that would be the angle measure.1111

If you have a variable there, then you are doing the opposite: you are looking for the angle measure.1118

Make sure you punch in inverse trigonometric functions.1126

The next one: Find sine of a, cosine of a, and tangent of a for each.1133

Here is where we are going to be using Soh-cah-toa.1140

This one right here is going to be "sin(a)," the angle measure, "equals"...don't forget: if you don't have the angle measure,1153

make sure that you write a variable right there; you can't leave it as "sin() ="; there has to be something there.1163

The sine of a equals opposite over hypotenuse; cosine of a is equal to adjacent over hypotenuse;1169

and then, tangent of a is going to be opposite over adjacent.1184

The first one: to find this, we are going to have to use the sine one: sine and sine.1195

Then, sin(a) here is going to be opposite (from a's point of view, what is the side opposite?1201

It is this, so what is the measure of that?)--it is 5, over the hypotenuse, which is 13.1218

Now, you are just going to leave it like that, because it is a fraction, and you can leave fractions: sin(a) = 5/13.1228

The next one, cosine of a: again, from A's point of view, it is going to be adjacent; there is the adjacent: 12,1235

the one next to the angle, over the hypotenuse, which is 13.1251

And then, tangent of a is opposite over adjacent: from this angle's point of view, opposite is 5; adjacent is 12.1257

Now, the next one: let's look for a right here, again, from A's point of view.1276

So then, sine of a is opposite (which is 3), over hypotenuse (which is 5).1281

Cosine of a equals adjacent (which is 4), over the hypotenuse (which is 5).1296

And then, tangent of a is opposite (3) over the adjacent (which is 4).1306

That is all they wanted you to find--just those things.1318

Now, if they wanted you to find the actual angle measure, that is different;1320

then you would have to use your calculator and do the inverse sine,1326

because again, this is not the angle measure, so you can't punch in the sine of this number.1330

So, you are going to have to do inverse sine, so that you find the angle measure.1335

But then, here, for this problem, you don't have to find the angle measure, because they just want you to find sin(a), cos(a), tan(a)--that is it.1340

We found sin(a), cos(a), and tan(a) for each of these triangles; that is it.1347

In the next example, it is going to ask you for the actual angle measure.1354

So, you are going to have to use these trigonometric ratios to actually find missing sides and angles.1359

You are actually solving for something there; in this, they just wanted you to actually just write down the ratio; that is it.1366

Next, find the value of x.1375

Like the previous example, we are not going to use...1383

well, in the previous example, we used all three trigonometric ratios, because they wanted us to find sin(a), cos(a), and tan(a).1387

For this one, we don't have to use all three; we just have to use whatever we need in order to find x.1395

We have to first look for an angle's point of view.1405

So, look for an angle: this is the angle that is given, so we are going to use this angle right here.1409

And then, what sides of the triangle are we working with?1416

Are we working with the opposite? No, we don't have the measure of the opposite one.1421

We have the adjacent, and we have the hypotenuse.1428

Now, if I were to write out Soh-cah-toa again, just so it is easier to see,1432

the three ratios are going to be sin(x) = opposite/hypotenuse; the next one, cos(x), equals adjacent/hypotenuse;1439

and then, the last one is tan(x), equal to opposite/adjacent.1459

Which one of these three would we use?1473

We don't have the opposite; we only have the adjacent, which is this side right here1480

(because that is what we are looking for, so we have to include that one) and the hypotenuse.1486

Which one uses, from this angle's point of view, the adjacent and the hypotenuse?1492

Opposite over hypotenuse...no; adjacent over hypotenuse: so then, we are going to have to use this one right here; we are going to have to use cosine.1498

Now, do we have the angle measure--do we have x?1509

Yes, we do, so now we are going to just start plugging in these numbers for the angle measure, the side adjacent, and the hypotenuse.1513

Cosine of 55 equals adjacent (what is the one adjacent? That is x), over (what is the hypotenuse?) 12.1522

Now, we are going to go to our calculator, and we are going to find x.1538

Cosine of 55: now, 55 is the angle measure, so I can just punch in cos(55); cosine of 55 is .5736.1546

