Home » Mathematics » Geometry » Law of Cosines

Mary Pyo

Mary Pyo

Law of Cosines

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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 (3)

0 answers

Post by Todd McKinney on December 9, 2014

You make it a lot easier to understand.
Thanks

1 answer

Last reply by: Mary Pyo
Sat Feb 4, 2012 12:48 AM

Post by Sanjay Nenawati on October 27, 2011

a-s-a is does have a angle side relationship, because 180 - a -a= other angle. then you will have a side angle side relationship.

Related Articles:

Law of Cosines

  • Law of Cosines: Let ΔABC be any triangle with a, b, and c representing the measures sides opposite angles with measures A, B, and C, respectively. Then,
  • Use the Law of Cosines when:
    • The triangle is a non-right triangle
    • The measures of an angle and the side opposite are not given

Law of Cosines

Determine whether the following statement is true or false.

In ∆ABC, a2 = b2 + c2 − 2bc cosA.
True.
Determine whether the following statement is true or false.

Using Law of Cosines is the only way to find m∠A.
False.
Determine which one should be filled in the blank.

Use the Law of Cosines when _____ of the following statements are true.
1. The triangle is a non - right triangle.
2. The measures of an angle and the side opposite are not given.

A. One of Them
B. Neither
C. Both
C

AB = 21, BC = 18, m∠B = 60, find AC.
  • AC2 = AB2 + BC2 − 2AB*BC*cosB
  • AC2 = 212 + 182 − 2*21*18*cos60
  • AC2 = 387
AC = 16.7.

Determine whether the Law of Cosines need to be used to find AC.
No.

Use the Law of Cosines to find m∠A.
  • 252 = 102 + 202 − 2*10*20*cosA
  • 625 = 500 − 400cosA
  • cosA = − [5/16]
m∠A = 108.2o
Determine whether the following statement is true or false.
In ∆ABC, if ∠A is an obtuse angle, then cosA is always negative.
True.
Determine which one should be used to find the missing measurement, Law of Sines or Law of Cosines.
Law of Sines.
Determine whether the following statement is true or false.
In triangle ABC, a and b are the opposite sides of ∠A and ∠B, if cosA > cosB, then a > b.
False.

Find CD.
  • AC2 = 142 + 102 − 2*14*10*cos110
  • AC2 = 392
  • AC = 19.8.
  • [(sin∠ACB)/14] = [sin110/19.8]
  • sin∠ACB = 0.664
  • m∠ACB = 41.6o
  • m∠CAD = m∠ACB = 41.6o
  • CD2 = 182 + 19.82 − 2*18*19.8*cos41.6
  • CD2 = 183
CD = 13.5

*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

Law of Cosines

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
  • Law of Cosines 0:35
    • Law of Cosines
  • Law of Cosines 6:22
    • Use the Law of Cosines When Both are True
  • Law of Cosines 8:35
    • Example: Law of Cosines
  • 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

Transcription: Law of Cosines

Welcome back to Educator.com.0000

We are going to continue this unit with the Law of Cosines.0002

We are still on triangles; now, using Law of Cosines is very similar to when we used Law of Sines.0007

We are using both of those for non-right triangles; it is very important to remember that there are different things0016

that you can use for right triangles, and then there are things that you can use for non-right triangles.0024

So, Law of Sines and Law of Cosines are used for non-right triangles.0030

The Law of Cosines consists of three different formulas, depending on what you need--what measure you have to find.0038

If you have a triangle ABC with sides a, b, c, and angles A, B, C, angles are capital, and the sides opposite them are the lowercase of that letter.0048

So, if this is angle A, then this would be side a; if this is angle B, this is side b, and this is side c.0063

It is really important to remember that the angle and the side opposite go together, like a pair: this angle and this side opposite, and so on.0073

If we are looking for, let's say, A, you can use this formula.0086

If you are given, let's say, a, b, and c, and you are looking for angle B, well, then, you have to use this one.0094

Or if you are, let's say, looking for b, and you are given sides c and a, and then you are given the angle C, then you would use this one.0105

You would only use one of these, depending on what you need.0117

That is a2 = b2 + c2 minus 2 times bc times cos(A).0121

If you look at this, this looks kind of like the Pythagorean theorem: a2 = b2 + c2, the other two sides squared.0132

