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Mary Pyo

Mary Pyo

Parallelograms

Slide Duration:

Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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
VII. 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
VIII. 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
IX. 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
X. 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
XI. 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
XII. 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 (2)

1 answer

Last reply by: patrick guerin
Thu Sep 25, 2014 6:19 AM

Post by Aurelio Da Costa on September 9, 2013

I cant find the practice questions. Can anyone tell me how? Thanks heaps - Aurelio

Parallelograms

  • Quadrilaterals are four-sided polygons
  • Parallelogram: A quadrilateral with two pairs of parallel sides
  • Properties of Parallelograms:
    • Opposite sides of a parallelogram are congruent
    • Opposite angles of a parallelogram are congruent
    • Consecutive angles in a parallelogram are supplementary
    • The diagonals of a parallelogram bisect each other

Parallelograms

Find which ones are quadrilaterals.
A, B and C are quadrilaterals.
Find which names are right for the parallelograms.

ABCD, BCDA, ACBD and ADBC
ABCD and BCDA.
Determine whether the following statement is true or false.
Consecutive angles in a parallelogram are congruent.
False.
Determine whether the following statement is true or false.
Oppostite angles in a parallelogram are congruent.
True.
Write all the congruent segments in the parallelogram.
AB ≅ CD , AD ≅ BC , AE ≅ CE , BE ≅ DE .
Write all the congruent angles in the parallelogram.
∠A ≅ C, ∠B ≅ ∠D
Write all the parallel segments in the parallelograms.
AB ||CD , AD ||BC .
ABCD, m∠BDC = 2x + 5, m∠ABD = 8, find x.
  • m∠BDC = m∠ABD
  • 2x + 5 = 8
x = 1.5
ABCD, m∠BAE = 45, m∠AEB = 55, find m∠BDC.
  • m∠ABD = 180 − m∠BAE − m∠AEB
  • m∠ABD = 180 − 45 − 55 = 80
  • m∠BDC = m∠ABD
m∠BDC = 80
Find the distance of each side to verify the parallelogram.
  • A( − 2, 1), B( − 3, − 3), C(3, − 2), D(4, 2).
  • AB = √{( − 3 − ( − 2))2 + ( − 3 − 1)2} = √{1 + 16} = √{17}
  • BC = √{(3 − ( − 3))2 + ( − 2 − ( − 3))2} = √{36 + 1} = √{37}
  • DC = √{(3 − 4)2 + ( − 2 − 2)2} = √{1 + 16} = √{17}
  • AD = √{(4 − ( − 2))2 + (2 − 1)2} = √{36 + 1} = √{37}
  • AB = CD, AD = BC
  • AB ≅ CD ,AD ≅ BC
AB = √{( − 3 − ( − 2))2 + ( − 3 − 1)2} = √{1 + 16} = √{17}
BC = √{(3 − ( − 3))2 + ( − 2 − ( − 3))2} = √{36 + 1} = √{37}
DC = √{(3 − 4)2 + ( − 2 − 2)2} = √{1 + 16} = √{17}
AD = √{(4 − ( − 2))2 + (2 − 1)2} = √{36 + 1} = √{37}
AB = CD, AD = BC AB ≅ CD ,AD ≅ BC
Quadrilatral ABCD is parallelogram.

*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

Parallelograms

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
  • Quadrilaterals 0:06
    • Four-sided Polygons
    • Non Examples of Quadrilaterals
  • Parallelograms 1:35
    • Parallelograms
  • Properties of Parallelograms 4:28
    • Opposite Sides of a Parallelogram are Congruent
    • Opposite Angles of a Parallelogram are Congruent
  • Angles and Diagonals 6:24
    • Consecutive Angles in a Parallelogram are Supplementary
    • The Diagonals of a Parallelogram Bisect Each Other
  • 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

Transcription: Parallelograms

Welcome back to Educator.com.0000

The next lesson is on parallelograms; and before we talk about parallelograms, let's talk about quadrilaterals.0002

A quadrilateral is a four-sided polygon; now, we know that a polygon is a shape that is closed off.0009

It is a shape with three or more sides that is completely closed, and each side has to be a straight line.0020