That equals...all of this is equal to this; that equals x/12.1568

Now, how do I solve for x here?--I am going to have to multiply the 12: multiply 12 on that side, and then multiply 12 to this side.1573

Then, x becomes...I just have to multiply: instead of clearing it, I can just leave that number, and then just multiply it to 12.1583

And I get 6.8829: so, this right here, this length, is 6.8829.1595

Again, to go over what we just did: cos(55)...now, why did we use cosine, and not sine and not tangent?1617

It is because we have to look at what we have.1625

And again, we don't have to use all three of them; we just have to use the one that we need,1629

unless it is like the previous problem, where it asks for all three; this one is not--it is just asking to find the missing values.1634

So, you have to look for cosine; you have to use cosine, because from this angle's point of view, you only have the adjacent and the hypotenuse.1641

So, you are going to use cos(x) = adjacent/hypotenuse; cos(55) =...adjacent is x; hypotenuse is 12.1656

You punch this into your calculator; you get this; then you multiply 12 to both sides, and you get the answer.1666

Now, I can find this third angle measure by subtracting it by 180, or I can just do 90 minus this number, and then I get this number.1675

I could do that, and then, if you want, you can use this angle's point of view.1687

This right here is actually going to be 35 degrees; and then, instead of using 55, you can use this one.1692

If you decide to use that one, then it is a different perspective, a different point of view.1705

So then, what would you have to use?1710

We have opposite (this is opposite), and then the hypotenuse; if you are going to use this angle, then you would have to use the sine,1713

because sin(35) is opposite over the hypotenuse; so you have options.1722

You don't have to do both; you just have to use one of them.1731

Now, because this is the angle that is given, it would just be easier, instead of doing more work, to look for this angle, and then go on from there.1733

But either way, that is an option for you, if you would like to just use that angle instead.1743

Let's do the next one: here we have...1751

And we are going to have to use this one; it is not like this one, where we have options,1756

where we can use this angle or this angle; for this one, we have to use this angle, because that is the angle that we are looking for.1760

And then, plus, we can't subtract it from 180 (because we don't know what this angle is) to find that angle.1765

We have to use this angle; from this angle's point of view, you have to work with the side opposite and the hypotenuse.1771

So, you have opposite, and you have hypotenuse; what are you going to use?1785

With opposite and hypotenuse, you are going to have to use sine, because that uses the side opposite and the hypotenuse.1790

The other ones use adjacent and the hypotenuse (no, we don't have the adjacent);1801

this one uses opposite, which we have, but no adjacent; so we have to use the sine.1806

Then, sine...and then, what would go as the angle measure?1814

We don't have the angle measure; that is what we are looking for.1820

So, we are going to put x right there; that equals...the side opposite is 21, over...what is the hypotenuse?...25.1822

Then, from here, I am going to just go to my calculator.1835

And again, don't punch in sine of this number, because then your calculator would think that this is the angle measure.1842

But it is not; so you are going to do inverse sine; that is 2nd and sine; you should see that sin-1.1851

And then, 21 divided by 25...and then, you should get...1865

That automatically just gives you x, because you already used the inverse sine.1875

57.1 degrees: that is this angle measure; 57.1 degrees--that is x.1881

That is it; just remember that you have to look for an angle.1897

That way, you know from which angle's perspective you have to look to see what you have.1905

And then, from there, based on what you do have, what sides are given, and what you are looking for,1912

you are going to have to pick one of these: sine of x, cosine of x, or tangent of x.1918

And then, using one of those ratios, you are just going to plug in the numbers, and then solve for whatever missing value you need to solve for.1924

Let's do the last example: this one is a little bit tricky, but it is not difficult at all.1932

Just remember that, when you are given points like this, and you have to find the measure of angle B,1944

now, when you are given this problem, you don't know if you have to use trigonometric functions yet.1953

So, let's just plug in the points first and see what we are dealing with.1963

The first one, A, is (-1,-5), right there; and then, B is -6 (1, 2, 3, 4, 5, 6) and -5; and then, C is -1, 1, 2, right there.1972

