Unit 3 Notes: Parallel Lines, Perpendicular Lines, and Angles 3-1 Transversal

Size: px
Start display at page:

Download "Unit 3 Notes: Parallel Lines, Perpendicular Lines, and Angles 3-1 Transversal"

Transcription

1 Unit 3 Notes: Parallel Lines, Perpendicular Lines, and Angles 3-1 Transversal REVIEW: *Postulates are Fundamentals of Geometry (Basic Rules) To mark line segments as congruent draw the same amount of tic marks on each one. Naming the angle in 4 ways: A 1) B 3) 1 2) ABC 4) CBA 1 * Transversal a line that intersects two unique lines at two different points B C There are five types of angles with regard to a transversal *Corresponding angles: Pairs of angles that are in the same location with regards to the transversal 1 and 5, 3 and 7, 2 and 6, 4 and 8 *Alternate Interior Angles: Pairs of angles on opposite sides of the transversal between the intersected lines. 3 and 6, 4 and 5 *Alternate Exterior Angles: Pairs of angles on opposite sides of the transversal outside the intersected lines. 1 and 8, 2 and 7 *Consecutive Interior Angles: Pairs of angles on the same side of the transversal Between the intersected line. aka same side interior 3 and 5, 4 and 6 *Consecutive Exterior Angles: Pairs of angles on the same side of the transversal outside the intersected lines. aka same side exterior 1 and 7, 2 and 8, 1

2 Using Transversal a Identify all pairs of Alternate Interior Angles: 4& 5, 3 & 6 Alternate Exterior Angles: 1 & 8, 2 & 7 Consecutive Interior Angles: 3 & 5, 4& 6 Consecutive Exterior Angles: 1 and 7, 2 and 8 Corresponding Angles: 1& 5, 2& 6, 3& 7, 4& 8 Using Transversal b Identify all pairs of Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Consecutive Exterior Angles Corresponding Angles Using Transversal c Identify all pairs of Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Consecutive Exterior Angles Corresponding Angles 2

3 3-2 Properties of Perpendicular Lines Perpendicular Lines: If two lines are perpendicular, then they intersect to form four right angles. The symbol for "is perpendicular to" is Linear Perpendicular Line Theorem: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. g g h h Perpendicular Complementary Theorem: If two sides of two adjacent acute angles are perpendicular, Then the angles are complementary. Developmental Proof of Linear Perpendicular Theorem: Statement Reason 1) 1 2 1) Given 2) 2) Given: 1 2, 1 & 2 are a linear pair. 3) 1 & 2 are a linear pair 3) Given 4) 4) Linear Pair Post Prove: g h 5) m 1 + m 1 = 180 5) Sub. Prop. of = g 6) 6) 7) m 1 = 90 7) Div. Prop of = 8) 8) 9) g h 9) Def. of Lines 1 2 h 3

4 3-3 Properties of Parallel Lines FOLLOWING THEOREMS ARE USEFUL IN PROOFS DEALING WITH PARALLEL LINES ***Corresponding Angles Postulate: - If a transversal intersects two parallel lines, then the corresponding angles formed are congruent. If a b, then all corresponding angles are congruent 1 5, 2 6, 3 7, 4 8 If m 1 = 52, using the corresponding angles postulate, Vertical angles theorem, and linear pair postulate, find the measure of the other angles. m 2= m 4= m 6= m 8= m 3= m 5= m 7= *Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then the Alternate Interior Angles are congruent. If a b, then all pairs of Alternate Interior Angles are congruent. 3 6, 4 5 If m 4 = (3x+4) and m 5 = 67, then what is the value of x? *Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then the Alternate Exterior Angles are congruent. If a b, then all pairs of alternate exterior angles are congruent. 1 8, 2 7 If m 1 = 32, then what is m 8? 4