But it is minus 2 times bc, and the cosine of A; so then, that is how it is going to work for all of them.0141

b2 = a2 + c2 - 2 times a and c, cosine of angle B.0149

c2 = a2 + b2 minus 2 times a times b, times the cosine of this angle C.0160

And that is the Law of Cosines.0170

Now, when you are given a non-right triangle, and you have to determine whether to use Law of Sines or Law of Cosines--0173

well, the Law of Sines would probably be the easier option, so if you can, you would use the Law of Sines.0183

But remember: the Law of Sines was sin(A) over side a = sin(B)/b and sin(C)/c.0191

So, in order for you to be able to use the Law of Sines, you would have to have an angle-side pair.0205

If you are given an angle-side pair, then you are always going to be given three measures.0215

Out of the six total, the three angles and three sides, you are going to be given three.0219

In order to use the Law of Sines for non-right triangles, out of those three, two of them would have to be an angle-side pair.0225

When you don't have an angle-side pair, and it is a non-right triangle, then you would have to use the Law of Cosines.0235

Again, for non-right triangles, you can use the Law of Sines or the Law of Cosines.0242

If you can, you will look to see if you have an angle-side pair; if you do, then you would use the Law of Sines.0250

But if all three measures that are given...none of them is an angle-side pair, then you would have to use the Law of Cosines.0258

It is either one or the other.0267

For example, let's say that the measure of angle B is 27 degrees, and let's say this over here is 5, and this over here is 10.0271

Well, here we have no angle-side pair, because angle C is not given; so then, we don't have that angle-side pair;0288

we have the measure of angle B, but no side b, so the angle-side pair is not given; and the same thing happens for this.0299

So, since we have no angle-side pair, we would have to use the Law of Cosines.0305

And see how we are given a and c, and we have angle B; so which one would we use?0310

Well, it depends on what we are looking for; if we are looking for this one right here--let's say we are looking for b--0321

that went first; then we would have to use this one right here, because this is what you are looking for.0331

You are given this; you are given this; and you are given the measure of angle B; so you would have to use this one.0337

Let's say we are looking for, let's see...once we find that, if we want to find, let's say...the measure of angle A;0344

then after we find B, we will have an angle-side pair; we will have the measure of angle B, and we will have side b.0359

So, from there, you can go ahead and use the Law of Sines, using this one and this one that we found, and then sin(A)/a.0366

Let's just do a few problems and see if we can get more familiar with this.0379

Use the Law of Cosines when both are true: the triangle is a non-right triangle (we know this one),0384

and then the measures of an angle and the side opposite are not given, meaning that there is no angle-side pair.0392

Here, if I draw this out, and this is a; this is b; and this is c; we know that, for us to use the Law of Cosines, these have to be true.0401

If these are true, then we have to use the Law of Cosines.0418

And from the three measures that they give you, there are going to be three different types of problems0421

where you are going to use the Law of Cosines, where both of these would be true.0428

If I am given a side, a side, and a side--the measure of all three sides of a non-right triangle--then that would be the Law of Cosines.0432

Also, if you have side-angle-side (with the one that we just did, that was side-angle side, because that was side b,0443

and then angle C, and then side a), that doesn't give you any angle-side pairs.0453

So, side-angle-side would be another one; and then, the third one is going to be angle-side-angle.0458

So, if you have an angle, a side, and an angle, that is also a case where you are not going to have any angle-side pairs.0468

These three are the types of problems, given the measures, where you are going to have to use the Law of Cosines: side-side-side...0479

and don't get this confused with the side-side-side postulates and the congruence theorems,0491

because that is when you are using two triangles, and you are trying to make them congruent.0499

With this, I am just doing this so that you can see that, if you are given the measure of a side, the measure of a side,0503

the measure of a side, of one triangle, then you are going to have to use the Law of Cosines, and so on.0509

OK, for this triangle ABC, we are going to look for side a, which is this side right here, the side opposite angle A.0516

I know that I have no angle-side pair; I have a side, an angle, and this side.0527

And so, using one of the formulas (I know that I have to use the Law of Cosines, so), here are the three formulas.0533

It is a2 = b2 + c2 - 2bc times the cosine of angle A.0543

Remember: it has to be the cosine, because it is the Law of Cosines.0555

And remember that all lowercase letters are the sides, and the capital letters are angles.0559

The second one is b2 = a2 + c2 (it is the other two sides), minus 2 times ac cosine of B.0569

And then, side c2 = a2 + b2 - 2ab cosine of angle C.0582

Those are your formulas; now, we are going to see which one we need to use.0597

I have side c, side b, and angle A; if I am looking for side a, well, I am given b, c, and angle A, and this is what I am looking for.0602

