  Mary Pyo

Angles and Parallel Lines

Slide Duration:

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
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
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
5:02
Example and Definition of Segment Addition Postulate
5:03
8:01
8:04
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
11:41
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
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
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
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
Parallelograms

29m 11s

Intro
0:00
0:06
Four-sided Polygons
0:08
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
11:19
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
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
8:25
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
5:50
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
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
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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• ## Related Books 0 answersPost by Khanh Nguyen on May 6, 2015I think question 1 in "Practice Questions" needs to be more specific.It asks nothing about postulates. 2 answers Last reply by: Mary PyoMon May 18, 2020 11:15 AMPost by Jeremy Cohen on August 27, 2014One of the practice questions says that 180-125=65.  This is incorrect, it's like three or four questions in.  Please correct 0 answersPost by reid brian on February 7, 2012ah yeah very good yeah 0 answersPost by Ahmed Shiran on June 7, 2011Interesting ! :-)

### Angles and Parallel Lines

• Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent
• Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent
• Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary
• Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent
• Perpendicular Transversal Theorem: In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other

### Angles and Parallel Lines

Describe the following as intersecting, paraller, or skew.
The lines seperating lanes on the road.
• Parallel.
• State the postulate or theorem that allows you to conclude that ∠1 ≅∠2.
• Alternate exterior angles theorem
State the postulate or theorem that allows you to conclude that ∠1 and ∠2 are supplementary. Consecutive interior angles theorem. m||n, p||q, m∠1 = 125o, find m∠2, m∠3, m∠4 and m∠5.
• m∠2 = m∠1 = 125o
• m∠3 + m∠2 = 180o
• m∠3 = 180o − 125o = 55o
• m∠5 + m∠2 = 180o
• m∠5 = 180o − m∠2 = 180o − 125o = 65o
m∠4 = m∠5 = 65o. m||n, p||q, m1 = 70o, m∠2 = x + 4, m∠3 = 2y + 4, m∠4 = 5z + 5, find x, y and z.
• 2∠ ≡ ∠1
• m∠2 = m∠1
• x + 4 = 70
• x = 66
• ∠3 ≡ ∠2
• m∠3 = m2
• 2y + 4 = 70
• y = 33
• ∠4 ≡ ∠1
• m∠4 = m∠1
• 5z + 5 = 70
z = 13.
Find two pairs of consecutive interior angles in the figure. ∠3& ∠5, ∠4& ∠6.
Find two pairs of alternate interior angles in the figure. .
∠3& ∠6, ∠5& ∠6.
Describe the following as intersecting, paraller, or skew.
The lines seperating lanes on the road.
Parallel
Find two pairs of corresonding angles in the figure. .
∠1& ∠5, ∠2& ∠6.
Find two pairs of alternate exterior angles in the figure. .
∠1 & ∠8, ∠2& ∠7.
Use sometimes, always, or never to fill the blank in the statement. If m||n and p⊥m, then p⊥n is ______ true.
Always.
BC ||DE , m4 = 60o, m∠3 = 70o, find m∠2. • ∠1 ≡ ∠4
• m∠1 = m4 = 60o
• m∠1 + m∠2 + m∠3 = 180o
m∠2 = 180o − 60o − 70o = 50o

*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.

### Angles and Parallel Lines

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
• Corresponding Angles Postulate 0:05
• Corresponding Angles Postulate
• Alternate Interior Angles Theorem 3:05
• Alternate Interior Angles Theorem
• Consecutive Interior Angles Theorem 5:16
• Consecutive Interior Angles Theorem
• Alternate Exterior Angles Theorem 6:42
• Alternate Exterior Angles Theorem
• Parallel Lines Cut by a Transversal 7:18
• Example: Parallel Lines Cut by a Transversal
• Perpendicular Transversal Theorem 14:54
• Perpendicular Transversal Theorem
• 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

### Transcription: Angles and Parallel Lines

Welcome back to Educator.com.0000

The next lesson is on angles and parallel lines.0002

OK, last lesson, we learned about the different special angle relationships, when we have a transversal.0007

The transversal with the other lines forms angles, and those pairs of angles have special relationships.0017

And one of them was the corresponding angles.0030

Now, the two lines that the transversal cuts through--remember: I said that the lines can be parallel, but they don't have to be.0040

