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Table of Contents

I. Tools of Geometry

Coordinate Plane

16m 41s

- Intro0:00
- The Coordinate System0:12
- Coordinate Plane: X-axis and Y-axis0:15
- Quadrants1:02
- Origin2:00
- Ordered Pair2:17
- Coordinate Plane2:59
- Example: Writing Coordinates3:01
- Coordinate Plane, cont.4:15
- Example: Graphing & Coordinate Plane4:17
- Collinear5:58
- Extra Example 1: Writing Coordinates & Quadrants7:34
- Extra Example 2: Quadrants8:52
- Extra Example 3: Graphing & Coordinate Plane10:58
- Extra Example 4: Collinear12:50

Points, Lines and Planes

17m 17s

- Intro0:00
- Points0:07
- Definition and Example of Points0:09
- Lines0:50
- Definition and Example of Lines0:51
- Planes2:59
- Definition and Example of Planes3:00
- Drawing and Labeling4:40
- Example 1: Drawing and Labeling4:41
- Example 2: Drawing and Labeling5:54
- Example 3: Drawing and Labeling6:41
- Example 4: Drawing and Labeling8:23
- Extra Example 1: Points, Lines and Planes10:19
- Extra Example 2: Naming Figures11:16
- Extra Example 3: Points, Lines and Planes12:35
- Extra Example 4: Draw and Label14:44

Measuring Segments

31m 31s

- Intro0:00
- Segments0:06
- Examples of Segments0:08
- Ruler Postulate1:30
- Ruler Postulate1:31
- Segment Addition Postulate5:02
- Example and Definition of Segment Addition Postulate5:03
- Segment Addition Postulate8:01
- Example 1: Segment Addition Postulate8:04
- Example 2: Segment Addition Postulate11:15
- Pythagorean Theorem12:36
- Definition of Pythagorean Theorem12:37
- Pythagorean Theorem, cont.15:49
- Example: Pythagorean Theorem15:50
- Distance Formula16:48
- Example and Definition of Distance Formula16:49
- Extra Example 1: Find Each Measure20:32
- Extra Example 2: Find the Missing Measure22:11
- Extra Example 3: Find the Distance Between the Two Points25:36
- Extra Example 4: Pythagorean Theorem29:33

Midpoints and Segment Congruence

42m 26s

- Intro0:00
- Definition of Midpoint0:07
- Midpoint0:10
- Midpoint Formulas1:30
- Midpoint Formula: On a Number Line1:45
- Midpoint Formula: In a Coordinate Plane2:50
- Midpoint4:40
- Example: Midpoint on a Number Line4:43
- Midpoint6:05
- Example: Midpoint in a Coordinate Plane6:06
- Midpoint8:28
- Example 18:30
- Example 213:01
- Segment Bisector15:14
- Definition and Example of Segment Bisector15:15
- Proofs17:27
- Theorem17:53
- Proof18:21
- Midpoint Theorem19:37
- Example: Proof & Midpoint Theorem19:38
- Extra Example 1: Midpoint on a Number Line23:44
- Extra Example 2: Drawing Diagrams26:25
- Extra Example 3: Midpoint29:14
- Extra Example 4: Segment Bisector33:21

Angles

42m 34s

- Intro0:00
- Angles0:05
- Angle0:07
- Ray0:23
- Opposite Rays2:09
- Angles3:22
- Example: Naming Angle3:23
- Angles6:39
- Interior, Exterior, Angle6:40
- Measure and Degrees7:38
- Protractor Postulate8:37
- Example: Protractor Postulate8:38
- Angle Addition Postulate11:41
- Example: Angle addition Postulate11:42
- Classifying Angles14:10
- Acute Angle14:16
- Right Angles14:30
- Obtuse Angle14:41
- Angle Bisector15:02
- Example: Angle Bisector15:04
- Angle Relationships16:43
- Adjacent Angles16:47
- Vertical Angles17:49
- Linear Pair19:40
- Angle Relationships20:31
- Right Angles20:32
- Supplementary Angles21:15
- Complementary Angles21:33
- Extra Example 1: Angles24:08
- Extra Example 2: Angles29:06
- Extra Example 3: Angles32:05
- Extra Example 4 Angles35:44

II. Reasoning & Proof

Inductive Reasoning

19m

- Intro0:00
- Inductive Reasoning0:05
- Conjecture0:06
- Inductive Reasoning0:15
- Examples0:55
- Example: Sequence0:56
- More Example: Sequence2:00
- Using Inductive Reasoning2:50
- Example: Conjecture2:51
- More Example: Conjecture3:48
- Counterexamples4:56
- Counterexample4:58
- Extra Example 1: Conjecture6:59
- Extra Example 2: Sequence and Pattern10:20
- Extra Example 3: Inductive Reasoning12:46
- Extra Example 4: Conjecture and Counterexample15:17

Conditional Statements

42m 47s

- Intro0:00
- If Then Statements0:05
- If Then Statements0:06
- Other Forms2:29
- Example: Without Then2:40
- Example: Using When3:03
- Example: Hypothesis3:24
- Identify the Hypothesis and Conclusion3:52
- Example 1: Hypothesis and Conclusion3:58
- Example 2: Hypothesis and Conclusion4:31
- Example 3: Hypothesis and Conclusion5:38
- Write in If Then Form6:16
- Example 1: Write in If Then Form6:23
- Example 2: Write in If Then Form6:57
- Example 3: Write in If Then Form7:39
- Other Statements8:40
- Other Statements8:41
- Converse Statements9:18
- Converse Statements9:20
- Converses and Counterexamples11:04
- Converses and Counterexamples11:05
- Example 1: Converses and Counterexamples12:02
- Example 2: Converses and Counterexamples15:10
- Example 3: Converses and Counterexamples17:08
- Inverse Statement19:58
- Definition and Example19:59
- Inverse Statement21:46
- Example 1: Inverse and Counterexample21:47
- Example 2: Inverse and Counterexample23:34
- Contrapositive Statement25:20
- Definition and Example25:21
- Contrapositive Statement26:58
- Example: Contrapositive Statement27:00
- Summary29:03
- Summary of Lesson29:04
- Extra Example 1: Hypothesis and Conclusion32:20
- Extra Example 2: If-Then Form33:23
- Extra Example 3: Converse, Inverse, and Contrapositive34:54
- Extra Example 4: Converse, Inverse, and Contrapositive37:56