When we have a polygon that is four-sided, that is a quadrilateral.0028

This is considered a quadrilateral; this is a quadrilateral; any type of shape where I have four sides is a quadrilateral.0033

Now, some non-examples, some examples that are not quadrilaterals, would be maybe something like that,0047

where it doesn't close; that is not a quadrilateral; or if I have something that overlaps like that,0056

if two sides overlap, this is not a quadrilateral; and if I have something that maybe is curved like that,0069

a side that is curved, that would not be a quadrilateral, as well.0079

Again, a quadrilateral is a four-sided polygon, something that is four-sided.0083

Each side has to be a straight line, and it is closed, and there are no overlapping sides.0089

A parallelogram is a special type of quadrilateral, meaning that it is a four-sided polygon, but it has two pairs of parallel sides.0098

That is why it is called a parallelogram, because there are two pairs of parallel sides.0108

To show that my sides are parallel, I can do this; that means that those sides are parallel.0117

To show that these sides are parallel, I am going to draw two of them.0126

These two are parallel, and then these two are parallel; so we have two pairs of parallel sides.0136

Now, if I label this as A, B, C, and D, then I can name this parallelogram by its symbol; that would be the symbol for a parallelogram.0142

Parallelogram ABCD: that is how I can name it, just like when I have a triangle; I can say triangle ABC.0159

In the same way, I can say that this is the parallelogram ABCD.0169

I can also say parallelogram BCDA; I could say parallelogram CDAB, and so on, as long as the four vertices that I name are in order.0173

So, I can say ADCB; I can say DABC; but I can't say parallelogram ACBD, because it skips.0192

You are skipping a vertex; when you label your parallelogram, make sure that they are in order--you are calling it in order.0210

It could be this way; it could be to the right, clockwise, or counterclockwise; it does not matter, as long as they are back-to-back.0222

This is something that you cannot say.0230

Now, when we talk about opposite pairs, we have two pairs of opposite sides and two pairs of opposite angles.0233

I know that side AB and side DC are opposite sides, and angle A with angle C are opposite angles.0243

And angle B with angle D are opposite angles.0258

Some properties of parallelograms: now, these properties are in the form of theorems,0270

just so you know that these are considered theorems.0278

The first property is that opposite sides of a parallelogram are congruent.0282

As long as we have a parallelogram, and you know that it is a parallelogram, they either tell you that it is a parallelogram,0287

or if they show you these symbols; the definition of a parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.0294

That is the definition of a parallelogram; that is not a property--that is what it means to be a parallelogram, the definition.0307

The properties are these, saying that, if this is a parallelogram, then opposite sides are congruent.0315

That means that this side with this opposite side are congruent, and this side with this side are congruent.0324

So then, if this is 5, then this will also be 5; if this is 8, then this is also 8.0333

Opposite sides are congruent; there will be two pairs--there is one pair, and this is the other pair.0341

Opposite angles of a parallelogram are congruent; that is the next property.0351

That means that this angle and this angle are congruent, and that angle with this angle are congruent.0357

So, as long as we have a parallelogram like that, then I know that opposite angles are congruent.0365

So far, opposite sides are congruent, and opposite angles are congruent.0378

Then, the next one: Consecutive angles in a parallelogram are supplementary.0386

So then, I am just going to show that...I don't have to do this, because I am already telling you that this is a parallelogram;0398

but just so you get used to seeing that opposite sides are parallel...0404

Consecutive angles of a parallelogram: that means this angle, so if I have angle 1 and 2, they are going to be supplementary.0409

And one way I can show you why this is true, or how this is true: if I extend this side out, and extend this side out,0420

then extend this side, then...now, I know that these two sides are parallel; this is like a transversal.0430

If you remember that one section, that chapter that we went over, about parallel lines being cut by a transversal,0442

then remember that we had special angle relationships; so remember how this angle right here,0450

and (I am actually going to call that angle 2, just so that it is the same as this diagram right here)0456

this is angle 1, and this is angle 2; now, we know that the name0460

for this angle relationship is consecutive interior angles, or same-side interior angles.0466