5 *Consecutive Interior Angles Theorem If a transversal intersects two parallel lines, then the Consecutive Interior Angles are supplementary. If a b, then all pairs of consecutive interior angles are supplementary. m 3 + m 5 =180 m 4 + m 6 =180 If m 4 = (4x+12) and m 6 = 120, then what is the value of x and m 4? *Consecutive Exterior Angles Theorem If a transversal intersects two parallel lines, then the Consecutive Exterior Angles are supplementary. If a b, then all pairs of consecutive interior angles are supplementary. m 1 + m 7 =180 m 2 + m 8 =180 If m 1 = (5x+10) and m 7 = 60, then what is the value of x and m 1? Developmental proof of the alternate interior angles theorem Hint: Use the corr s Postulate Statements 1) a b 1) Reasons 2) 4 1 2) 3) 1 5 3) 4) 4 5 4) 5

6 Examples using geometric shapes: Find the value of x. 3-4 Proving Lines Parallel Converse theorems about transversals and parallel lines ***Converse to the corresponding angles postulate If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel *** If 1 5, then a b If m 1 = 2x + 66 and m 5 = 8x -24, then what is the value of x? and what is the measure of each angle? ***Converse to the Alternate interior angles theorem -If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.*** If 2 5, then a b If m 2 = 3x -4 and m 5 = 2x + 16, then what is the value of x?... and what is the measure of each angle? 6

7 ***Converse to the Alternate Exterior Angles Theorem-If two lines and a transversal form alternate exterior angles that are, then the lines are *** If 3 7, then a b If m 3 = 4x - 5 and m 7 = 2x + 37, what is the value of x? ***Converse to the Consecutive interior angles theorem -If two lines and a transversal form consecutive interior angles that are, then the lines are.*** If m 4 + m 5 =, then If m 4 = 2x -4 and m 5 = 3x 16, then what is the value of x?... and what is the measure of each angle? ***Converse to the Consecutive Exterior Angles Theorem -If two lines and a transversal form consecutive exterior angles that are, then the lines are.*** If m 1 + m 7 =, then 1 a 7 b If m 1 = 6x - 14 and m 7 = 3x 13, then what is the value of x? and what is the measure of each angle? 7

8 3-5 Transversal Parallel and Perpendicular Line Theorems Transitive Parallel Lines Theorem *** If two lines are parallel to the same line, then they are parallel to each other.*** If a b and b c, then a c. Parking Lot Theorem ***If two coplanar lines are perpendicular to the same line, then they are parallel to each other.*** If a d and b d, then a b Developmental proof of the parking lot theorem. Statements Reasons 1) a d, b d 1) 2) 1 and 2 are right s 2) 3) 1 2 3) 4) a b 4) Statements Reasons 1) a b, b c, and c d 1) 2) a c 2) 3) a d 3) 8

9 Converse to the Parking Lot Theorem ***In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.*** If a b and a d, then b d A ladder is an excellent example of all the theorems comparing perpendiculars with parallels. What can you conclude about a ladder for each given: 1) Given: The rungs are perpendicular to one side: a. Conclusion: 2) Given : Each side is perpendicular to the top rung: a. Conclusion: 3) Given: Each rung is parallel to the top rung: a. Conclusion: With the given information, we need to find the relationship between line a and line d. (DRAW A PICTURE) 1) a b, b c, c d What do we know about a and d? 2) a b, b c, c d What do we know about a and d? 3) a b, b c, c d What do we know about a and d? Indirect Proof Given: m 1 m 2 Prove: line k is not perpendicular to line m. 1 2 k m 9

10 3-6 Lines in the Coordinate Plane Finding an Equation of a Line The Point-Slope Equation A non-vertical line with slope m and containing a point(x 1, y 1 ) has the point-slope equation y y 1 = m(x x 1 ) Example 1 Write an equation for the line with slope 3 that contains the point (5, 2). Express the equation in slope-intercept form. y y 1 = m(x x 1 ) y 2 = 3(x 5) y 2 = 3x 15 y = 3x 13 Try This. Write an equation for each line with the given point and slope. Express the equation in slope-intercept form. a. (3, 5), m = 6 b. (1, 4), m = 2 3 We can also use the point-slope equation to find an equation of a line if we know any two points on the line. Example 2. We first find any two points on the line. Use (1, 1) and (2, 3). We next find the slope. m = = 2 1 = 2 10