So, even if the lines are not parallel, you are still going to have corresponding angles.0050

But then, now the postulate is saying that, if the lines are parallel, then the corresponding angles are congruent.0056

If these lines are parallel (let's say that they are parallel lines), then each pair of corresponding angles is congruent--only if the lines are parallel.0068

If we don't have parallel lines--if I have lines like this and like this--they are not parallel;0083

they don't look parallel, but I have a transversal--let's say 1 and 2: these angles are corresponding angles, but they are not congruent.0093

They are not congruent, but they are still corresponding; angles 1 and 2 are corresponding angles, but they are not congruent.0105

They are just called corresponding angles; so be very careful--only if the lines are parallel, then you can see that corresponding angles are congruent.0112

They are the same; they have the same measure; they are congruent.0123

Since these lines are parallel, I can say that angles 1 and 2 are congruent.0127

So, angle 1 is congruent to angle 2; and it goes with all of the pairs of corresponding angles,0137

like this one and this one--they are congruent...this one and this one, and this one and this one.0145

Each of the pairs of corresponding angles is congruent only if the lines are parallel--that is very, very important.0152

And that is a postulate; a postulate, remember, is any statement (such as this) that we can assume to be true.0159

It doesn't have to be proved; if it is a theorem (the next few are actually going to be theorems),0170

then they have to be proved in order for you to be able to use them, because it is not true until it is proven.0176

The next one: here is a theorem; now, we are not going to prove these theorems now, but they are shown in your textbooks.0186

The alternate interior angles theorem--just so you know, some kind of proof has to be shown for the theorems0198

in order for them to be counted as true and correct, and then, that is when we can use them.0208

But for now, since they are proven in your book, we are just going to go ahead and use them.0214

The alternate interior angles theorem says that, if two parallel lines are cut by a transversal0219

(meaning, if the two lines that are cut by a transversal are parallel), then each pair of alternate interior angles is congruent.0225

Again, from the last lesson: if I have two lines...now, I know I am repeating myself a lot,0238

but that is so that you will understand this, because I have seen a lot of students make careless mistakes with these,0245

always thinking that these are congruent; in this case, if I tell you that these lines are not parallel,0258

or if I don't even say anything about them being parallel, then you don't assume that they are parallel.0265

We just have to assume that they are not parallel; then we can't say that angle 1 and angle 2 are congruent.0272

We can say that they are alternate interior angles; that is the relationship; but they are not congruent in this case.0277

So, for lines being parallel (now I am telling you that the lines are parallel), then alternate interior angles0286

(let's say that this is angle 1 and angle 2)...angle 1 is congruent to angle 2.0296

If the lines that are cut by a transversal are parallel, then alternate interior angles are congruent; and that is the theorem, the other one.0306

The next one is the consecutive interior angles theorem: If two lines that are cut by a transversal are parallel,0317

then (this is the tricky part--not tricky, but this is the part that students really make mistakes on) the consecutive interior angles0331

are not congruent; they are supplementary--this is very important.0344

Consecutive interior angles, we know, are angles that are on the same side, like these two angles right here.0351

And they are both on the inside, the interior.0357

So, angles 1 and 2 are consecutive interior angles; but then, they are not congruent--they are supplementary.0362

Only if the lines are parallel, then consecutive interior angles are supplementary.0369

See how the other ones that we just went over are congruent: these are not congruent--they are supplementary.0375

You have to say that the measure of angle 1, plus the measure of angle 2, equals (supplementary means) 180.0383

That means that this angle measure, plus this angle measure, equals 180--very important.0391

And the next one: Alternate exterior angles, if the lines are parallel, are congruent.0404

So, here is a pair of alternate exterior angles; angle 1 is congruent to angle 2.0415

And that also works for this pair of alternate exterior angles, like 3 and 4; those will be congruent, also.0425

Here we have parallel lines that are cut by a transversal.0442

If AB (let's say that this is A, and here is point B--and these are the points, not the angles;0445

here is point C and point D...then AB is a line, so it is line AB) is parallel to line CD,0460

and line CA is parallel to line DB (and then I am going to add these parallel markers;0471

that means that these two are parallel lines, and then for these--this is another pair of parallel lines,0484

so that means that I have to draw two of them for these, because it is another pair), find the values of x and y.0488

