  Mary Pyo

Slope of 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 Jeremy Cohen on August 27, 2014where is the -6 coming from 3 answersLast reply by: Jing ChenSat Aug 26, 2017 11:25 AMPost by jeeyeon lim on December 31, 2012How do I know which coordinate is x1 , y1 and x2, y2k? Do I just choose on randomly?

### Slope of Lines

• Slope = the ratio between the vertical rise and horizontal run
• The slope m of a line containing two points with coordinates (x1, y1) and (x2, y2) is given by the formula • Slope postulates:
• Two non-vertical lines have the same slope if and only if they are parallel
• Two non-vertical lines are perpendicular if and only if the product of their slopes is -1

### Slope of Lines

A line passes through points A(2, 8) and B (6, − 9), find the slope of this line.
m = [( − 9 − 8)/(6 − 2)] = [( − 17)/4].
Find the slope of the line. • A(4, 3), B( − 4, 0)
m = [(0 − 3)/( − 4 − 4)] = [3/8].
Points A ( − 6, 5) and B (4, 3) are on line p. Decide the slope of this line is positive or negative.
• Graph the points on a coordinate plane
• The slope is negative.
Line p passes through points A(2, 3) and B(5, 9), line q passes through points C( − 1, 4) and D(1, 8). Decide whether line p a nd q are parallel.
• slope of line p: m = [(9 − 3)/(5 − 2)] = [6/3] = 2
• slope of line q: m = [(8 − 4)/(1 − ( − 1))] = [4/2] = 2.
Lines p and q are parallel.
Given points A(1, 5), B(4, − 4), C( − 2, 3) and D(3, 2). Decide whether is perpendicular to .
• Slope of line AB: m1 = [( − 4 − 5)/(4 − 1)] = [( − 9)/3] = − 3
• slope of line CD: m2 = [(2 − 3)/(3 − ( − 2))] = − [1/5].
• m1*m2 = [3/5] − 1
Line AB is not perpendicular to line CD.
Line p passes through points (2 − x, 9) and (4, 4), line q passes through points (3, 2) and (1, 6), line p and line q are parallel, find x.
• The slope of line q is :m = [(6 − 2)/(1 − 3)] = − 2
• so the slope of line p is also − 2.
• the slope of line p is: m = [(4 − 9)/(4 − (2 − x))] = − 2
• − 5 = − 2(4 − (2 − x))
• − 5 = − 2(2 + x)
• 2.5 = 2 + x
x = 0.5.
Find the slope of the line that passes through points (2, 9) and (6, − 3).
m = [( − 3 − 9)/(6 − 2)] = [( − 12)/4] = − 3.
Line p is perpendicular to line q, line p passes through points ( − 3, 4) and (4, − 10), find the slope of line q.
• the slope of line p is : m = [( − 10 − 4)/(4 − ( − 3))] = [( − 14)/7] = − 2
the slope of line q is [1/2].
A line passes through points (x + 3, 4) and (5, 7), and the slope is − 3. Determin the value of x.
• The slope of the line is: m = [(7 − 4)/(5 − (x + 3))] = − 3
• [3/(2 − x)] = − 3
• [1/(2 − x)] = − 1
• x − 2 = 1
x = 3.
Find the slope of a line passes through points ( − 5, 8) and (2, 8).
m = [(8 − 8)/(2 − ( − 5))] = 0

*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.

### Slope of 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
• Definition of Slope 0:06
• Slope Equation
• Slope of a Line 3:45
• Example: Find the Slope of a Line
• Slope of a Line 8:38
• More Example: Find the Slope of a Line
• Slope Postulates 12:32
• Proving Slope Postulates
• Parallel or Perpendicular Lines 17:23
• Example: Parallel or Perpendicular Lines
• Using Slope Formula 20:02
• Example: Using Slope Formula
• 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

### Transcription: Slope of Lines

Hello--welcome back to Educator.com.0000

The next lesson is on the slope of lines; this might be a little bit of a review for you from algebra.0002

But this whole lesson is going to be on slope.0010

Slope is the ratio between the vertical and the horizontal, or we can say "rise over run."0014

Rise, we know, is going up and down; and then, the run is going left and right.0027

So, when we talk about slope, we are talking about the vertical change and the horizontal change.0033