So, I have to use the first one: that is going to be a2 = b2...0618

our b is 21, so 212, plus...c is 14, squared, minus 2 times 21 times 14, cosine of 60.0629

From here, it is all calculator math.0646

Now, if your calculator is the type that doesn't perform everything in one screen, then just go ahead and solve it in chunks.0652

But keep in mind--remember the order of operations: we have to multiply all of those together.0663

And this right here is the cosine of 60, so you would have to punch in cos(60) on your calculator to find that number.0672

You cannot separate the cosine and the angle measure.0682

It is going to be 2 times 21 times 14 times whatever this number is, so it is four numbers; and cos(60) is actually 1/2,0686

so it will be those four, multiplied; you are going to square this; you can add these together,0694

because here, multiplying this, this is far away from this right here; so you can solve this, and they won't affect each other.0702

Just remember that you have to multiply the 2 by all of those numbers before subtracting the 2.0709

So, be careful not to just solve it all in the order of this plus this minus 2; it has to be minus the whole thing.0716

I am going to just go ahead and do 21; and I do have a calculator on my screen: 212 is 441,0725

plus...142 is 196, minus...and then, let's do all of that, too.0735

2 times 21 times 14 times cos(60), .5, equals 294, so a2 is going to be 441 + 196 - 294, so 343.0750

Then, the square root of that is 18.5, so that is the measure of side a, 18.5.0795

All right, we are going to work on some more examples.0817

Determine whether the Law of Sines or the Law of Cosines should be used to solve each triangle.0820

We are not going to actually solve them; we are just going to determine if we can use the Law of Sines or the Law of Cosines.0826

Remember that both the Law of Sines and the Law of Cosines are used for non-right triangles.0835

And you want to look to see if you have an angle-side pair.0841

If you do, then you can go ahead and use the Law of Sines, because the Law of Sines is a lot easier.0847

If you don't, then you would have to use the Law of Cosines.0852

Now, if they say to solve the triangle, we went over that before; solving the triangle just means looking for all of the unknown measures.0856

So, in this case, they have given you this side, this side, and this angle.0866

To solve the triangle, we would have to solve for the measure of angle B, the measure of angle C, and AB, because those are all unknown measures.0870

The first one: we have a non-right triangle, and let's see how we have the measure of angle A, and we have side a, and then we have side b.0885

So, we do have an angle-side pair; since we have an angle-side pair...0900

now, I am not going to actually solve it, but I just want you to see that the Law of Sines is this.0904

Remember: you don't use all three of these ratios--you would just use two of them to create a proportion, whichever two you need.0917

We know that we are going to use the angle A and side a; it is going to be sin(30)/7 =...0925

and then, this is side b; so then, we would be looking for angle B; it would be sin(B)/11.0937

And then, you would solve it that way; you would have to solve sin(30) on your calculator; you can multiply that by 11, cross-multiplying.0948

And then, you are going to divide the 7.0958

And then, when you solve sin(B), remember: you have to use the inverse sine, the sin-1.0963

It is probably the 2nd key, and then "sin" on your calculator; use the inverse sine to find the measure of angle B.0969

That is the Law of Sines for this one; and then, for this one, we are given all of the side measures, but no angles.0981

That means that we have no angle-side pairs, so we would have to use the Law of Cosines--the Law of Sines here, and the Law of Cosines here.0994

And since you have three formulas to work with, you would have to choose whichever one you need,1008

depending on what angle or what measure...what side...whatever it is that you are looking for.1014

That is Example 1; for the next example, we are going to use more Law of Cosines to find missing measures.1021

For the first one, I want to look for the measure of angle B.1032

I am going to label that as x, just so that you know that that is the missing measure.1036

And then, before we start, let's write out all of the formulas again.1042

That is a2 = b2 + c2 - 2bc cosine of angle A.1048

b2 = a2 + c2 - 2ac cos(B).1061

c2 = a2 + b2 - 2ab cos(C); there are the formulas.1072

And then, to find the measure of angle B, given all three sides, I want to see what I want to use--which formula I would use.1084

So, for this one right here, I would have to use the second formula, because that is the only one that gives me angle B.1094

If you are looking for an angle measure, then you would have to pick and choose your formula, depending on this right here.1108

These are the only parts of the formulas that give you the angle measures, so it has to be the second one.1115

It is going to be b2, which is 52, equals a2 (is 13 squared), plus 122, minus 2ac cosine of angle B.1127

This is going to be 25 = 169 + 144 - 2...here, let's solve this part out on the calculator.1154