So then, here we have 80; and then I need to take a look at x.0500

If I look for a relationship between this one and another one, even though these two have a relationship,0508

this has a variable x, and this has a variable z.0517

I would rather use this relationship, 4x and 80, because, if I am going to compare them, at least this one doesn't have another variable.0521

So, it is easier to solve; so then, if I look at these two, I am only dealing with this line, line AB, line CD, and line BD.0530

That means that line AC, I am going to ignore, because it is not involved in this pair of relationships.0543

Remember from the last lesson: you look at the pair, and when you have the special pair, it only has three lines involved.0549

It only has line AB, line CD, and line BD involved; the other lines that are there--cover them up.0560

Those lines are there for another pair of relationships, so just cover it up.0567

You don't need this line for this pair, so just ignore it.0573

And then, to solve it, the theorem (and the relationship between these two: they are alternate interior angles,0583

because BD would be the transversal between these two lines) says that if the two lines0596

cut by a transversal are parallel (which they are--we know that because it gives us that in here),0604

if the lines are parallel, then alternate interior angles are congruent.0611

Since the lines are parallel, I can say that these two angles are congruent.0618

Then, they are congruent, so 4x = 80; and I divide by 4: x = 20.0621

There is my x-value; and then, for my y, let's look at this one.0638

Well, with this one, I know that, since I have an 80 here, 80 is also congruent to this angle right here,0651

because they are corresponding, and I know that these two lines are parallel.0667

If these two lines are parallel, here is my transversal; that means that this angle right here and this angle right here are corresponding.0670

And as long as the two lines that are cut by the transversal are parallel, then corresponding angles are congruent.0680

So then, I can just write an 80 in here; and then, between this and this, they are vertical.0685

Now, I could have just done this angle right here to this right here; so there are many ways to look at it.0694

You can look at corresponding angles; if you didn't really see the alternate exterior angles--0700

if that is kind of hard for you to see--then you can just say that, OK, they are corresponding, and then these two are vertical.0707

And vertical angles, remember, are always congruent.0711

So, you can say that these two are the same, because they are vertical.0717

Or you can say that this and this are the same, because they are alternate exterior angles.0722

And those are the same, as long as the two lines are parallel.0727

So, either way: 4y + 10 = 80; then 4y = 70; so y = 35/2.0730

And that is just 70/4, and then you just simplify it to 35/2.0755

Now, it doesn't ask for the value of z, but let's just go ahead and solve it.0765

We know that 6z and 80 have a relationship.0774

Now, I know that this is 80, because we found x; x is 20; and 4 times x is 80;0783

and also because they are alternate interior angles, so whatever this is, this has to have the same measure.0790

So, either way we look at it: we can look at it as 6z with this one right here,0798

or we can look at this one with this one right here--same relationship, same value,0801

which also means that this one is also the same as 4x; this angle and this angle have the same measure.0806

Either way, the 6z with this angle right here are consecutive interior angles, or same-side interior angles.0814

Now, if the lines are parallel (which they are), then consecutive interior angles are supplementary--not congruent, but supplementary,0826

which means that I can't make them equal to each other.0839

Consecutive interior angles are the only ones that are not congruent from the special pairs of angles.0844

Supplementary--that means that I have to make 6z + 80 equal to 180.0851

6z = 100; z = 100/6, and then I can just simplify this to 50/3, and that is it; that is z.0859

OK, the last theorem from this section in this lesson is the perpendicular transversal theorem.0897

Perpendicular, we know, are two lines that intersect to form a right angle.0904

So, if I have a line like this and a line like this, and they form a right angle, then they are perpendicular.0911

But then, here we have a transversal involved; so in a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other.0918

Here are my parallel lines; I am going to show it by doing that.0932

If my transversal, which is this line right here, is perpendicular to just one of the lines0936

(it doesn't matter which one), as long as these lines are parallel (they have to be parallel),0943

if it is perpendicular to one of the lines, then it has to be perpendicular to the other line.0951

If this is perpendicular to this line, then it is going to be perpendicular to this line, as well.0960

And that is the perpendicular transversal theorem.0967

Now, if the two lines are not parallel (let's say like this), and then I tell you that this line is perpendicular to this line,0969

it is not going to be perpendicular to this line, because these lines are not parallel.0983