And we know that they are supplementary; as long as the lines are parallel, then these angles are supplementary.0474

So, it is the same thing here: these are consecutive interior angles.0479

Then these are supplementary; they are all the consecutive angles in a parallelogram.0486

So, if I know that these two lines are parallel, this is a transversal;0493

then, my interior angles are supplementary, in the same way that consecutive interior angles are supplementary.0497

Then, these two are supplementary; this angle with this angle is supplementary;0507

this angle with this angle is supplementary; and this angle with this angle is supplementary.0513

Opposite angles are congruent, but consecutive angles are supplementary.0517

Diagonals of a parallelogram bisect each other.0524

If I draw diagonals, those are my diagonals; now, they are not congruent--diagonals are not congruent.0528

That is a common mistake made by students--saying that these diagonals are congruent.0541

Diagonals of a parallelogram are not congruent; obviously, it looks like this...from here...0549

if you were to walk from this point to this point, that looks further than this point to this point.0554

So, you can't say that this diagonal is the same as this diagonal.0563

Diagonals are not congruent; they actually just bisect each other--that means that this diagonal cuts this diagonal in half;0572

this got cut in half; and then, this diagonal cuts this diagonal in half, so that means these are congruent.0582

The two diagonals are not congruent to each other; but the two halves of each diagonal are congruent.0591

So then, only this part and this part of the same diagonal are congruent; this part and this part of the same diagonal are congruent.0601

Another mistake is saying that this part, this half, of this diagonal is congruent to this half of the other diagonal.0608

No; only the diagonal is cut in half--each diagonal is just cut in half; that is all you have to think of.0615

They just bisect each other, and that is it.0621

We went over four properties so far, before we start our next example.0625

The first property of parallelograms...0631

Well, first of all, the definition of parallelogram says that we have a quadrilateral (meaning that it is a four-sided polygon)0633

with two pairs of opposite sides parallel; that is a parallelogram.0642

The properties: there are four of them--the first one is that opposite sides are congruent;0649

opposite angles are congruent; consecutive angles of a parallelogram are supplementary;0657

and the last one is that diagonals bisect each other--those are four properties.0664

So, using those properties, let's continue with our examples.0671

The first one: complete each statement about the parallelogram.0677

I know that I have a parallelogram; BC (there is that side) is congruent to what?0681

You are going to complete the statement: BC is congruent to what?--the side opposite, so AD.0688

Now, I know that this sounds really simple; it looks like the only other side that could be congruent to BC is AD.0697

But sometimes parallelograms are not so stretched out like that; sometimes this side and this side could look a lot closer in length.0704

Just don't assume; if it is just a parallelogram, even if side BC and this side right here, CD, look like they are congruent, you can't assume that.0720

So, all we can assume from parallelograms when it comes to their sides is that opposite sides are congruent.0733

AE is congruent to BE? No, AE is congruent to DE? No, AE is congruent to CE.0742

This diagonal is just cut in half; so AE is congruent to CE, the other half of the same diagonal.0753

AD is parallel to what? AD is parallel to BC.0763

And then, angle BCD is congruent to the angle opposite.0770

Now, we are just talking about the whole angle, not just these parts.0777

So then, the whole angle, BCD, is congruent to angle D (since B and D are corresponding)...DAB.0782

That is Example 1; the next example is to find the values of x, y, and z of the parallelogram.0798

Here is x; there is y; and there is z.0807

Since I know that opposite sides are parallel and congruent (these sides are congruent;0812

these sides are congruent), I can say that z...let's just say x =, y =, and z =; and that way,0822

I have all of my answers right here...so I know that z = 11.0835

Now, for x, if you look here, remember: we know that opposite sides are parallel.0840

So, if I just extend these out, see how these are my parallel lines, and this is my transversal.0848

If I were to draw that out again, there is BC; this is AD; and then, here is my transversal, CA; this is C, and this is A.0860

Now, if this is parallel, and this is 40, and this is x, what can I say about x?0874

If lines are parallel, then alternate interior angles are congruent.0884

So, if this is 40, this and this are alternate interior angles, so x has to be 40 degrees.0892