Remember that you can square this; you can square that; exponents come first, always, always, by the order of operations.1171

And once you do that, make sure that you do not subtract 2; it is not this number, subtract 2, because 2 is multiplied to all of this.1180

So, it would have to be 2 times 13 times 12...and then, for this one, cos(B)...1193

we are looking for angle B, so this is our variable; but we cannot separate the cosine and the B, because they have to go together.1199

It has to be the cosine of an angle measure.1208

Think of this whole thing as the variable; that is what you are solving for, so think of that whole thing as x.1211

Here, it would be minus 2 times 13 times 12; these three numbers, multiplied together, is going to be the coefficient of this variable.1222

2 times 13 times 12 is 312, cosine of B; I will circle this again.1235

Then, this is going to be 25; it equals this plus this; it is going to be 313, minus 312, cosine of B.1254

Be careful, again: do not subtract these two numbers; this is not 25 = 1cos(B); do not subtract them; it is this minus this whole thing.1269

But what I can do is subtract this over to the other side.1280

If I subtract 313, I am going to get...that is going to be -288, which is equal to negative (don't forget the negative sign) 312 cosine of B.1288

Now, from here, again, if I circle this, and that is the variable, then this is the coefficient;1310

I would have to divide this number to both sides, so that I can solve for this variable.1317

Divide this by -312; divide this by -312 (I'll write that over).1326

Your number is going to be -288, divided by -312: 0.923 = cos(B).1340

Now, remember, again: we are solving for angle B; that means we have to find the inverse cosine,1363

because here, if we have this angle measure, then we can go ahead and punch in the cosine of that angle measure.1372

But since that is the actual number that we are looking for, you would have to use the inverse cosine, don't forget--1381

because when you punch in a number after sin, cos, or tan, then the calculator is going to think1387

that that number is the angle measure, because it is always the cosine, sine, or tangent of an angle measure.1395

Just make sure you don't punch in the cosine of this number here, or else your calculator is going to think that that is the angle measure.1403

But instead, we are doing the cosine of the unknown angle measure; it equals...and that is the answer.1412

You would have to use the inverse cosine; that is cos-1; on your calculator, you might have to press the 2nd key and cosine.1420

It depends on your calculator; it might have its own separate button for it--just look for it.1436

See how you just punch that in with this number; and then it is going to be 22.6...let's see, let's just round it to hundredths.1445

So, that would be the measure of angle B, 22.6 degrees.1462

The next one: we are given angle, side, and angle; we have no angle-side pairs.1473

So, let's say that I want to look for two unknown measures for this one.1487

If I have this as the unknown measure--this is the first unknown measure--remember...1498

now, you can use the Law of Cosines, but it is a lot of work, and it does take quite some time.1504

Here, we actually don't have to use the Law of Cosines, because this is the third angle of a triangle.1511

So, try to remember that, if you are looking for the third angle of a triangle, then you can use the angle sum theorem,1518

where all three angles of a triangle add up to 180; so it is just going to be 180 = 40 degrees + 59 + x.1528

So, all three angles together add up to 180.1548

This is 99 + x; if we subtract it, that is going to be 81; it equals x, so the measure of angle C is 81 degrees.1550

Now, I also want to look for, let's say, side b.1580

So, in this case, we would have to use the Law of Cosines; which one would I use?1588

If I am looking for this here, for this one, do I have all the sides--what sides do I have?1598

I have this side right here; so here, I have 14; this is unknown; and then, for this one, let's see...1611

I have all three angles, so I don't have to worry about that; I have side c; I have this; I have this; the first one I have...1622

And then, this is what I am looking for; and then, I have the measure of angle A, and I have this, again.1638

I wouldn't be able to use that one, because I only have that side right there.1648

Now, if you notice that all three of these don't work, it is because here, now that I found this angle measure, I have an angle-side pair.1660

I would use Law of Sines, instead of Law of Cosines.1674

Let's save ourselves a lot of work; and then here, since the measure of angle C is now 81,1679

in order for me to look for side b, I can use the Law of Sines, because of this angle-side pair.1684

Here, I am going to do this with the sine of angle B.1694

If you have something like this, you wouldn't have to use the Law of Cosines, because you can just find the third angle measure.1709

Now, if you want, you could; but again, this would be a lot easier.1718

This here, the sine of C, the sine of 81, over 14, equals the sine of 59, over b.1725

So, the sine of 81...we are going to cross-multiply, so bsin(81) = 14sin(59).1738

I can divide the sine of 81 (remember: don't forget that you can't separate them)...so b would equal all of this, over sin(81).1757