In this case, don't assume that it is perpendicular to both--that is only if the lines are parallel.0990

Let's do a few examples: State the postulate or theorem that allows you conclude that angle 1 is congruent to angle 2.0999

Now, remember: the only postulate was the corresponding angles one: that is the one where you have the angles in the same corner,1007

in the same position, in the same corner of the intersection--that is the corresponding angles postulate.1015

Everything else--the consecutive interior angles theorem, the alternate interior angles theorem,1026

the alternate exterior angles theorem--those are all theorems; so the only one is the corresponding angles postulate.1034

Here, what postulate or theorem allows you to conclude that angle 1 is congruent to angle 2?1043

We know that this is our transversal line, because it is the one that cuts through two or more lines.1051

Then, angle 1 and angle 2 are alternate exterior angles.1056

Now, if these two lines are parallel, then we can conclude that angle 1 is congruent to angle 2; let me show that these two lines are parallel, too.1063

Then, this would be the alternate exterior angles theorem.1073

And this one right here--we know that these are corresponding angles.1097

And the only way that the postulate will make them congruent (the only way we can apply the postulate) is if these two lines are parallel, which they are.1106

So, I can say that, by the corresponding angles postulate, angle 1 is congruent to angle 2.1114

All right, the next one: In the figure, line e is parallel to line f.1134

So, let me show this; it doesn't matter which way--I can just do like this, or I can just do like that.1142

AB is parallel to CD, so this one is parallel to this; and the measure of angle 1 is 73.1149

I am going to write that in blue; so this is 73, right here.1158

Find the measure of the numbered angles.1165

All of the numbered angles is what it is asking for.1169

Let's look at this: to look for the measure of angle 2, I know that angle 1 and angle 2 are supplementary, because they are a linear pair.1175

They form a line, and a line is 180 degrees.1190

So, all linear pairs are supplementary; so since linear pairs are supplementary, and these are a linear pair,1196

I can say that 73 plus the measure of angle 2 equals 180.1208

And then, to find the measure of angle 2, I have to subtract the 73; so the measure of angle 2 equals 107.1218

And then, the next one: the measure of angle 3--well, if you look at this, we know that these two lines are parallel.1235

This line intersects both of the parallel lines; so this is a transversal--this line segment AB is a transversal,1246

which means that angle 2 and angle 3 are alternate interior angles.1256

And by the alternate interior angles theorem, since the lines are parallel, we know that these angles are congruent.1263

Since the measure of angle 2 is 107, I can say that the measure of angle 3 is 107.1272

And then, the measure of angle 4: it is also alternate interior angles with angle 1, so by that theorem, again,1284

since the two lines are parallel, those two will be the same; so it is 73.1303

Then, the measure of angle 5 is corresponding with angle 5; angle 5 and angle 1 are corresponding,1314

because it is as if I extend this line segment, just to help me out here: these two lines are parallel;1324

here is my transversal; can you see that?--this is a line, and this is a line; here is that transversal;1332

angle 1 and angle 5 are corresponding, so if this is 73, then the measure of angle 5 has to be 73.1344

And then, the measure of angle 6--you can say that angle 6 is also corresponding with angle 3.1354

So, if you extend this out again, there is my intersection, angle 3, and then my intersection, angle 6.1368

The measure of angle 3 is 107, so the measure of angle 6 is also 107.1378

Angle 7 is alternate interior angles with angle 6, so that has to be the same, since the lines are parallel.1386

And the two lines involved would be this line and this line--can you see that?--this line and this line, and here is my transversal.1399

These two lines are parallel, so angle 6 and angle 7 are congruent by the alternate interior angles theorem.1408

And then, the last one, the measure of angle 8: it is supplementary with angle 6, because it is a linear pair.1421

Or it is alternate interior angles with angle 5, or it is corresponding with angle 4; there are a lot of different relationships going on here.1431

If you want to use the alternate interior angles theorem with angle 5 and angle 8, then it is going to be 73.1447

If you want to look at the corresponding angles postulate with angle 4, then it is also 73.1457

If you want to say that it is supplementary with angle 6 (it is a supplement to angle 6), then it is 180 - 107, which is 73.1463