So, here is the triangle; I know that this is a right angle.2001

I know that because here, this point was (-1,-5); this is -5; and this point right here was (-6,-5).2013

So then, these two points are on the same y-value right there, so then it makes a horizontal line.2028

And then here, since this is (-1,-5), and this is (-1,2), they share the same x-coordinate, so this is a vertical line.2039

A horizontal and a vertical--they are perpendicular.2050

Now, what was this point? This point was A; this was B; and then, this right here is C.2055

And we want to find the measure of angle B; so then, this is the angle that we have to find.2067

Now, this is our variable; and then, let's look for side length.2076

Because here, it is perfectly horizontal or perfectly vertical, we wouldn't have to use the distance formula--we can just count.2087

Now, when it comes to BC, I am going to have to use the distance formula if I want to find BC.2097

Why?--because it is going diagonally, and you can't start counting to see the distance when it is going diagonally.2104

When it is going horizontally or vertically, then you can; you can just count the units.2113

So here, this is (-1,-5), and this one is (-6,-5); from -6 all the way to -5, this is 5 units.2120

If you have a bigger graph, you can start counting: it would just be from here, 1, 2, 3, 4, 5; so this is 5.2136

Then, this right here, the vertical--we can count that also, so here is -5: 1, 2, 3, 4, 5, 6, 7; this is 7.2146

AB is 5; AC is 7; we don't know BC.2160

Now, if you want, you can go ahead and solve for BC by using the distance formula.2165

The distance formula is x2, the second x, minus the first x, squared, plus (y2 - y1)2.2170

That is the distance formula; so you can just, using the coordinates for B and using the coordinates for C,2183

just plug it into this formula to find the distance of B to C.2188

But I don't think I will need it; now, let me re-draw this triangle...here is A, B, C.2194

This is what I am looking for, here; this is 5, and this is 7.2209

Now, if I am looking for this angle measure, what do I have to work with?2216

I have the side opposite, which is 7, and I have the side adjacent.2220

So, I know that I am going to use...Soh-cah-toa: from here, which one uses opposite and adjacent?2231

That would be this one right here, so we are going to use tangent.2246

That means that tangent of...I am just going to use b for the variable, because that is the angle that they want us to find...2250

tangent of b equals opposite (what is the side opposite? It is 7), and...what is the side adjacent? 5.2260

So then, here is my equation: tan(b) = 7/5.2274

Then, you just go straight to your calculator.2279

Now, remember: don't forget; don't push tan(7/5), because that is not the angle measure; b would be the angle measure.2284

Push 2nd, tan, to get the inverse tangent, 7/5.2290

I just close the parentheses; then, the calculator knows that I am giving them this answer, and that I want the angle measure.2299

The angle measure that it gave me was 54.46 degrees.2313

And that is it; now, when do we use Soh-cah-toa--when do we use trigonometric functions?2321

First of all, you must have a right triangle; they must be right triangles.2335

And #2: when you are dealing with angles and sides together (you are given an angle measure,2344

but you are looking for a side, using that angle measure; or given sides, you have to look for an angle measure)--2354

anything that uses a combination of angles and sides of a right triangle is when you are going to use Soh-cah-toa.2360

What if they give you two sides, and they just want you to find the other side, the missing side, the third side?2368

Well, we don't have to use Soh-cah-toa, because no angles are involved.2377

It is just only the sides; so let's say they gave you that this is 5 and this is 7, and they wanted you to find the missing side.2381

Now, do we have to use Soh-cah-toa here--do we have to use trigonometric functions here?2397

No, because no angles are involved here; then what can we use?2401

We can use the Pythagorean theorem, remember, because 52, a2, plus b2, equals c2.2409

Again, you are only using Soh-cah-toa when you are given sides, and then they want you to find the angle measure;2418

or given the angle measure, to find the missing side; and so on.2426

We are going to continue trigonometric functions next lesson.2431

For now, thank you for watching Educator.com.2435

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