For slope, we use m; so the slope of a line containing two points with coordinates (x1,y1)0044

and (x2,y2) is given by this formula right here.0057

Now, (x1,y1), these numbers right here, and (x2,y2), those numbers, are very different from exponents.0061

They are not to the power of anything; it is just saying that it is the first x and the first y.0073

So, we know that all points are (x,y), and so, this right here is just saying that this is the first x and the first y.0083

And this is also x and y, but you are just saying that it is the second point; it is the second x and the second y.0094

Because you have two x's and two y's, you are just differentiating the points; this is the first (x,y) point, and this is the second (x,y) point.0103

It doesn't matter which one you label as first and which one you label as second.0112

You are just talking about two different points.0116

And when you have two points, then the slope is going to be (y2 - y1)/(x2 - x1).0118

You are going to subtract the y's, and that is going to be your vertical change, because y, you know, is going up and down.0130

And x2 - x1 is your horizontal change, the difference of the x's, which is going horizontally.0139

Now, it doesn't matter...like I said, if you are going to subtract this point, this second y, from this y,0150

then you have to make sure that you subtract your x's in the same order.0163

If you are going to subtract y2 - y1, then it has to be x2 - x1 for the denominator.0168

It can't be (y2 - y1)/(x1 - x2).0174

If you do y2 - y1 over here, you cannot do x1 - x2.0179

You can't switch; it has to be subtracted in the same order, or else you are going to get the wrong answer.0185

And this right here is just saying that x1 and x2, these numbers, can't equal each other,0190

because if they do, then this denominator is going to become 0.0197

If x1 is 5, and x2 is 5, then it is just going to be 5 - 5, and that is going to be 0.0201

And when we have a fraction, you can't have 0 in the denominator, or else it is going to be undefined.0212

So, that is what it is saying right here: they should not equal each other, or else you are going to have an undefined slope.0218

Let's find the slope of these lines: here we have (-4,-2) and (5,3).0229

Now, you can do this two ways: you can use the slope formula by doing (y2 - y1)/(x2 - x1);0239

if you have a coordinate plane, and it is marked out for you--you have grids that show each unit--0252

then you can count; you can just go from one point to the other point,0262

and you can just count your vertical change and count your horizontal change; you could do it that way.0269

But since these are not labeled--each unit is not labeled out--let's just use the slope formula.0275

Here, if I make this (x1,y1), (x2,y2), then my slope is going to be (-2 - 3)...0285

so then, this value is (y2 - y1, which is 3), over (-4 - 5).0301

Now, I could do (3 - -2); I can go that way if I want, but if I do that, if I choose to do this one first,0312

(3 - -2), then I have to do (5 - -4); you have to be in the same order.0321

If you do 3 minus -2, then you can't go with (-4 - 5); you can't go the other way then.0327

It doesn't matter which one you start with; but when you do your x, you have to do it in the same order.0338

This one right here is -5/-9, which is just 5/9.0345

Now, without solving slope, if you look at the line, you should be able to tell if the slope is going to be positive, negative, 0, or undefined.0358

For this one, since the slope measures how slanted a line is, how tilted a line is, if we look at this line,0374

imagine a stick man (I like to call him "stick man," because I can only draw stick figures) walking on this line.0388

Now, he can only walk from left to right, because let's say you read this--you have to read from left to right.0398

So then, it can only go from left to right; he is walking uphill, and that would be a positive slope.0407

This is a positive slope; if the stick man is walking uphill, it is a positive slope.0416

If the stick man is walking downhill, like the next one (again, he can only walk from left to right)--he is going to walk downhill, so this is a negative slope.0426

Without even solving, I know that my slope is going to be negative.0439

This is positive; it is positive 5/9.0443

Now, the slope for this one--I know, before I even solve it, that it is going to be negative.0446

So, after I do solve it, if I get a positive answer, then I know that I did something wrong, because it has to be negative.0451

For this one, the slope is 5 - -4; and make sure that you are going to find the difference of the y's for your numerator.0463

Don't do your x's first; the numerator is y's; the denominator is x's.0475

I went from this to this, so then I have to do -2 - 3.0482

So then, this is...minus a negative is the same thing as a plus, so 5 + 4 is 9, and then -2 - 3 is -5.0489