And then, for y...well, I know that y and this one are congruent, and angle B, but they don't give me angle B, so I don't know--0903

I can't say that y is the same as this angle, because I don't have that angle.0915

I have to find y in another way, so let's see...I know that x is 40; this is 40; what is this angle all together?0920

This is 65 + 40; this is 105 degrees; I know that consecutive angles are supplementary, so if this is 105, then this has to be the supplement to 105.0934

So, the measure of angle y equals 180 - 105; or you can say 105 + the measure of angle y equals 180; you can do whichever.0954

Here, this is 75; so the measure of angle y is 75 degrees.0971

Again, all I did here to find y is find the whole angle right here, because that is what makes up this angle of this parallelogram;0980

and then this angle with this angle is supplementary.0989

The next one: Find the distance of each side to verify the parallelogram.0997

I need to find the distance of AB and the distance of BC, this distance and this distance,1003

and then compare them, because I know that opposite sides are congruent.1008

So then, as long as the distance of A to D, this right here, and this, are congruent,1013

and then this and this are congruent, then that is verifying that it is a parallelogram.1020

Now, they are just telling you that it is a parallelogram; you are just verifying.1024

You are just using the distance formula to just show...you are just writing down...1028

now, you know that it is going to be the same, because it is a parallelogram.1036

So, they are just asking you to verify that.1039

The reason why I am saying this is because, in the next lesson, we are actually going to prove parallelograms.1043

We are going to use the distance formula and other theorems and such to actually prove that a quadrilateral is a parallelogram.1047

But this is not proving; you are not proving that it is a parallelogram--you are just verifying,1057

meaning that it is a parallelogram, but you are just showing it; just actually write the numbers to show that it is a parallelogram.1061

The distance formula, we know, is the square root of (x1 - x2)...and this doesn't mean x squared,1068

or anything like that; it is just the first x, minus the second x, squared; plus y1, the first y, minus the second y, squared.1082

So then, let's find the distance of AB; AB is going to be the square root of...1094

Oh, before we begin, let's actually write out the points of what these are.1104

B is 1, 2, 3; that is (-3,2); C is 1, comma, 1, 2, 3; D is at 1, 2, 3, comma, -1, -2; and A is at -1, -1, 2, 3.1109

That way, I have all of my points.1140

So, if I am trying to find the distance of A to B, then I am going to have to use these points.1142

x1 (it doesn't matter which one you use) is -1, minus -3, squared, plus y1, the first y, -3, minus 2, squared.1147

-1 minus -3; now, if you remember from algebra, minus a negative makes a plus.1169

If you have two negative signs next to each other like that, that just makes both of them into a plus.1179

-1 + 3 is 2; that is 2 squared; plus...-3 - 2; that is -5, squared; this is 4 plus 25; and this is the square root of 29.1185

And this we can just leave like that, unless your teacher wants you to round it to the nearest hundredth, or some decimal number.1214

Then you would have to change this to a decimal.1225

Otherwise, you can just leave it like that; I am pretty sure that you can probably just leave it like that.1230

Then, I am going to find the side opposite to show that they are congruent: CD.1235

CD is, let's see...we can use whichever number, so let's do 3, the first x, minus...we will call that the second x, squared,1243

plus the first y, minus the second y, squared; so here, this is 3 - 1, is 2; that is 2 squared, plus -2 - 3 is -5, squared;1255

that is 4 + 25, which is the square root of 29.1275

Since these are congruent, we know that these sides are congruent.1287

Let's do BC: BC = √[(-3 [the first x] - 1)2 + (2 - 3)2].1292

The square root of...-3 - 1 is -4, squared, plus...this is -1, squared; equals...16 + 1, which equals the square root of 17.1309

Then, AD, the side opposite that, is going to be -1 - 3, squared, plus -3 - -2, squared;1330

the square root of...-1 - 3 is -4, squared; plus...minus a negative...they both make a plus, so this is -1, squared;1350

4 times 4 is 16, plus 1, which is the square root of 17.1366

Now, see how this is equal to this, and this is equal to this.1374

So then, I know that AB is congruent to CD, and BC is congruent to AD.1384

OK, for the next example, we are going to use the same diagram, the same parallelogram.1396