Point, Line, and Plane Postulates

17m 24s

- Intro0:00
- What are Postulates?0:09
- Definition of Postulates0:10
- Postulates1:22
- Postulate 1: Two Points1:23
- Postulate 2: Three Points2:02
- Postulate 3: Line2:45
- Postulates, cont..3:08
- Postulate 4: Plane3:09
- Postulate 5: Two Points in a Plane3:53
- Postulates, cont..4:46
- Postulate 6: Two Lines Intersect4:47
- Postulate 7: Two Plane Intersect5:28
- Using the Postulates6:34
- Examples: True or False6:35
- Using the Postulates10:18
- Examples: True or False10:19
- Extra Example 1: Always, Sometimes, or Never12:22
- Extra Example 2: Always, Sometimes, or Never13:15
- Extra Example 3: Always, Sometimes, or Never14:16
- Extra Example 4: Always, Sometimes, or Never15:03

Deductive Reasoning

36m 3s

- Intro0:00
- Deductive Reasoning0:06
- Definition of Deductive Reasoning0:07
- Inductive vs. Deductive2:51
- Inductive Reasoning2:52
- Deductive reasoning3:19
- Law of Detachment3:47
- Law of Detachment3:48
- Examples of Law of Detachment4:31
- Law of Syllogism7:32
- Law of Syllogism7:33
- Example 1: Making a Conclusion9:02
- Example 2: Making a Conclusion12:54
- Using Laws of Logic14:12
- Example 1: Determine the Logic14:42
- Example 2: Determine the Logic17:02
- Using Laws of Logic, cont.18:47
- Example 3: Determine the Logic19:03
- Example 4: Determine the Logic20:56
- Extra Example 1: Determine the Conclusion and Law22:12
- Extra Example 2: Determine the Conclusion and Law25:39
- Extra Example 3: Determine the Logic and Law29:50
- Extra Example 4: Determine the Logic and Law31:27

Proofs in Algebra: Properties of Equality

44m 31s

- Intro0:00
- Properties of Equality0:10
- Addition Property of Equality0:28
- Subtraction Property of Equality1:10
- Multiplication Property of Equality1:41
- Division Property of Equality1:55
- Addition Property of Equality Using Angles2:46
- Properties of Equality, cont.4:10
- Reflexive Property of Equality4:11
- Symmetric Property of Equality5:24
- Transitive Property of Equality6:10
- Properties of Equality, cont.7:04
- Substitution Property of Equality7:05
- Distributive Property of Equality8:34
- Two Column Proof9:40
- Example: Two Column Proof9:46
- Proof Example 116:13
- Proof Example 223:49
- Proof Example 330:33
- Extra Example 1: Name the Property of Equality38:07
- Extra Example 2: Name the Property of Equality40:16
- Extra Example 3: Name the Property of Equality41:35
- Extra Example 4: Name the Property of Equality43:02

Proving Segment Relationship

41m 2s

- Intro0:00
- Good Proofs0:12
- Five Essential Parts0:13
- Proof Reasons1:38
- Undefined1:40
- Definitions2:06
- Postulates2:42
- Previously Proven Theorems3:24
- Congruence of Segments4:10
- Theorem: Congruence of Segments4:12
- Proof Example10:16
- Proof: Congruence of Segments10:17
- Setting Up Proofs19:13
- Example: Two Segments with Equal Measures19:15
- Setting Up Proofs21:48
- Example: Vertical Angles are Congruent21:50
- Setting Up Proofs23:59
- Example: Segment of a Triangle24:00
- Extra Example 1: Congruence of Segments27:03
- Extra Example 2: Setting Up Proofs28:50
- Extra Example 3: Setting Up Proofs30:55
- Extra Example 4: Two-Column Proof33:11

Proving Angle Relationships

33m 37s

- Intro0:00
- Supplement Theorem0:05
- Supplementary Angles0:06
- Congruence of Angles2:37
- Proof: Congruence of Angles2:38
- Angle Theorems6:54
- Angle Theorem 1: Supplementary Angles6:55
- Angle Theorem 2: Complementary Angles10:25
- Angle Theorems11:32
- Angle Theorem 3: Right Angles11:35
- Angle Theorem 4: Vertical Angles12:09
- Angle Theorem 5: Perpendicular Lines12:57
- Using Angle Theorems13:45
- Example 1: Always, Sometimes, or Never13:50
- Example 2: Always, Sometimes, or Never14:28
- Example 3: Always, Sometimes, or Never16:21
- Extra Example 1: Always, Sometimes, or Never16:53
- Extra Example 2: Find the Measure of Each Angle18:55
- Extra Example 3: Find the Measure of Each Angle25:03
- Extra Example 4: Two-Column Proof27:08

III. Perpendicular & Parallel Lines

Parallel Lines and Transversals

37m 35s

- Intro0:00
- Lines0:06
- Parallel Lines0:09
- Skew Lines2:02
- Transversal3:42
- Angles Formed by a Transversal4:28
- Interior Angles5:53
- Exterior Angles6:09
- Consecutive Interior Angles7:04
- Alternate Exterior Angles9:47
- Alternate Interior Angles11:22
- Corresponding Angles12:27
- Angles Formed by a Transversal15:29
- Relationship Between Angles15:30
- Extra Example 1: Intersecting, Parallel, or Skew19:26
- Extra Example 2: Draw a Diagram21:37
- Extra Example 3: Name the Figures24:12
- Extra Example 4: Angles Formed by a Transversal28:38

Angles and Parallel Lines

41m 53s

- Intro0:00
- Corresponding Angles Postulate0:05
- Corresponding Angles Postulate0:06
- Alternate Interior Angles Theorem3:05
- Alternate Interior Angles Theorem3:07
- Consecutive Interior Angles Theorem5:16
- Consecutive Interior Angles Theorem5:17
- Alternate Exterior Angles Theorem6:42
- Alternate Exterior Angles Theorem6:43
- Parallel Lines Cut by a Transversal7:18
- Example: Parallel Lines Cut by a Transversal7:19
- Perpendicular Transversal Theorem14:54
- Perpendicular Transversal Theorem14:55
- Extra Example 1: State the Postulate or Theorem16:37
- Extra Example 2: Find the Measure of the Numbered Angle18:53
- Extra Example 3: Find the Measure of Each Angle25:13
- Extra Example 4: Find the Values of x, y, and z36:26