So, this is -9/5; and I have a negative slope, so that is my answer.0502

A couple more: here I have a horizontal line; my slope is (y2 - y1)...(-3 - -3), over (-6 - 4); this is 0,0519

because -3 + 3 is 0; this is -10; well, 0 over anything is always 0; so the slope here is 0.0541

Now, let's bring back the stick man: if stick man is walking on this, he is not walking uphill or downhill; he is just walking on a flat surface.0554

If he is walking on a flat surface, since slope measures how slanted a line is, it is not slanted at all--it is just flat; that is why the slope is 0.0563

Whenever it is flat, it is a horizontal line, and the slope will be 0--always.0573

It doesn't matter if it is up here or down here; as long as it is a horizontal line, your slope is going to be 0.0580

The next one: 4 - -4...be careful with the negatives: it is 4 minus -4; -2 - -2; change those to a plus--0592

minus a negative is also a plus--so 4 + 4 is 8, over -2 + 2...is 0.0615

Now, look at the difference between this one and this one.0624

In this one, the 0 is in the numerator; if it is in the numerator, it makes it just 0; 0 is a number, just like 5, 6, -3;0628

those are all numbers, and 0 is a number; so your answer for this problem, your slope, is 0; you have a slope; it is 0.0640

And in this problem, we cannot have 0 in the denominator--it is just not possible.0650

So, since you did come up with a 0 in the denominator, this is going to be undefined.0657

You can also say "no slope"; in this case, you do have a slope--the slope is 0; in this case, you do not have a slope.0669

There is no slope; it doesn't exist; it is undefined, because the denominator is 0.0680

So, my answer is just "undefined."0685

And then, to bring back the stick man: since it is a man, it can't do this--this is like walking up a wall.0689

Stick man can't walk up a wall; it is not possible--it can't do it.0702

He can't walk up a wall; he is not Spiderman; he can't walk up a wall.0706

So, in this case, this man can't do this; if he can't walk up this wall, it is undefined; it can't be done; there is no slope.0711

If he is walking on a horizontal--no slant at all--it is 0.0724

If he has to walk up a wall (which is impossible), then it is an undefined slope.0731

If the stick man is walking uphill, it is a positive slope; downhill is a negative slope; a horizontal line is 0; a vertical line, like a wall, is undefined.0737

You can't walk up a wall.0749

A couple of postulates: If we have two non-vertical lines that have the same slope, then those lines are parallel,0754

because again, slope measures how slanted a line is.0769

So, if I have two lines that are slanted exactly the same way, then they are going to be parallel.0775

Again, two lines that have the same slope are parallel.0788

And this part right here: "if and only if"--now, we went over conditionals, if/then statements;0794

to change this one right here (let's go over this...number 1)...two non-vertical lines have the same slope if and only if they are parallel.0807

This just means that this conditional and its converse are both true.0819

It is basically two statements, two conditionals in one.0833

I can say, "If two lines have the same slope, then they are parallel."0840

And this would be the converse: I can say, "If two lines are parallel, then they have the same slope."0865

The statement and its converse are both true: this is true, and this is true.0893

So, just instead of writing each of those conditionals separately, you can write them together by "if and only if."0897

It just means that this statement and its converse (converse means, remember, that you switch the hypothesis and the conclusion) are both true.0908

So then, you can just use "if and only if."0919

If two lines have the same slope, then the two lines are parallel.0925

Or you can say, "If two lines are parallel, then they have the same slope."0930

Either way, parallel means same slope; same slope means parallel.0934

Now, the next postulate is talking about perpendicular lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1.0939

Now, "the product of their slopes is -1"--that means that, if, first of all, I have a line like this,0957

and I have a line like this, let's say they are perpendicular; and the slope of this line, let's say, is 1/20970

(it has to be positive; it is going uphill); then the slope of this line is going to be the negative reciprocal,0979

meaning that you are going to make it negative; if it is negative already, then you are going to make it positive;0990

so, the slope of this line will be negative...and then the reciprocal of it will be 2/1.0997

So then, it is saying that the product of their slopes is going to be -1; so 1/2 times -2 is -1.1006

Just think of it as: If you have two perpendicular lines, then the slopes are going to be negative reciprocals of each other.1023

And if you multiply those two slopes, then you should get -1, always.1032

OK, parallel or perpendicular lines: you are given points A, B, C, and D; you want to determine if line AB is parallel or perpendicular to CD.1040