I am going to put 12 over 0.988...so b is going to be 12.5.1785

Again, here we have a side and a side and a side given; and so, we are going to have to use this one, because the unknown measure would be angle B.1824

For this one, when you are given angle-side-angle, then just solve it by finding the third angle measure,1835

and then using the Law of Sines, using the angle-side pair.1844

For this next one, we are going to solve the triangle.1850

Remember: to solve the triangle means to solve for all unknown measures, angles and sides.1853

Here, I want to list out all of the different measures that I am going to look for: the measure of angle B, the measure of angle C, and then side a.1861

And that is just so that I don't forget to solve for something.1875

Now, we have no angle-side pairs; we have the measure of angle A, but we don't have side a...and here, and here.1880

So, there are no angle-side pairs; that means that I am going to have to use the Law of Cosines.1888

The three formulas, again, would be a2 = b2 + c2 - 2 times b times c times the cosine of angle A.1895

Then, b2 = a2 + c2 - 2 times a times c times the cosine of angle B.1909

c2 = a2 + b2 - 2 times a times b cosine of angle C.1921

First, in order for me to solve for this angle measure, I want this sine here.1935

If I want to solve for angle B, then I am going to have to know all of the measures of these sides.1941

Since we are given the two sides, side c and side b, I am going to look for side a.1951

Then, since I have the measure of angle A, I want to use this first one, because that is what goes in there.1962

It is going to be a2 = 202 + 242 - 2(b)(c)(cos(47)).1971

a2 = 400 + 576 -...don't forget that you cannot subtract this number with this;1994

it is going to be 2 times 20 times 24 times the cosine of 47.2012

You are going to multiply all four of those numbers together...that number, and this...so this is 654.72.2017

And then, now that you have multiplied all of that, you are just going to take this, add that, and subtract that.2047

400...add the 576, and subtract 654.72; you are going to get 321.28.2061

And then, to find a, we are going to have to take the square root of that; you are going to get 17.92.2084

So, I am going to write that answer here, for this one...92; and then I need to solve for these two.2095

So then, the measure of angle B...now that I have this, if I write this in here...2104

Keep in mind that, for any given problem where you have to use the Law of Cosines, you only have to use it once--you don't have to keep using it.2115

So, here, if I am solving the triangle, then I have to find the measures of three unknowns.2122

I don't have to use the Law of Cosines for each one of these; I only have to use it once.2128

And then, once you use it once, you are going to end up with an angle-side pair.2136

So, since I solved for a, I have side a, and I have angle A; there is my angle-side pair.2141

If you have an angle-side pair, then you know to use the Law of Sines: so sin(A)/a2150

equals the sine of angle B, over side b, which is equal to sin(C) over side c.2158

Since we have the angle-side pair of angle A and side a, we are going to use this one.2168

And then, if we solve for angle B first, then I am going to use that one.2173

I will put it down here: sin(47)/17.92 = sin(B)/20.2184

So, you are going to cross-multiply; that will be 20sin(47) = 17.92sin(B).2203

This is going to be, using your calculator...remember that this is sine, not cosine...it is this number, and then divided by this number;2216

this is your variable that you are solving for; that is the coefficient; so I need to divide this number...17.92.2239

I get 0.816; I am just going to double-check that really quickly.2263

47...the sine of 47 degrees...multiply that by 20 (you can do 20 times sin(47); it depends on what your calculator can do); divide it by 17.92.2272

So, .816 = the sine of angle B; now again, here we are looking for the angle measure, so we are going to have to use the inverse sine.2294

That means that you are going to hit the 2nd button; and that is going to give us B; the measure of angle B is 54.71.2309

We still have one more to solve for, the measure of angle C.2335

Now, we want to look for the easiest possible way to solve for this.2344

If you want, you can go ahead and use the Law of Sines again, because you have two different angle-side pairs.2348

But for this one, since I have two angle measures, I just want to look for the third one using the angle sum theorem.2355

Keep that in mind: if you have two angles, that would be the easiest way to look for a third angle.2363

The measure of angle C is going to be 180 minus 47 added to 54.71.2370

180 minus...and then, what are these two?...that will be 101.71; that means that the measure of angle C is going to be 78.29.2390

180 minus 101.71...the measure of angle C is 78.29.2421

All three of these are your answers, because these are the unknown measures of the triangle.2436

When it says to solve the triangle, then you are going to be looking for all of the unknown measures.2444

The easiest way would be to just first write out what you have to find, and then do one at a time.2451