Slope of Lines

44m 6s

- Intro0:00
- Definition of Slope0:06
- Slope Equation0:13
- Slope of a Line3:45
- Example: Find the Slope of a Line3:47
- Slope of a Line8:38
- More Example: Find the Slope of a Line8:40
- Slope Postulates12:32
- Proving Slope Postulates12:33
- Parallel or Perpendicular Lines17:23
- Example: Parallel or Perpendicular Lines17:24
- Using Slope Formula20:02
- Example: Using Slope Formula20:03
- Extra Example 1: Slope of a Line25:10
- Extra Example 2: Slope of a Line26:31
- Extra Example 3: Graph the Line34:11
- Extra Example 4: Using the Slope Formula38:50

Proving Lines Parallel

25m 55s

- Intro0:00
- Postulates0:06
- Postulate 1: Parallel Lines0:21
- Postulate 2: Parallel Lines2:16
- Parallel Postulate3:28
- Definition and Example of Parallel Postulate3:29
- Theorems4:29
- Theorem 1: Parallel Lines4:40
- Theorem 2: Parallel Lines5:37
- Theorems, cont.6:10
- Theorem 3: Parallel Lines6:11
- Extra Example 1: Determine Parallel Lines6:56
- Extra Example 2: Find the Value of x11:42
- Extra Example 3: Opposite Sides are Parallel14:48
- Extra Example 4: Proving Parallel Lines20:42

Parallels and Distance

19m 48s

- Intro0:00
- Distance Between a Points and Line0:07
- Definition and Example0:08
- Distance Between Parallel Lines1:51
- Definition and Example1:52
- Extra Example 1: Drawing a Segment to Represent Distance3:02
- Extra Example 2: Drawing a Segment to Represent Distance4:27
- Extra Example 3: Graph, Plot, and Construct a Perpendicular Segment5:13
- Extra Example 4: Distance Between Two Parallel Lines15:37

IV. Congruent Triangles

Classifying Triangles

28m 43s

- Intro0:00
- Triangles0:09
- Triangle: A Three-Sided Polygon0:10
- Sides1:00
- Vertices1:22
- Angles1:56
- Classifying Triangles by Angles2:59
- Acute Triangle3:19
- Obtuse Triangle4:08
- Right Triangle4:44
- Equiangular Triangle5:38
- Definition and Example of an Equiangular Triangle5:39
- Classifying Triangles by Sides6:57
- Scalene Triangle7:17
- Isosceles Triangle7:57
- Equilateral Triangle8:12
- Isosceles Triangle8:58
- Labeling Isosceles Triangle9:00
- Labeling Right Triangle10:44
- Isosceles Triangle11:10
- Example: Find x, AB, BC, and AC11:11
- Extra Example 1: Classify Each Triangle13:45
- Extra Example 2: Always, Sometimes, or Never16:28
- Extra Example 3: Find All the Sides of the Isosceles Triangle20:29
- Extra Example 4: Distance Formula and Triangle22:29

Measuring Angles in Triangles

44m 43s

- Intro0:00
- Angle Sum Theorem0:09
- Angle Sum Theorem for Triangle0:11
- Using Angle Sum Theorem4:06
- Find the Measure of the Missing Angle4:07
- Third Angle Theorem4:58
- Example: Third Angle Theorem4:59
- Exterior Angle Theorem7:58
- Example: Exterior Angle Theorem8:00
- Flow Proof of Exterior Angle Theorem15:14
- Flow Proof of Exterior Angle Theorem15:17
- Triangle Corollaries27:21
- Triangle Corollary 127:50
- Triangle Corollary 230:42
- Extra Example 1: Find the Value of x32:55
- Extra Example 2: Find the Value of x34:20
- Extra Example 3: Find the Measure of the Angle35:38
- Extra Example 4: Find the Measure of Each Numbered Angle39:00

Exploring Congruent Triangles

26m 46s

- Intro0:00
- Congruent Triangles0:15
- Example of Congruent Triangles0:17
- Corresponding Parts3:39
- Corresponding Angles and Sides of Triangles3:40
- Definition of Congruent Triangles11:24
- Definition of Congruent Triangles11:25
- Triangle Congruence16:37
- Congruence of Triangles16:38
- Extra Example 1: Congruence Statement18:24
- Extra Example 2: Congruence Statement21:26
- Extra Example 3: Draw and Label the Figure23:09
- Extra Example 4: Drawing Triangles24:04

Proving Triangles Congruent

47m 51s

- Intro0:00
- SSS Postulate0:18
- Side-Side-Side Postulate0:27
- SAS Postulate2:26
- Side-Angle-Side Postulate2:29
- SAS Postulate3:57
- Proof Example3:58
- ASA Postulate11:47
- Angle-Side-Angle Postulate11:53
- AAS Theorem14:13
- Angle-Angle-Side Theorem14:14
- Methods Overview16:16
- Methods Overview16:17
- SSS16:33
- SAS17:06
- ASA17:50
- AAS18:17
- CPCTC19:14
- Extra Example 1:Proving Triangles are Congruent21:29
- Extra Example 2: Proof25:40
- Extra Example 3: Proof30:41
- Extra Example 4: Proof38:41

Isosceles and Equilateral Triangles

27m 53s

- Intro0:00
- Isosceles Triangle Theorem0:07
- Isosceles Triangle Theorem0:09
- Isosceles Triangle Theorem2:26
- Example: Using the Isosceles Triangle Theorem2:27
- Isosceles Triangle Theorem Converse3:29
- Isosceles Triangle Theorem Converse3:30
- Equilateral Triangle Theorem Corollaries4:30
- Equilateral Triangle Theorem Corollary 14:59
- Equilateral Triangle Theorem Corollary 25:55
- Extra Example 1: Find the Value of x7:08
- Extra Example 2: Find the Value of x10:04
- Extra Example 3: Proof14:04
- Extra Example 4: Proof22:41