So, to determine if the two lines are parallel or perpendicular, then you have to compare their slopes.1058

So, for line AB, I need to use points A and B.1066

If I find the slope using these two points, the slope of AB is going to be -6 - 0, y2 - y1, over x2 - x1.1070

And that is -6/-3, so we have 2.1093

And then, the slope of CD is y2, -3, minus -4, over 4 - 2; this is 1/2, so this is positive 1 over 2.1100

Did I get that right?--yes.1133

In this case, it is going to be neither, because here we have AB; (-6 - 0)/(-2 - 1) becomes positive 2.1138

And then here, this is -3 - -4, and 4 - 2; for this, I get positive 1/2.1157

Now, it looks like they are going to be perpendicular, but remember: they have to be the negative reciprocal of each other.1168

If this is 2, this is 2/1, and the inverse, or the reciprocal, is 1/2; but they are not negative--it is not negated.1176

So, if I multiply 2/1 times 1/2, I am only going to get 1, not -1; so this is neither.1188

OK, find the value of x if the line that passes through this point and this point is perpendicular to the line that passes through (-1,6) and (-2,8).1204

We are given our two points that we have to find the slope of.1220

But from those two points, one of the values, the x-value, is missing.1227

That means that we need the slope.1235

They didn't just give us the slope in this problem; they didn't just hand it to us.1239

We have to actually solve for the slope, because we know that it is perpendicular to a line that passes through these two points.1243

So, basically, the points that I have to use are (x,4) and (-3,3).1255

I have to find x; and this is another line, and I am just going to use that line to find the slope,1266

because I have my slope that I need that has a relationship with the slope of this line.1278

So, to find the slope of this line right here that passes through these points,1285

I am going to do (6 - 8), (y2 - y1), over (-1 - -2); this is -2/1, so the slope is -2.1293

But since I know that my line is perpendicular to this line, my slope is going to be the negative reciprocal of this slope.1307

If this slope is -2, then my slope is going to be positive 1/2, positive one-half.1322

That is what I need to use: the slope is positive 1/2.1333

Now, using the slope formula, I know that this is (y2 - y1)/(x2 - x1).1338

Well, I can just fill everything in, except for this, and then use that as x1.1350

1/2 is my m; y2...if this is (x1,y1), (x2,y2),1356

y2 is 3, minus 4, over -3, minus x; since I don't know this value, which is what I am solving for, I can just leave it like that.1365

And then, from here, I have to solve this out.1384

I can solve this out a couple of ways: first of all, since this is a fraction equaling a fraction, I can use proportions.1388

I can make (-3 - x) times 1 equal to 2 times 3 minus 4; or let me just do this--let me just simplify this first.1399

1/2 = -1/(-3 - x); or I can just multiply...1411

I have a variable; the variable that I am solving for is x, and that is in the denominator.1423

If I want to solve for the variable, it cannot be in the denominator, so I have to move it out of the denominator.1427

I can do that by multiplying both sides or the whole thing by -3 - x;1433

or again, since this is like a fraction equaling a fraction, like a proportion, I can just do that.1439

So, just make -3 - x equal to...and then, I am just multiplying it this way...equaling this; it is -3 - x = -2;1444

if I add 3, then I get -x = 1; x = -1.1459

So again, you are going to find your slope.1469

They might not just hand you the slope; they might not tell you what the slope is directly.1476

So then, you have to find it this way; they will give you another line that has a relationship with your line, your slope.1483

So, you have to find the slope of that other line, and then use that slope to find your slope.1496

And then, plug it all into the slope formula; and then from there, you just solve.1503

Let's do a few examples: Find the slope of the line passing through the points.1510

Again, here is the slope formula; this equals...it doesn't matter which one I use first, so I will just use (-2 - 5) first.1517

That means that I have to use this one first; so it is (3 - -4).1536

This becomes -7/7, which is equal to -1; and then, for this one, the slope is (0 - 6)/(-7 - -7).1542

This is going to be -6/0; 0 is in the denominator, which means that I have an undefined slope.1564

And that just means that the line that is passing through these points is going to be a vertical line; vertical lines have undefined slopes.1580

Find the slope of each line: now, they don't give you the points--they just show me the lines.1593