First, always look to see if you can use the Law of Sines or the Law of Cosines.2460

In our case, we had to use the Law of Cosines, because we had no angle-side pair.2464

We are going to look for that side; and then, from there, you only have to use Law of Cosines once in a single problem.2470

We use the Law of Cosines; and then, since that gave us an angle-side pair, we went ahead and used the Law of Sines to find one of the angle measures.2480

Then, since you are given two angle measures, you can solve for the third angle measure by the angle sum theorem--that is solving the triangle.2489

And the fourth example: In parallelogram ABCD, AD is 8; AB is 11; and the measure of angle A is 110; find the measure of each diagonal of the parallelogram.2501

First, I am going to draw a parallelogram and label it ABCD.2515

AD is 8; AB is 11; the measure of angle A is 110; and then, find the measure of each diagonal.2529

That is the first diagonal; by drawing this diagonal in, we now have this triangle here.2548

If this is 11, and this is 8, then this is what we are looking for; this would be side a.2557

And then, in order to solve for that, I know it is a non-right triangle...do I have an angle-side pair?2566

No, I don't; so then, I have to use the Law of Cosines.2576

And remember: it is a2 = b2 + c2 - 2 bc cosine of angle A.2581

Here, a2 is what we are looking for; and even though it is not sides b and c, it would just be the other two sides.2602

There are three sides total; it is just all three, written out here.2612

So, a2 = b2...we are going to say 82...2616

+ c2, just the third side, 112, minus 2 times those two sides cosine of angle A.2624

It would be the angle with that side.2638

82 is 64, plus 112 is 121, minus...this would be 176cos(A).2646

Then, from here, remember that this is what we are solving for.2664

a2 = 185 - 176...OK, we are solving for side a; this would be 110; let me circle the variable right here;2671

then this is the cosine of 110; make sure you do not subtract these two numbers.2699

So here, we are going to just solve this out on the calculator...and then...a2 = 185 - -60.196.2705

a2 = 245.196; and then, a would be √245.196...15.67.2756

So, if I were to round to the hundredths, it would be 15.67; that is BD.2805

Now, they want us to find the measure of the other diagonal.2816

Now, this, obviously, doesn't look like it is 110; if you want to draw it so that it would fit better, you can draw it the other way, like this.2823

This was A, B, C, D; and it is the same thing, but just if you want to have it more scaled, this would be 110;2838

since this side is a little bit shorter...8...11...that is 110, and then we found that this right here is 15.67.2851

BD, one of the diagonals, is 15.67; now, if I erase that, and I look for the other diagonal right here,2861

well, I know that this is 11; and this is the side that I am looking for.2883

Here, this angle measure is...we have 8 and 11; and then, do we know any of the angle measures?2895

Well, what do we know about parallelograms?2909

If this is 110 (let me erase this diagonal, so that it is easier to see), we know that this angle right here,2914

with this angle, are supplementary, because if you were to draw these a little bit longer--2927

extend these sides and the side right here--then remember: we have same-side interior angles, or consecutive interior angles.2934

So then, with parallel lines and a transversal, if this is 110, then this angle right here would be that.2945

So, using that, let me erase this, and then redraw the diagonal right there.2953

I am going to be looking for the side AC; and it doesn't matter what we label it.2961

I can redraw just that triangle...70...and then I label this a, b, and c...8, 11, and this is what we are looking for.2971

Just for the sake of our formulas, I can use this same formula again, because this would be side a.2984

Then a2 would be b2, which is 112, plus 82, minus 2(11)(8) cosine of angle A.2993

So, a2 = 121 +...82 is 64, minus 176 cosine of 70 degrees.3013

a2 = 185 - 176cos(70); and then, you can solve this part right here.3032

Make sure that you multiply this number by this number on the calculator, instead of subtracting it first.3044

The cosine of 70, times 176, is minus 60.196; and then, take that and subtract the 180; a2 = 124.80; so a is going to equal 11.17.3052

That side is the other diagonal, and that is AC; that would be 11.17; those are the measures of the two diagonals of this parallelogram.3113

And remember how we initially drew this parallelogram.3128

Now, this is just so you have a visual; if you need to draw it more to scale, you can redraw it,3132

depending on what sides you have and what is bigger and what is smaller.3139

And then, we know that the measure of angle A was 110, so that would be an obtuse angle; so you can draw it in that way.3144

And here, I just redrew this triangle right here, so that it would be easier to see what we are dealing with, what we are using.3152

That is it for this lesson; thank you for watching Educator.com.3161

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