V. Triangle Inequalities

Special Segments in Triangles

43m 44s

- Intro0:00
- Perpendicular Bisector0:06
- Perpendicular Bisector0:07
- Perpendicular Bisector4:07
- Perpendicular Bisector Theorems4:08
- Median6:30
- Definition of Median6:31
- Median9:41
- Example: Median9:42
- Altitude12:22
- Definition of Altitude12:23
- Angle Bisector14:33
- Definition of Angle Bisector14:34
- Angle Bisector16:41
- Angle Bisector Theorems16:42
- Special Segments Overview18:57
- Perpendicular Bisector19:04
- Median19:32
- Altitude19:49
- Angle Bisector20:02
- Examples: Special Segments20:18
- Extra Example 1: Draw and Label22:36
- Extra Example 2: Draw the Altitudes for Each Triangle24:37
- Extra Example 3: Perpendicular Bisector27:57
- Extra Example 4: Draw, Label, and Write Proof34:33

Right Triangles

26m 34s

- Intro0:00
- LL Theorem0:21
- Leg-Leg Theorem0:25
- HA Theorem2:23
- Hypotenuse-Angle Theorem2:24
- LA Theorem4:49
- Leg-Angle Theorem4:50
- LA Theorem6:18
- Example: Find x and y6:19
- HL Postulate8:22
- Hypotenuse-Leg Postulate8:23
- Extra Example 1: LA Theorem & HL Postulate10:57
- Extra Example 2: Find x So That Each Pair of Triangles is Congruent14:15
- Extra Example 3: Two-column Proof17:02
- Extra Example 4: Two-column Proof21:01

Indirect Proofs and Inequalities

33m 30s

- Intro0:00
- Writing an Indirect Proof0:09
- Step 10:49
- Step 22:32
- Step 33:00
- Indirect Proof4:30
- Example: 2 + 6 = 85:00
- Example: The Suspect is Guilty5:40
- Example: Measure of Angle A < Measure of Angle B6:06
- Definition of Inequality7:47
- Definition of Inequality & Example7:48
- Properties of Inequality9:55
- Comparison Property9:58
- Transitive Property10:33
- Addition and Subtraction Properties12:01
- Multiplication and Division Properties13:07
- Exterior Angle Inequality Theorem14:12
- Example: Exterior Angle Inequality Theorem14:13
- Extra Example 1: Draw a Diagram for the Statement18:32
- Extra Example 2: Name the Property for Each Statement19:56
- Extra Example 3: State the Assumption21:22
- Extra Example 4: Write an Indirect Proof25:39

Inequalities for Sides and Angles of a Triangle

17m 26s

- Intro0:00
- Side to Angles0:10
- If One Side of a Triangle is Longer Than Another Side0:11
- Converse: Angles to Sides1:57
- If One Angle of a Triangle Has a Greater Measure Than Another Angle1:58
- Extra Example 1: Name the Angles in the Triangle From Least to Greatest2:38
- Extra Example 2: Find the Longest and Shortest Segment in the Triangle3:47
- Extra Example 3: Angles and Sides of a Triangle4:51
- Extra Example 4: Two-column Proof9:08

Triangle Inequality

28m 11s

- Intro0:00
- Triangle Inequality Theorem0:05
- Triangle Inequality Theorem0:06
- Triangle Inequality Theorem4:22
- Example 1: Triangle Inequality Theorem4:23
- Example 2: Triangle Inequality Theorem9:40
- Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle12:00
- Extra Example 2: Finding the Third Side of a Triangle13:34
- Extra Example 3: Always True, Sometimes True, or Never True18:18
- Extra Example 4: Triangle and Vertices22:36

Inequalities Involving Two Triangles

29m 36s

- Intro0:00
- SAS Inequality Theorem0:06
- SAS Inequality Theorem & Example0:25
- SSS Inequality Theorem4:33
- SSS Inequality Theorem & Example4:34
- Extra Example 1: Write an Inequality Comparing the Segments6:08
- Extra Example 2: Determine if the Statement is True9:52
- Extra Example 3: Write an Inequality for x14:20
- Extra Example 4: Two-column Proof17:44

VI. Quadrilaterals

Parallelograms

29m 11s

- Intro0:00
- Quadrilaterals0:06
- Four-sided Polygons0:08
- Non Examples of Quadrilaterals0:47
- Parallelograms1:35
- Parallelograms1:36
- Properties of Parallelograms4:28
- Opposite Sides of a Parallelogram are Congruent4:29
- Opposite Angles of a Parallelogram are Congruent5:49
- Angles and Diagonals6:24
- Consecutive Angles in a Parallelogram are Supplementary6:25
- The Diagonals of a Parallelogram Bisect Each Other8:42
- Extra Example 1: Complete Each Statement About the Parallelogram10:26
- Extra Example 2: Find the Values of x, y, and z of the Parallelogram13:21
- Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram16:35
- Extra Example 4: Slope of Parallelogram23:15

Proving Parallelograms

42m 43s

- Intro0:00
- Parallelogram Theorems0:09
- Theorem 10:20
- Theorem 21:50
- Parallelogram Theorems, Cont.3:10
- Theorem 33:11
- Theorem 44:15
- Proving Parallelogram6:21
- Example: Determine if Quadrilateral ABCD is a Parallelogram6:22
- Summary14:01
- Both Pairs of Opposite Sides are Parallel14:14
- Both Pairs of Opposite Sides are Congruent15:09
- Both Pairs of Opposite Angles are Congruent15:24
- Diagonals Bisect Each Other15:44
- A Pair of Opposite Sides is Both Parallel and Congruent16:13
- Extra Example 1: Determine if Each Quadrilateral is a Parallelogram16:54
- Extra Example 2: Find the Value of x and y20:23
- Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram24:05
- Extra Example 4: Two-column Proof30:28

Rectangles

29m 47s

- Intro0:00
- Rectangles0:03
- Definition of Rectangles0:04
- Diagonals of Rectangles2:52
- Rectangles: Diagonals Property 12:53
- Rectangles: Diagonals Property 23:30
- Proving a Rectangle4:40
- Example: Determine Whether Parallelogram ABCD is a Rectangle4:41
- Rectangles Summary9:22
- Opposite Sides are Congruent and Parallel9:40
- Opposite Angles are Congruent9:51
- Consecutive Angles are Supplementary9:58
- Diagonals are Congruent and Bisect Each Other10:05
- All Four Angles are Right Angles10:40
- Extra Example 1: Find the Value of x11:03
- Extra Example 2: Name All Congruent Sides and Angles13:52
- Extra Example 3: Always, Sometimes, or Never True19:39
- Extra Example 4: Determine if ABCD is a Rectangle26:45