You can look at it in many different ways.1475

That is it: see how all of the angle measures are either 73 or 107.1482

Since all of these lines are parallel--these pairs are parallel, and those two pairs of lines are parallel--1488

they are going to have only two different numbers, because all of their relationships are congruent or supplementary.1498

So, it is either going to be congruent, or it is just going to be a supplement to it.1507

Another example: BC is parallel to DE (that is already shown); the measure of angle 1 is 61 (this is 61);1514

the measure of angle 2 is 43; and the measure of angle 3 is 35.1527

This one is going to be a little bit more difficult, because we have lines that are closing in on the sides.1539

And sometimes it is going to be a little confusing, or a little bit hard to see the lines that you need to see.1549

And you are going to have to ignore these.1561

So, look at angle...let's see...3 and angle 4; if you look at BE as a transversal, and these two1564

as the lines that the transversal is intersecting, 3 and 4 are alternate interior angles.1584

But these two lines are not parallel--those two lines that the transversal is intersecting are not parallel.1594

So, you can't assume that they are congruent; you can't say that they are congruent, because look: the two lines are intersecting.1602

Even though they are alternate interior angles, you can't apply the theorem saying that they are congruent, because the lines are not parallel.1612

You have to be very careful; you can't say that angle 4 is 35 degrees.1620

OK, so what can we say? We know that this line segment right here is parallel to this line right here.1626

I can say that the measure of angle 5...because look at this: this angle 5 and angle 2 are alternate interior angles;1641

now, let's see if we can apply the theorem and say that they are congruent.1658

Here is my transversal; here are the two lines that the transversal is intersecting; are the two lines parallel?1663

Yes, they are parallel; now, ignore this side and this side, AD and AE, because you don't need them.1671

It is as if they are not even there; cover it up.1681

Angle 5 doesn't involve those lines; angle 2 doesn't involve those lines.1684

So, all you have to see is this right here; here is BC; there is angle 5 and angle 2.1690

Here, these are parallel; here is 5, and here is 2.1700

So then, these are alternate interior angles, and they are congruent, because their lines are parallel.1708

The measure of angle 5 would be 43.1716

And then, from here, I can say that the measure of angle 7...if you look at angle 7 and angle 1, I have a transversal;1721

there is my angle 7, and there is my angle 1; these two lines are parallel.1752

See how it is only involving the three lines, this line, this line, and this line.1760

Ignore BE; see how I didn't draw it, because it is not involved.1766

Ignore all of the other lines; just look at those three lines for angle 1 and angle 7.1769

If it helps, you can draw it again; this one is a little bit hard to see using this diagram,1774

so if it helps you like this, then just draw it again, just using those three lines.1779

Angle 7 and angle 1 are corresponding; and since the lines are parallel, I can use the postulate to say that angle 1 and angle 7 are congruent.1785

So, the measure of angle 7 is 61.1794

So then, the ones that I found: this is 43; this is 61.1799

OK, to find the measure of angle 4, I can say that, because all these three angles right here form a linear pair,1807

that the measure of angle 7 plus the measure of angle 5 plus the measure of angle 4--they are all going to add up to 180,1824

because they form a straight line; all three angles right here are going to form a 180-degree angle.1833

You can say that the measure of angle...not 1....4, plus 61, plus 43, equals 180.1844

The measure of angle 4, plus 104, equals 180; you subtract the 104, so the measure of angle 4 equals 76; here is 76.1861

All of this is the one that I found, so I will write this in red: 76.1890

And then, let's look at some other ones: now, if you look at angle 8, angle 8 also involves this parallel line.1895

But this one is a little bit harder to see, because you have angle 8 like that; what is this angle right here?1915

This is angles 2 and 3 together; it is this angle and this whole thing.1930

Now, ignore this line; you are just involving this line, this transversal, and this bottom line DE.1936

So, this BE is not there; so it would just be this whole angle together.1946

So then, see how this angle right here and this angle right here are corresponding.1954

But this has another line coming out of it like this to separate it into angles 2 and 3.1961

All I have to do is add up angles 2 and 3, and that is going to be my angle 8.1970

This is going to be 78 degrees, and since the lines are parallel, the corresponding postulate says that they are congruent; that equals 78.1975

I will write that here: 78; and then, this 78 and angle 6 are going to form a linear pair.1993