And we are just going to verify that it is a parallelogram by using slope.1404

Now, how would we use slope to verify the parallelogram?1409

If I find the slope of BC and the slope of AD, they are parallel; we know that they are parallel, because it is a parallelogram.1413

So, if we find the slope of this, and we find the slope of this, they should be the same,1424

because we know that two lines, when they are parallel, have the same slope.1429

The same thing works here: we are going to find the slope of this, and find the slope of this, to verify that it is a parallelogram.1436

Now, if you want, you can find the points of each one of these, like we did in the previous example.1448

Find the coordinates of each; and you can use the slope formula.1455

The slope formula is y2 - y1, meaning the difference of the y's, over the difference of the x's.1460

You could do it that way, using the point of this and using the point of that, and then solve it that way.1471

Or, since we actually have a coordinate plane, and we have all of the points there, we can just do rise over run,1477

meaning we can just count how many we are rising and how many we are running.1488

So then, the slope of BC: to go from this point to this point, you are going to rise one,1492

meaning you are going to go up one (and rise just means how many you are going to go up and down).1499

Run is how many you are going to go left and right.1504

We are going up 1; since we are going up 1, that is a positive 1, over...1, 2, 3, 4; and that is going to the right, so we know that it is a positive number.1507

Whenever you go up, it is a positive, because remember: the y-axis...see how when you go up, you are going positive.1519

But when you go down, you are going negative; see how these numbers are all negative.1527

For the x-axis, the same thing happens: when you go right, you are going towards positive numbers;1531

when you go to the left, you are going towards the negative numbers.1536

So, if you go up, you are going positive; if you go right, you go positive; going down is negative; left is negative.1539

So, we went up; that is a positive 1; and 1, 2, 3, 4--you went to the right 4; that is a positive 4.1545

That means that the slope of BC is 1/4.1552

For CD, we are going 2 to the right (that is 2), over...how many are we going down?...1, 2, 3, 4, 5.1558

That is a negative 5, because we went down 5; the slope of CD is -2/5.1572

Now, it doesn't matter if you go from D to C--it will be the same thing.1584

If we count from D up to C, we are going to go up 5 (that is a positive 5)...1589

Oh, I'm sorry; I did it the wrong way.1597

Let's see, let me just try to draw this over again; OK.1607

Let's try that again: I am going from C to D.1611

Now, I did my rise and run; I did it run over rise--now, be careful not to make that same mistake, OK? I'm sorry about that.1616

Now, we are going run...if we go and run 2, I have to write that as my denominator, 2,1622

because that is the run; that is the bottom part; that is the denominator.1630

And then, I am going down 5; so that is a rise.1633

It is actually more helpful if you just do the rise, the up and down, first.1640

So then, I would have to go down first, and then to the right; that would be a lot easier, and I wouldn't have made that mistake.1644

So then, it is better to just go down...rise first (meaning up and down first), and then you go left and right.1650

Here, you can go down 1, 2, 3, 4, 5; that is -5, over 2; that is a positive 2.1657

The same thing happens if I go from D to C; I am going to go up 5 (that is a positive 5), over left 2 (that is a negative 2).1665

It is the same exact thing.1675

Now, the slope of AD will be...let's go from A to D; I am going to go up 1; that is a positive 1, over 1, 2, 3, 4; that is a positive 1/4.1679

See how they are the same.1693

Then, the slope of AB is going to be...I go from A to B.1696

So then, that would be from A..."rise"--going up 1, 2, 3, 4, 5, positive 5, over left 2; that is negative 2; and that is the same as that.1703

So then again, I am going to show that this is parallel to this; they are parallel.1716

And this is the same as this, which means that these are parallel.1723

We are just verifying that they are parallel, and that this is, in fact, a parallelogram.1727

In the next lesson, we are actually going to do pretty much the same thing,1734

except we are going to prove that they are parallelograms.1737

Using the properties, and using the theorems, we are going to actually prove that a quadrilateral is a parallelogram.1742

That is it for this lesson; we will see you soon--thank you for watching Educator.com.1748

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