Squares and Rhombi

39m 14s

- Intro0:00
- Rhombus0:09
- Definition of a Rhombus0:10
- Diagonals of a Rhombus2:03
- Rhombus: Diagonals Property 12:21
- Rhombus: Diagonals Property 23:49
- Rhombus: Diagonals Property 34:36
- Rhombus6:17
- Example: Use the Rhombus to Find the Missing Value6:18
- Square8:17
- Definition of a Square8:20
- Summary Chart11:06
- Parallelogram11:07
- Rectangle12:56
- Rhombus13:54
- Square14:44
- Extra Example 1: Diagonal Property15:44
- Extra Example 2: Use Rhombus ABCD to Find the Missing Value19:39
- Extra Example 3: Always, Sometimes, or Never True23:06
- Extra Example 4: Determine the Quadrilateral28:02

Trapezoids and Kites

30m 48s

- Intro0:00
- Trapezoid0:10
- Definition of Trapezoid0:12
- Isosceles Trapezoid2:57
- Base Angles of an Isosceles Trapezoid2:58
- Diagonals of an Isosceles Trapezoid4:05
- Median of a Trapezoid4:26
- Median of a Trapezoid4:27
- Median of a Trapezoid6:41
- Median Formula7:00
- Kite8:28
- Definition of a Kite8:29
- Quadrilaterals Summary11:19
- A Quadrilateral with Two Pairs of Adjacent Congruent Sides11:20
- Extra Example 1: Isosceles Trapezoid14:50
- Extra Example 2: Median of Trapezoid18:28
- Extra Example 3: Always, Sometimes, or Never24:13
- Extra Example 4: Determine if the Figure is a Trapezoid26:49

VII. Proportions and Similarity

Using Proportions and Ratios

20m 10s

- Intro0:00
- Ratio0:05
- Definition and Examples of Writing Ratio0:06
- Proportion2:05
- Definition of Proportion2:06
- Examples of Proportion2:29
- Using Ratio5:53
- Example: Ratio5:54
- Extra Example 1: Find Three Ratios Equivalent to 2/59:28
- Extra Example 2: Proportion and Cross Products10:32
- Extra Example 3: Express Each Ratio as a Fraction13:18
- Extra Example 4: Fin the Measure of a 3:4:5 Triangle17:26

Similar Polygons

27m 53s

- Intro0:00
- Similar Polygons0:05
- Definition of Similar Polygons0:06
- Example of Similar Polygons2:32
- Scale Factor4:26
- Scale Factor: Definition and Example4:27
- Extra Example 1: Determine if Each Pair of Figures is Similar7:03
- Extra Example 2: Find the Values of x and y11:33
- Extra Example 3: Similar Triangles19:57
- Extra Example 4: Draw Two Similar Figures23:36

Similar Triangles

34m 10s

- Intro0:00
- AA Similarity0:10
- Definition of AA Similarity0:20
- Example of AA Similarity2:32
- SSS Similarity4:46
- Definition of SSS Similarity4:47
- Example of SSS Similarity6:00
- SAS Similarity8:04
- Definition of SAS Similarity8:05
- Example of SAS Similarity9:12
- Extra Example 1: Determine Whether Each Pair of Triangles is Similar10:59
- Extra Example 2: Determine Which Triangles are Similar16:08
- Extra Example 3: Determine if the Statement is True or False23:11
- Extra Example 4: Write Two-Column Proof26:25

Parallel Lines and Proportional Parts

24m 7s

- Intro0:00
- Triangle Proportionality0:07
- Definition of Triangle Proportionality0:08
- Example of Triangle Proportionality0:51
- Triangle Proportionality Converse2:19
- Triangle Proportionality Converse2:20
- Triangle Mid-segment3:42
- Triangle Mid-segment: Definition and Example3:43
- Parallel Lines and Transversal6:51
- Parallel Lines and Transversal6:52
- Extra Example 1: Complete Each Statement8:59
- Extra Example 2: Determine if the Statement is True or False12:28
- Extra Example 3: Find the Value of x and y15:35
- Extra Example 4: Find Midpoints of a Triangle20:43

Parts of Similar Triangles

27m 6s

- Intro0:00
- Proportional Perimeters0:09
- Proportional Perimeters: Definition and Example0:10
- Similar Altitudes2:23
- Similar Altitudes: Definition and Example2:24
- Similar Angle Bisectors4:50
- Similar Angle Bisectors: Definition and Example4:51
- Similar Medians6:05
- Similar Medians: Definition and Example6:06
- Angle Bisector Theorem7:33
- Angle Bisector Theorem7:34
- Extra Example 1: Parts of Similar Triangles10:52
- Extra Example 2: Parts of Similar Triangles14:57
- Extra Example 3: Parts of Similar Triangles19:27
- Extra Example 4: Find the Perimeter of Triangle ABC23:14

VIII. Applying Right Triangles & Trigonometry

Pythagorean Theorem

21m 14s

- Intro0:00
- Pythagorean Theorem0:05
- Pythagorean Theorem & Example0:06
- Pythagorean Converse1:20
- Pythagorean Converse & Example1:21
- Pythagorean Triple2:42
- Pythagorean Triple2:43
- Extra Example 1: Find the Missing Side4:59
- Extra Example 2: Determine Right Triangle7:40
- Extra Example 3: Determine Pythagorean Triple11:30
- Extra Example 4: Vertices and Right Triangle14:29

Geometric Mean

40m 59s

- Intro0:00
- Geometric Mean0:04
- Geometric Mean & Example0:05
- Similar Triangles4:32
- Similar Triangles4:33
- Geometric Mean-Altitude11:10
- Geometric Mean-Altitude & Example11:11
- Geometric Mean-Leg14:47
- Geometric Mean-Leg & Example14:18
- Extra Example 1: Geometric Mean Between Each Pair of Numbers20:10
- Extra Example 2: Similar Triangles23:46
- Extra Example 3: Geometric Mean of Triangles28:30
- Extra Example 4: Geometric Mean of Triangles36:58