Right here, 78 +...now, since this is angle 6 right here, you can look at angle 6 and this angle right here,2013

78 degrees, because that is angles 2 and 3 combined; they are going to be consecutive interior angles.2029

And they are supplementary; so you can just do angle 6 + 78 = 180, which is the same thing as looking at this.2038

These are supplementary, so angle 6 and angle 8 (78) are going to add up to 180.2049

It is the same thing: the measure of angle 6, plus 78, is going to equal 180.2055

The measure of angle 6: if I subtract 78, then you get 102, so this is 102.2070

Now, with angle 9, to find the measure of angle 9, that is actually going to involve using the triangles,2088

because the only relationship that this angle has with any of the other angles is that it forms within the triangle.2099

And see how angle 9 is not supplementary; it doesn't form a linear pair; there is no transversal involved with angle 9.2111

It is just these two angles, or those two right there.2120

Angle 9 is actually going to involve what is called the triangle sum theorem, where the three angles of a triangle are going to add up to 180.2125

So, we haven't gone over that yet; if you want, you can just say that the measure of angle 9, plus 61 (this angle),2134

plus the 78, is going to equal 180, and then find the measure of angle 9 that way.2146

You can also look at this big triangle and say that this angle, plus this angle, plus this angle, are going to add up to 180.2151

You can also look at it as this triangle right here, saying the measure of angle 9 plus 75, together, and then 3, are going to be 180.2163

And then, find the measure of angle 9 that way.2175

So, for now, we are just going to solve for these; and that is it for this problem.2179

The last example: Find the values of x, y, and z.2186

Here you have three lines: now, these three lines are going to be parallel.2192

I am going to make them parallel, so that I can solve for these values.2199

Now, the only angle that is given is right here, 118.2203

If you look at this, again, we have four lines involved; and to form special angle relationships, you only need three lines.2209

You need the transversal and the two lines that it intersects to form those pairs of angles.2219

Whichever lines you are using, always keep them in mind, and then look at what line you are not going to use,2228

and ignore that line, since we have four and we only need three.2238

Using this angle right here, 118, I can say that now this one right here and 11z + 8 are corresponding.2245

And then, this one right here and this one right here are alternate exterior angles, because it is involving these three lines, and not this one right here.2262

These would be alternate exterior angles.2273

Or, if I ignore this middle line, and I just say that this transversal with this line and this line2276

(again, ignoring the middle line--pretending it is not there), then 118, this angle right here, with x, would be alternate exterior angles.2290

Imagine if you have a line, a line...here is your transversal; the middle line is not there; this is x, and then this is 118.2303

You see that it is alternate exterior angles.2314

So then, I can say that x is equal to 118, because the lines are parallel.2319

And so then, I can apply the alternate exterior angles theorem, saying that that relationship, that pair, is congruent.2325

The next one: let's look at z; this one right here, 11z + 8, is going to equal 118.2337

Why?--because, if I look at this line, with this line and this transversal, they are going to be corresponding angles.2346

And then, since the lines are parallel, the corresponding angles postulate says that they are congruent.2358

11z + 8 = 118; so if you subtract the 8, 11z = 110; z = 10.2366

There is my x; there is my z; and then, I have to find y now.2387

For my y, I can say that this angle with 118--they are not congruent, remember, because they are going to form a linear pair.2392

They form a line, so they are going to be supplementary.2407

You can also note that this angle is 118, remember, because we said that they were corresponding--this one with this one.2412

So, since this is 118, this angle with this angle would be consecutive interior angles.2421

And if the lines are parallel, then the theorem says that they are supplementary, not congruent.2430

So, either way, 3y + 2 =...not 180; you have to say that this whole thing, plus the 118, is going to equal 180.2436

3y + 2 = 62, and then, if you subtract the 2, then 3y is going to equal 60; y is going to equal 20.2456

x is 118; y is 20; and z is 10; just remember to keep looking for those relationships between the pairs.2474

You can also definitely use the linear pair, if they are supplementary; you can definitely use that.2482

If they are vertical, definitely use that, because you know that vertical angles are congruent.2490

So, any of those things--you have a lot of different concepts that you learn that will help you solve these types of problems.2499

That is it for this lesson; thank you for watching Educator.com.2510

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