11 We can now use point-slope equation to find an equation for the line. We can use either point. Using (1, 1) may make the computation easier. y y 1 = m(x x 1 ) y 1 = 2(x 1) y 1 = 2x 2 y = 2x 1 Try This. Write an equation for each line in slope-intercept form. g. h. 11

12 3-7 Slopes of Parallel and Perpendicular Lines Parallel lines are lines in the same plane that never intersect. All vertical lines are parallel. Non-vertical lines that are parallel are precisely those that have the same slope and different y- intercepts. The graphs below are for the linear equations y = 2x + 5 and y = 2x 3. The slope of each line is 2 and the y-intercepts are 5 and -3 so these lines are parallel. Example 1. Determine whether the lines of y = -3x + 4 and 6x + 2y = -10 are parallel. We must find each equation for y. y = -3x + 4 6x + 2y = -10 2y = -6x 10 y = -3x 5 The graphs of these lines have the same slope and different y-intercepts. Thus they are parallel. Try This. Decide whether the graphs of the equations are parallel. 1) 3x y = -5 and 5y 15x = 10 2) 4y = -12x + 16 and y = 3x + 4 Perpendicular Lines are lines that intersect to form a 90 angle (or a right angle). A vertical line and a horizontal line are perpendicular. Algebraically, the product of the slopes of perpendicular lines is -1. The slopes are 2 and 1 2 have a product of -1 so these lines are perpendicular. 12

13 Example 2 Tell whether the graphs of 3y = 9x + 3 and 6y + 2x = 6 are perpendicular lines. We first solve for y in each equation to find the slopes. 3y = 9x + 3 6y + 2x = 6 The slopes are 3 and 1 3 Y= 3x + 1 6y = -2x + 6 Y = 1 3 x + 1 The products of the slopes of these lines is 3(- 1 ) = -1. Thus the lines are perpendicular. 3 Try This. Tell whether the graphs of the equations are perpendicular. 1) 2y x = 2 and y = -2x + 4 2) 4y = 3x + 12 and -3x + 4y 2 = 0 Example 3 Write an equation for the line containing (1, 2) and perpendicular to the line y = 3x 1. The slope of the line y = 3x 1. The slope of the line y = 3x 1 is 3. The negative reciprocal of 3 is y y 1 = m(x x 1 ) y 2 = 1 (x 1) 3 y 2 = 1 3 x y = 1 3 x

14 Try This. Write an equation for the line containing the given point and perpendicular to the given line. a. (3, 2); y = 2x + 4 b. (-1, -3); x + 2y = 8 14

2 and 6 4 and 8 1 and 5 3 and 7

2 and 6 4 and 8 1 and 5 3 and 7 Geo Ch 3 Angles formed by Lines Parallel lines are two coplanar lines that do not intersect. Skew lines are that are not coplanar and do not intersect. Transversal is a line that two or more lines at different

More information

2. Write the point-slope form of the equation of the line passing through the point ( 2, 4) with a slope of 3. (1 point)

2. Write the point-slope form of the equation of the line passing through the point ( 2, 4) with a slope of 3. (1 point) Parallel and Perpendicular Lines Unit Test David Strong is taking this assessment. Multiple Choice 1. Which construction is illustrated above? a segment congruent to a given segment an angle congruent

More information

GEOMETRY Angles and Lines NAME Transversals DATE Per.

GEOMETRY Angles and Lines NAME Transversals DATE Per. GEOMETRY Angles and Lines NAME t l p 1 2 3 4 5 6 7 8 1. a) Which are the angles that are on the same side but opposite and interior to each exterior angle? 1 7 b) What letter do they appear to form? 2.