Special Right Triangles

37m 57s

- Intro0:00
- 45-45-90 Triangles0:06
- Definition of 45-45-90 Triangles0:25
- 45-45-90 Triangles5:51
- Example: Find n5:52
- 30-60-90 Triangles8:59
- Definition of 30-60-90 Triangles9:00
- 30-60-90 Triangles12:25
- Example: Find n12:26
- Extra Example 1: Special Right Triangles15:08
- Extra Example 2: Special Right Triangles18:22
- Extra Example 3: Word Problems & Special Triangles27:40
- Extra Example 4: Hexagon & Special Triangles33:51

Ratios in Right Triangles

40m 37s

- Intro0:00
- Trigonometric Ratios0:08
- Definition of Trigonometry0:13
- Sine (sin), Cosine (cos), & Tangent (tan)0:50
- Trigonometric Ratios3:04
- Trig Functions3:05
- Inverse Trig Functions5:02
- SOHCAHTOA8:16
- sin x9:07
- cos x10:00
- tan x10:32
- Example: SOHCAHTOA & Triangle12:10
- Extra Example 1: Find the Value of Each Ratio or Angle Measure14:36
- Extra Example 2: Find Sin, Cos, and Tan18:51
- Extra Example 3: Find the Value of x Using SOHCAHTOA22:55
- Extra Example 4: Trigonometric Ratios in Right Triangles32:13

Angles of Elevation and Depression

21m 4s

- Intro0:00
- Angle of Elevation0:10
- Definition of Angle of Elevation & Example0:11
- Angle of Depression1:19
- Definition of Angle of Depression & Example1:20
- Extra Example 1: Name the Angle of Elevation and Depression2:22
- Extra Example 2: Word Problem & Angle of Depression4:41
- Extra Example 3: Word Problem & Angle of Elevation14:02
- Extra Example 4: Find the Missing Measure18:10

Law of Sines

35m 25s

- Intro0:00
- Law of Sines0:20
- Law of Sines0:21
- Law of Sines3:34
- Example: Find b3:35
- Solving the Triangle9:19
- Example: Using the Law of Sines to Solve Triangle9:20
- Extra Example 1: Law of Sines and Triangle17:43
- Extra Example 2: Law of Sines and Triangle20:06
- Extra Example 3: Law of Sines and Triangle23:54
- Extra Example 4: Law of Sines and Triangle28:59

Law of Cosines

52m 43s

- Intro0:00
- Law of Cosines0:35
- Law of Cosines0:36
- Law of Cosines6:22
- Use the Law of Cosines When Both are True6:23
- Law of Cosines8:35
- Example: Law of Cosines8:36
- Extra Example 1: Law of Sines or Law of Cosines?13:35
- Extra Example 2: Use the Law of Cosines to Find the Missing Measure17:02
- Extra Example 3: Solve the Triangle30:49
- Extra Example 4: Find the Measure of Each Diagonal of the Parallelogram41:39

IX. Circles

Segments in a Circle

22m 43s

- Intro0:00
- Segments in a Circle0:10
- Circle0:11
- Chord0:59
- Diameter1:32
- Radius2:07
- Secant2:17
- Tangent3:10
- Circumference3:56
- Introduction to Circumference3:57
- Example: Find the Circumference of the Circle5:09
- Circumference6:40
- Example: Find the Circumference of the Circle6:41
- Extra Example 1: Use the Circle to Answer the Following9:10
- Extra Example 2: Find the Missing Measure12:53
- Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle15:51
- Extra Example 4: Find the Circumference of Each Circle19:24

Angles and Arc

35m 24s

- Intro0:00
- Central Angle0:06
- Definition of Central Angle0:07
- Sum of Central Angles1:17
- Sum of Central Angles1:18
- Arcs2:27
- Minor Arc2:30
- Major Arc3:47
- Arc Measure5:24
- Measure of Minor Arc5:24
- Measure of Major Arc6:53
- Measure of a Semicircle7:11
- Arc Addition Postulate8:25
- Arc Addition Postulate8:26
- Arc Length9:43
- Arc Length and Example9:44
- Concentric Circles16:05
- Concentric Circles16:06
- Congruent Circles and Arcs17:50
- Congruent Circles17:51
- Congruent Arcs18:47
- Extra Example 1: Minor Arc, Major Arc, and Semicircle20:14
- Extra Example 2: Measure and Length of Arc22:52
- Extra Example 3: Congruent Arcs25:48
- Extra Example 4: Angles and Arcs30:33

Arcs and Chords

21m 51s

- Intro0:00
- Arcs and Chords0:07
- Arc of the Chord0:08
- Theorem 1: Congruent Minor Arcs1:01
- Inscribed Polygon2:10
- Inscribed Polygon2:11
- Arcs and Chords3:18
- Theorem 2: When a Diameter is Perpendicular to a Chord3:19
- Arcs and Chords5:05
- Theorem 3: Congruent Chords5:06
- Extra Example 1: Congruent Arcs10:35
- Extra Example 2: Length of Arc13:50
- Extra Example 3: Arcs and Chords17:09
- Extra Example 4: Arcs and Chords19:45

Inscribed Angles

27m 53s

- Intro0:00
- Inscribed Angles0:07
- Definition of Inscribed Angles0:08
- Inscribed Angles0:58
- Inscribed Angle Theorem 10:59
- Inscribed Angles3:29
- Inscribed Angle Theorem 23:30
- Inscribed Angles4:38
- Inscribed Angle Theorem 34:39
- Inscribed Quadrilateral5:50
- Inscribed Quadrilateral5:51
- Extra Example 1: Central Angle, Inscribed Angle, and Intercepted Arc7:02
- Extra Example 2: Inscribed Angles9:24
- Extra Example 3: Inscribed Angles14:00
- Extra Example 4: Complete the Proof17:58

Tangents

26m 16s

- Intro0:00
- Tangent Theorems0:04
- Tangent Theorem 10:05
- Tangent Theorem 1 Converse0:55
- Common Tangents1:34
- Common External Tangent2:12
- Common Internal Tangent2:30
- Tangent Segments3:08
- Tangent Segments3:09
- Circumscribed Polygons4:11
- Circumscribed Polygons4:12
- Extra Example 1: Tangents & Circumscribed Polygons5:50
- Extra Example 2: Tangents & Circumscribed Polygons8:35
- Extra Example 3: Tangents & Circumscribed Polygons11:50
- Extra Example 4: Tangents & Circumscribed Polygons15:43