And I have to see what points the lines are going through to find the slope.1602

Let's see, let's look for the slope of n first; here is n.1612

Now, remember: for slope, I can do this two ways: I can find two points on this line, like this and like this--1618

those are two points on the line (or here is another point; it doesn't matter--any two points on the line);1630

you can find the coordinates of the points and use the slope formula.1636

For two points, find the coordinates and use the slope formula.1642

Or an easier way, in this problem: Since we have all of these grids marked out for us, I can just1646

(because slope is rise over run, the vertical change over the horizontal change; rise is how many it is going up or down,1656

and then run is how many is going left or right)--whenever I go up (here, this is the positive y-axis), any time I am counting upwards,1668

it is a positive number; if I am counting downwards, then it is a negative, because I am going smaller.1683

If you are going up, it is a positive number; if you are going down, if you are counting down, then it is a negative number.1690

The same thing for x: if you are moving to the right, it is a positive number; if you are moving to the left, it is a negative number,1696

because it is getting larger as you go to the right; and as you go to the left, you are moving towards the negative numbers.1703

You are getting smaller, so it is negative if you are going to the left.1709

To find the slope of n, I am just going to do rise over run; I am going to just count my vertical and horizontal change.1715

You go from any point to any other point on the line.1724

I can start from here; I am going to go one up, because it is on this right here.1729

My vertical change: I only went up one; remember, up is positive, so to find the slope of n, it is positive 11740

(that is my rise), and then I am going 1, 2 to the right--that is positive 2, so the slope is 1/2.1754

Now, remember: I can also go from any point to any other point on the line.1767

So, if I start from this point, let's say I am going to go from this point, and then (I didn't see this point, so) to this point;1773

then I can go down 2 (remember: down 2 is negative 2), over...then I am going to go 1, 2, 3, 4.1781

And that is to the left, so that is negative 4; -2/-4 is the same thing as 1/2.1797

It doesn't matter how you go from whichever point to any other point on the line, as long as both points are on the line,1808

and as long as you make it a positive number going up, a negative number going down, positive to go right, and negative to go left.1817

You are going to get the same answer; you are going to get the same slope.1826

The slope of n is 1/2; then the slope of p (let me use red for this one): let's see, I have a point here, and I have another point here.1830

So again, I can go from this point to that point, or I can go from that point to that point; it doesn't matter.1853

Let's start right here: I am going to go 1, 2, 3; I went up 3, so that is a positive 3.1858

And then, from here, I go to the left 1, which is a negative 1.1869

3/-1 is the same thing as -3, so my slope of p is -3.1876

Or I could go from this point to this point; that would be to the right one (that is positive 1), over down 3 (1, 2, 3);1883

oh, I'm sorry; I did horizontal over vertical, which is wrong; so I have to go this way--vertical first.1893

1, 2, 3: that is a -3, and to the right 1--that is positive 1; so this is also -3, the same thing.1902

The next line is line q (I will use red for this one, too): this is a vertical line.1916

Automatically, I know that that has an undefined slope. I can also just...1932

Now, I know that this line is not really completely lined up with the grid, but sometimes when you transfer1939

this into this program, or move things into this program, it might shift a little bit.1947

But think of this line as being on 2 right here, as x being 2.1951

Let's say I have this point right here, and then any other point--that point right there.1960

All that I am doing is: my vertical change is going down to -2; my horizontal is nothing: 0.1964

I am not moving to the right or left at all; that is 0.1975

So, we have a 0 in the denominator; this is an undefined slope.1979

And again, I knew that because this is a vertical line; the stick man can't walk up that line; so it is an undefined slope.1989

And the last one, for line l: any two points...1999

Again, this line is shifted a little bit, but I can just do that if I want.2007

Vertical change first: the vertical change is 0, because I am not moving vertically; to get from this point to this point, I don't go up or down at all.2014

So, it is 0 over...and then, I can move 1, 2, 3 to the right; so no matter what the bottom number is, my slope will be 0.2021

The slope of the l is 0; again, it is a horizontal line, so it is not slanted; it is not going uphill or downhill--nothing.2035

It is just horizontal; then the slope is 0.2044

The next example: Graph the line that satisfies each description; slope is 2/3 and passes through (-1,0).2052

You just have to graph this first one; let's say I am going to graph it right up here.2066