More information

Unit 2A: Angle Pairs and Transversal Notes

Unit 2A: Angle Pairs and Transversal Notes Unit 2A: Angle Pairs and Transversal Notes Day 1: Special angle pairs Day 2: Angle pairs formed by transversal through two nonparallel lines Day 3: Angle pairs formed by transversal through parallel lines

More information

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles.

Geometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles. Geometry Definitions, Postulates, and Theorems Chapter : Parallel and Perpendicular Lines Section.1: Identify Pairs of Lines and Angles Standards: Prepare for 7.0 Students prove and use theorems involving

More information

Geometry Tutor Worksheet 4 Intersecting Lines

Geometry Tutor Worksheet 4 Intersecting Lines Geometry Tutor Worksheet 4 Intersecting Lines 1 Geometry Tutor - Worksheet 4 Intersecting Lines 1. What is the measure of the angle that is formed when two perpendicular lines intersect? 2. What is the

More information

Geometry CP Constructions Part I Page 1 of 4. Steps for copying a segment (TB 16): Copying a segment consists of making segments.

Geometry CP Constructions Part I Page 1 of 4. Steps for copying a segment (TB 16): Copying a segment consists of making segments. Geometry CP Constructions Part I Page 1 of 4 Steps for copying a segment (TB 16): Copying a segment consists of making segments. Geometry CP Constructions Part I Page 2 of 4 Steps for bisecting a segment

More information

Notes Formal Geometry Chapter 3 Parallel and Perpendicular Lines

Notes Formal Geometry Chapter 3 Parallel and Perpendicular Lines Name Date Period Notes Formal Geometry Chapter 3 Parallel and Perpendicular Lines 3-1 Parallel Lines and Transversals and 3-2 Angles and Parallel Lines A. Definitions: 1. Parallel Lines: Coplanar lines

More information

Warmup pg. 137 #1-8 in the geo book 6 minutes to finish

Warmup pg. 137 #1-8 in the geo book 6 minutes to finish Chapter Three Test Friday 2/2 Warmup pg. 137 #1-8 in the geo book 6 minutes to finish 1 1 and 5, 2 and 5 3 and 4 1 and 2 1 and 5, 2 and 5 division prop of eq Transitive prop of congruency 16 = 4x x = 4

More information

3.2 Homework. Which lines or segments are parallel? Justify your answer with a theorem or postulate.

3.2 Homework. Which lines or segments are parallel? Justify your answer with a theorem or postulate. 3.2 Homework Which lines or segments are parallel? Justify your answer with a theorem or postulate. 1.) 2.) 3.) ; K o maj N M m/ll = 180 Using the given information, which lines, if any, can you conclude

More information

GH Chapter 3 Quiz Review (3.1, 3.2, 3.4, 3.5)

GH Chapter 3 Quiz Review (3.1, 3.2, 3.4, 3.5) Name: Class: Date: SHOW ALL WORK GH Chapter 3 Quiz Review (3.1, 3.2, 3.4, 3.5) Match each vocabulary term with its definition. (#1-5) a. parallel lines b. parallel planes c. perpendicular lines d. skew

More information

GEOMETRY POSTULATES AND THEOREMS. Postulate 1: Through any two points, there is exactly one line.

GEOMETRY POSTULATES AND THEOREMS. Postulate 1: Through any two points, there is exactly one line. GEOMETRY POSTULATES AND THEOREMS Postulate 1: Through any two points, there is exactly one line. Postulate 2: The measure of any line segment is a unique positive number. The measure (or length) of AB

More information

GEOMETRY R Unit 2: Angles and Parallel Lines

GEOMETRY R Unit 2: Angles and Parallel Lines GEOMETRY R Unit 2: Angles and Parallel Lines Day Classwork Homework Friday 9/15 Unit 1 Test Monday 9/18 Tuesday 9/19 Angle Relationships HW 2.1 Angle Relationships with Transversals HW 2.2 Wednesday 9/20

More information

3-2 Proving Lines Parallel. Objective: Use a transversal in proving lines parallel.