Secants, Tangents, & Angle Measures

27m 50s

- Intro0:00
- Secant0:08
- Secant0:09
- Secant and Tangent0:49
- Secant and Tangent0:50
- Interior Angles2:56
- Secants & Interior Angles2:57
- Exterior Angles7:21
- Secants & Exterior Angles7:22
- Extra Example 1: Secants, Tangents, & Angle Measures10:53
- Extra Example 2: Secants, Tangents, & Angle Measures13:31
- Extra Example 3: Secants, Tangents, & Angle Measures19:54
- Extra Example 4: Secants, Tangents, & Angle Measures22:29

Special Segments in a Circle

23m 8s

- Intro0:00
- Chord Segments0:05
- Chord Segments0:06
- Secant Segments1:36
- Secant Segments1:37
- Tangent and Secant Segments4:10
- Tangent and Secant Segments4:11
- Extra Example 1: Special Segments in a Circle5:53
- Extra Example 2: Special Segments in a Circle7:58
- Extra Example 3: Special Segments in a Circle11:24
- Extra Example 4: Special Segments in a Circle18:09

Equations of Circles

27m 1s

- Intro0:00
- Equation of a Circle0:06
- Standard Equation of a Circle0:07
- Example 1: Equation of a Circle0:57
- Example 2: Equation of a Circle1:36
- Extra Example 1: Determine the Coordinates of the Center and the Radius4:56
- Extra Example 2: Write an Equation Based on the Given Information7:53
- Extra Example 3: Graph Each Circle16:48
- Extra Example 4: Write the Equation of Each Circle19:17

X. Polygons & Area

Polygons

27m 24s

- Intro0:00
- Polygons0:10
- Polygon vs. Not Polygon0:18
- Convex and Concave1:46
- Convex vs. Concave Polygon1:52
- Regular Polygon4:04
- Regular Polygon4:05
- Interior Angle Sum Theorem4:53
- Triangle5:03
- Quadrilateral6:05
- Pentagon6:38
- Hexagon7:59
- 20-Gon9:36
- Exterior Angle Sum Theorem12:04
- Exterior Angle Sum Theorem12:05
- Extra Example 1: Drawing Polygons13:51
- Extra Example 2: Convex Polygon15:16
- Extra Example 3: Exterior Angle Sum Theorem18:21
- Extra Example 4: Interior Angle Sum Theorem22:20

Area of Parallelograms

17m 46s

- Intro0:00
- Parallelograms0:06
- Definition and Area Formula0:07
- Area of Figure2:00
- Area of Figure2:01
- Extra Example 1:Find the Area of the Shaded Area3:14
- Extra Example 2: Find the Height and Area of the Parallelogram6:00
- Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices10:11
- Extra Example 4: Find the Area of the Figure14:31

Area of Triangles Rhombi, & Trapezoids

20m 31s

- Intro0:00
- Area of a Triangle0:06
- Area of a Triangle: Formula and Example0:07
- Area of a Trapezoid2:31
- Area of a Trapezoid: Formula2:32
- Area of a Trapezoid: Example6:55
- Area of a Rhombus8:05
- Area of a Rhombus: Formula and Example8:06
- Extra Example 1: Find the Area of the Polygon9:51
- Extra Example 2: Find the Area of the Figure11:19
- Extra Example 3: Find the Area of the Figure14:16
- Extra Example 4: Find the Height of the Trapezoid18:10

Area of Regular Polygons & Circles

36m 43s

- Intro0:00
- Regular Polygon0:08
- SOHCAHTOA0:54
- 30-60-90 Triangle1:52
- 45-45-90 Triangle2:40
- Area of a Regular Polygon3:39
- Area of a Regular Polygon3:40
- Are of a Circle7:55
- Are of a Circle7:56
- Extra Example 1: Find the Area of the Regular Polygon8:22
- Extra Example 2: Find the Area of the Regular Polygon16:48
- Extra Example 3: Find the Area of the Shaded Region24:11
- Extra Example 4: Find the Area of the Shaded Region32:24

Perimeter & Area of Similar Figures

18m 17s

- Intro0:00
- Perimeter of Similar Figures0:08
- Example: Scale Factor & Perimeter of Similar Figures0:09
- Area of Similar Figures2:44
- Example:Scale Factor & Area of Similar Figures2:55
- Extra Example 1: Complete the Table6:09
- Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures8:56
- Extra Example 3: Find the Unknown Area12:04
- Extra Example 4: Use the Given Area to Find AB14:26

Geometric Probability

38m 40s

- Intro0:00
- Length Probability Postulate0:05
- Length Probability Postulate0:06
- Are Probability Postulate2:34
- Are Probability Postulate2:35
- Are of a Sector of a Circle4:11
- Are of a Sector of a Circle Formula4:12
- Are of a Sector of a Circle Example7:51
- Extra Example 1: Length Probability11:07
- Extra Example 2: Area Probability12:14
- Extra Example 3: Area Probability17:17
- Extra Example 4: Area of a Sector of a Circle26:23

XI. Solids

Three-Dimensional Figures

23m 39s

- Intro0:00
- Polyhedrons0:05
- Polyhedrons: Definition and Examples0:06
- Faces1:08
- Edges1:55
- Vertices2:23
- Solids2:51
- Pyramid2:54
- Cylinder3:45
- Cone4:09
- Sphere4:23
- Prisms5:00
- Rectangular, Regular, and Cube Prisms5:02
- Platonic Solids9:48
- Five Types of Regular Polyhedra9:49
- Slices and Cross Sections12:07
- Slices12:08
- Cross Sections12:47
- Extra Example 1: Name the Edges, Faces, and Vertices of the Polyhedron14:23
- Extra Example 2: Determine if the Figure is a Polyhedron and Explain Why17:37
- Extra Example 3: Describe the Slice Resulting from the Cut19:12
- Extra Example 4: Describe the Shape of the Intersection21:25