Now, just a sketch will do; let's say...here is my x; here is my y; (-1,0) is right there.2075

My slope is 2/3; so again, this is rise over run.2103

Now, I can use the same concept, the positive going up and negative going down, positive to the right and negative to the left.2110

The top number, the rise, to go up and down: I have a positive 2--that means I am going to go up 2, because it is positive.2120

From here, I am going to go 1, 2; and then, I have 3 that I am going to move to the left or to the right;2130

but since it is a positive 3, I am going to move to the right: 1, 2, 3.2139

Now, from this point, I can go down if I want to, because 2/3, that slope, is the same thing as -2/-3.2150

So, if I go -2, I am going to go down 2: 1, 2; and then, -3 is to the left, so 1, 2, 3; there is my line, right there.2163

This is the second one: it passes through point (3,1) and is parallel to AB with A at this point and B at that point.2182

Again, they don't give us our slope; they just give us the point that we need to use.2194

Our line is going through this point, and we don't have the slope of our line;2201

instead, they give us the slope of another line, line AB; and they say that it is parallel to it.2205

So, as long as we find the slope of AB, since it is parallel, we know that our slope will just be the same as this slope.2211

The slope of AB is (4 - 3)/(-1 - -2), which is 1 over...this is 1...so 1.2222

Now, since our slope is parallel, again, our slope is 1.2238

And then, this is our point; so we have point (3,1), and the slope is 1.2247

To graph (x,y), (3,1), it is 1, 2, 3; and 1; there is our point that our line is passing through.2258

And then, our slope is 1; 1 is the same thing as 1/1, so positive 1 is up 1, to the right 1; also, positive 1 is up 1.2291

And then again, you can just do -1/-1; that is the same thing as 1.2308

So, from here, I can go down 1, left 1; down 1 is -1; left 1 is -1.2313

And that is going to be a line like that.2322

The last example: Determine the value of x so that a line through the points has the given slope.2331

Again, they give us a slope, and then we have to find the missing value, which is x.2338

Since we know that the slope formula is (y2 - y1)/(x2 - x1),2345

if I make this (this is (x,y), and this is also (x,y)) my first point, and this is my second point--2354

so that is (x2,y2)--then my slope is y2, which is -5, minus 1, over x2, -3, minus x.2366

Now, again, you can turn this into a proportion; or I can just multiply this out to both sides.2385

I can multiply this to this right here to cancel it out, and then multiply to the other side and distribute that; I could solve it that way.2397

Or I could just make this a proportion: 2/1 = -6/(-3 - x); so to continue right here, it is going to be 2(-3 - x) = -6.2406

This is -6 - 2x, and you just distribute that; that equals -6; -2x = 0; x = 0.2430

So, we have 0 as our x for this problem.2444

OK, the next one: again, plugging everything into the slope formula, we have 4/3 = (-2 - -6)/(x - 7).2450

So, 4/3 =...this is -2 + 6, so this is 4, over x - 7.2472

OK, well, in this problem, again, you can multiply x - 7 to both sides to get it out of the denominator,2482

because since you are solving for x, it cannot be in the denominator.2492

Or you can cross-multiply using proportions, because it is a fraction equaling a fraction.2497

Or, since this 4 numerator equals this 4 numerator, then the denominator has to equal the denominator, so you can make 3 equal to x - 7.2505

So, let's just solve it out, multiplying: I can multiply this side by (x - 7) like that,2518

and multiply this side by (x - 7); then, it is going to be 4/3x minus 28/3 equaling 4.2529

Now, this is actually probably the harder way to do it; but I just wanted to show you how to multiply it to both sides.2553

This is a binomial that, again, you are just multiplying to both sides.2561

But the easiest way would be to make (x - 7) equal to 3: because 4 = 4 (the top), then 3 = x - 7 there.2567

So, let's just continue out this way: if I add 28/3 to both sides, this is going to be the same thing as 12/3 + 28/3,2577

and that is just because I need a common denominator; that is equal to 4/3x.2593

That equals...this is 40/3...if I multiply the 3's to both sides, this will be 4x = 40; x = 10.2605

So here, x = 10, this value right here.2626

And if we solved it the other way, 3 = x - 7, then you would just add 7, so x would be 10.2632

That is it for this lesson; thank you for watching Educator.com.2642

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