3-2 Proving Lines Parallel. Objective: Use a transversal in proving lines parallel. 3-2 Proving Lines Parallel Objective: Use a transversal in proving lines parallel. Objectives: 1) Identify angles formed by two lines and a transversal. 2) Prove and use properties of parallel. Page 132

More information

Unit 5, Lesson 5.2 Proving Theorems About Angles in Parallel Lines Cut by a Transversal

Unit 5, Lesson 5.2 Proving Theorems About Angles in Parallel Lines Cut by a Transversal Unit 5, Lesson 5.2 Proving Theorems About Angles in Parallel Lines Cut by a Transversal Think about all the angles formed by parallel lines intersected by a transversal. What are the relationships among

More information

Given: Prove: Proof: 2-9 Proving Lines Parallel

Given: Prove: Proof: 2-9 Proving Lines Parallel Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 5. Find x so that m n. Identify the postulate or theorem you used.

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Chapter 3 Maintaining Mathematical Proficiency Find the slope of the line.. y. y 3. ( 3, 3) y (, ) (, ) x x (, ) x (, ) ( 3, 3)... (, ) y (0, 0) 8 8 x x 8 8 y (, ) (, ) y (, ) (, 0) x Write an equation

More information

If lines m and n are parallel, we write. Transversal: A line that INTERSECTS two or more lines at 2

If lines m and n are parallel, we write. Transversal: A line that INTERSECTS two or more lines at 2 Unit 4 Lesson 1: Parallel Lines and Transversals Name: COMPLEMENTARY are angles to add up to 90 SUPPLEMENTARY are angles to add up to 180 These angles are also known as a LINEAR PAIR because they form

More information

Quarter 1 Study Guide Honors Geometry

Quarter 1 Study Guide Honors Geometry Name: Date: Period: Topic 1: Vocabulary Quarter 1 Study Guide Honors Geometry Date of Quarterly Assessment: Define geometric terms in my own words. 1. For each of the following terms, choose one of the

More information

3-1 Study Guide Parallel Lines and Transversals

3-1 Study Guide Parallel Lines and Transversals 3-1 Study Guide Parallel Lines and Transversals Relationships Between Lines and Planes When two lines lie in the same plane and do not intersect, they are parallel. Lines that do not intersect and are

More information

Given: Prove: Proof: 5-6 Proving Lines Parallel

Given: Prove: Proof: 5-6 Proving Lines Parallel Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 5. SHORT RESPONSE Find x so that m n. Show your work. 1. and are

More information

Unit 6: Connecting Algebra and Geometry Through Coordinates

Unit 6: Connecting Algebra and Geometry Through Coordinates Unit 6: Connecting Algebra and Geometry Through Coordinates The focus of this unit is to have students analyze and prove geometric properties by applying algebraic concepts and skills on a coordinate plane.

More information

GEOMETRY APPLICATIONS

GEOMETRY APPLICATIONS GEOMETRY APPLICATIONS Chapter 3: Parallel & Perpendicular Lines Name: Teacher: Pd: 0 Table of Contents DAY 1: (Ch. 3-1 & 3-2) SWBAT: Identify parallel, perpendicular, and skew lines. Identify the angles

More information

3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines

3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines 3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x + 2 3. y = x 3 4. y = 4 x + 3 3 November 2, 2015 3.4 Proofs with Perpendicular Lines Geometry 3.5 Equations of Parallel and Perpendicular Lines Day

More information

Given: Prove: Proof: 3-5 Proving Lines Parallel

Given: Prove: Proof: 3-5 Proving Lines Parallel Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 6. PROOF Copy and complete the proof of Theorem 3.5. 1. Given: j

More information

3.3 Prove Lines are Parallel

3.3 Prove Lines are Parallel Warm-up! Turn in your proof to me and pick up a different one, grade it on our 5 point scale! If it is not a 5 write on the paper what they need to do to improve it. Return to the proof writer! 1 2 3.3