Surface Area of Prisms and Cylinders

38m 50s

- Intro0:00
- Prisms0:06
- Bases0:07
- Lateral Faces0:52
- Lateral Edges1:19
- Altitude1:58
- Prisms2:24
- Right Prism2:25
- Oblique Prism2:56
- Classifying Prisms3:27
- Right Rectangular Prism3:28
- Oblique Pentagonal Prism6:26
- Right Hexagonal Prism7:14
- Lateral Area of a Prism7:42
- Lateral Area of a Prism7:43
- Surface Area of a Prism13:44
- Surface Area of a Prism13:45
- Cylinder16:18
- Cylinder: Right and Oblique16:19
- Lateral Area of a Cylinder18:02
- Lateral Area of a Cylinder18:03
- Surface Area of a Cylinder20:54
- Surface Area of a Cylinder20:55
- Extra Example 1: Find the Lateral Area and Surface Are of the Prism21:51
- Extra Example 2: Find the Lateral Area of the Prism28:15
- Extra Example 3: Find the Surface Area of the Prism31:57
- Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder34:17

Surface Area of Pyramids and Cones

26m 10s

- Intro0:00
- Pyramids0:07
- Pyramids0:08
- Regular Pyramids1:52
- Regular Pyramids1:53
- Lateral Area of a Pyramid4:33
- Lateral Area of a Pyramid4:34
- Surface Area of a Pyramid9:19
- Surface Area of a Pyramid9:20
- Cone10:09
- Right and Oblique Cone10:10
- Lateral Area and Surface Area of a Right Cone11:20
- Lateral Area and Surface Are of a Right Cone11:21
- Extra Example 1: Pyramid and Prism13:11
- Extra Example 2: Find the Lateral Area of the Regular Pyramid15:00
- Extra Example 3: Find the Surface Area of the Pyramid18:29
- Extra Example 4: Find the Lateral Area and Surface Area of the Cone22:08

Volume of Prisms and Cylinders

21m 59s

- Intro0:00
- Volume of Prism0:08
- Volume of Prism0:10
- Volume of Cylinder3:38
- Volume of Cylinder3:39
- Extra Example 1: Find the Volume of the Prism5:10
- Extra Example 2: Find the Volume of the Cylinder8:03
- Extra Example 3: Find the Volume of the Prism9:35
- Extra Example 4: Find the Volume of the Solid19:06

Volume of Pyramids and Cones

22m 2s

- Intro0:00
- Volume of a Cone0:08
- Volume of a Cone: Example0:10
- Volume of a Pyramid3:02
- Volume of a Pyramid: Example3:03
- Extra Example 1: Find the Volume of the Pyramid4:56
- Extra Example 2: Find the Volume of the Solid6:01
- Extra Example 3: Find the Volume of the Pyramid10:28
- Extra Example 4: Find the Volume of the Octahedron16:23

Surface Area and Volume of Spheres

14m 46s

- Intro0:00
- Special Segments0:06
- Radius0:07
- Chord0:31
- Diameter0:55
- Tangent1:20
- Sphere1:43
- Plane & Sphere1:44
- Hemisphere2:56
- Surface Area of a Sphere3:25
- Surface Area of a Sphere3:26
- Volume of a Sphere4:08
- Volume of a Sphere4:09
- Extra Example 1: Determine Whether Each Statement is True or False4:24
- Extra Example 2: Find the Surface Area of the Sphere6:17
- Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters7:25
- Extra Example 4: Find the Surface Area and Volume of the Solid9:17

Congruent and Similar Solids

16m 6s

- Intro0:00
- Scale Factor0:06
- Scale Factor: Definition and Example0:08
- Congruent Solids1:09
- Congruent Solids1:10
- Similar Solids2:17
- Similar Solids2:18
- Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither3:35
- Extra Example 2: Determine if Each Statement is True or False7:47
- Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume10:14
- Extra Example 4: Find the Volume of the Larger Prism12:14

XII. Transformational Geometry

Mapping

14m 12s

- Intro0:00
- Transformation0:04
- Rotation0:32
- Translation1:03
- Reflection1:17
- Dilation1:24
- Transformations1:45
- Examples1:46
- Congruence Transformation2:51
- Congruence Transformation2:52
- Extra Example 1: Describe the Transformation that Occurred in the Mappings3:37
- Extra Example 2: Determine if the Transformation is an Isometry5:16
- Extra Example 3: Isometry8:16

Reflections

23m 17s

- Intro0:00
- Reflection0:05
- Definition of Reflection0:06
- Line of Reflection0:35
- Point of Reflection1:22
- Symmetry1:59
- Line of Symmetry2:00
- Point of Symmetry2:48
- Extra Example 1: Draw the Image over the Line of Reflection and the Point of Reflection3:45
- Extra Example 2: Determine Lines and Point of Symmetry6:59
- Extra Example 3: Graph the Reflection of the Polygon11:15
- Extra Example 4: Graph the Coordinates16:07

Translations

18m 43s

- Intro0:00
- Translation0:05
- Translation: Preimage & Image0:06
- Example0:56
- Composite of Reflections6:28
- Composite of Reflections6:29
- Extra Example 1: Translation7:48
- Extra Example 2: Image, Preimage, and Translation12:38
- Extra Example 3: Find the Translation Image Using a Composite of Reflections15:08
- Extra Example 4: Find the Value of Each Variable in the Translation17:18

Rotations

21m 26s

- Intro0:00
- Rotations0:04
- Rotations0:05
- Performing Rotations2:13
- Composite of Two Successive Reflections over Two Intersecting Lines2:14
- Angle of Rotation: Angle Formed by Intersecting Lines4:29
- Angle of Rotation5:30
- Rotation Postulate5:31
- Extra Example 1: Find the Rotated Image7:32
- Extra Example 2: Rotations and Coordinate Plane10:33
- Extra Example 3: Find the Value of Each Variable in the Rotation14:29
- Extra Example 4: Draw the Polygon Rotated 90 Degree Clockwise about P16:13

Dilation

37m 6s

- Intro0:00
- Dilations0:06
- Dilations0:07
- Scale Factor1:36
- Scale Factor1:37
- Example 12:06
- Example 26:22
- Scale Factor8:20
- Positive Scale Factor8:21
- Negative Scale Factor9:25
- Enlargement12:43
- Reduction13:52
- Extra Example 1: Find the Scale Factor16:39
- Extra Example 2: Find the Measure of the Dilation Image19:32
- Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation26:18
- Extra Example 4: Graphing Polygon, Dilation, and Scale Factor32:08

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1 answer

Last reply by: Professor Pyo

Thu Jan 2, 2014 4:36 PM

Post by Mirza Baig on December 6, 2013

I have question "do we always have draw a point ????"