More information

Geometry Unit 3 Equations of Lines/Parallel & Perpendicular Lines

Geometry Unit 3 Equations of Lines/Parallel & Perpendicular Lines Geometry Unit 3 Equations of Lines/Parallel & Perpendicular Lines Lesson Parallel Lines & Transversals Angles & Parallel Lines Slopes of Lines Assignment 174(14, 15, 20-37, 44) 181(11-19, 25, 27) *TYPO

More information

a triangle with all acute angles acute triangle angles that share a common side and vertex adjacent angles alternate exterior angles

a triangle with all acute angles acute triangle angles that share a common side and vertex adjacent angles alternate exterior angles acute triangle a triangle with all acute angles adjacent angles angles that share a common side and vertex alternate exterior angles two non-adjacent exterior angles on opposite sides of the transversal;

More information

(1) Page #1 24 all. (2) Page #7-21 odd, all. (3) Page #8 20 Even, Page 35 # (4) Page #1 8 all #13 23 odd

(1) Page #1 24 all. (2) Page #7-21 odd, all. (3) Page #8 20 Even, Page 35 # (4) Page #1 8 all #13 23 odd Geometry/Trigonometry Unit 1: Parallel Lines Notes Name: Date: Period: # (1) Page 25-26 #1 24 all (2) Page 33-34 #7-21 odd, 23 28 all (3) Page 33-34 #8 20 Even, Page 35 #40 44 (4) Page 60 61 #1 8 all #13

More information

When two (or more) parallel lines are cut by a transversal, the following angle relationships are true:

When two (or more) parallel lines are cut by a transversal, the following angle relationships are true: Lesson 8: Parallel Lines Two coplanar lines are said to be parallel if they never intersect. or any given point on the first line, its distance to the second line is equal to the distance between any other

More information

Geometry Cheat Sheet

Geometry Cheat Sheet Geometry Cheat Sheet Chapter 1 Postulate 1-6 Segment Addition Postulate - If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC. Postulate 1-7 Angle Addition Postulate -

More information

Geometry Midterm Review Vocabulary:

Geometry Midterm Review Vocabulary: Name Date Period Geometry Midterm Review 2016-2017 Vocabulary: 1. Points that lie on the same line. 1. 2. Having the same size, same shape 2. 3. These are non-adjacent angles formed by intersecting lines.

More information

5-5 Angles and Parallel Lines. In the figure, m 1 = 94. Find the measure of each angle. Tell which postulate(s) or theorem (s) you used.

5-5 Angles and Parallel Lines. In the figure, m 1 = 94. Find the measure of each angle. Tell which postulate(s) or theorem (s) you used. In the figure, m 1 = 94 Find the measure of each angle Tell which postulate(s) or theorem (s) you used 1 3 4 In the figure, angles 3 are corresponding Use the Corresponding Angles Postulate: If two parallel

More information

Geo - CH3 Prctice Test

Geo - CH3 Prctice Test Geo - CH3 Prctice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Identify the transversal and classify the angle pair 11 and 7. a. The transversal

More information

3-2 Angles and Parallel Lines. In the figure, m 1 = 94. Find the measure of each angle. Tell which postulate(s) or theorem (s) you used.

3-2 Angles and Parallel Lines. In the figure, m 1 = 94. Find the measure of each angle. Tell which postulate(s) or theorem (s) you used. In the figure, m 1 = 94. Find the measure of each angle. Tell which postulate(s) or theorem (s) you used. 7. ROADS In the diagram, the guard rail is parallel to the surface of the roadway and the vertical

More information

Chapter 1-2 Points, Lines, and Planes

Chapter 1-2 Points, Lines, and Planes Chapter 1-2 Points, Lines, and Planes Undefined Terms: A point has no size but is often represented by a dot and usually named by a capital letter.. A A line extends in two directions without ending. Lines

More information

Semester Test Topic Review. Correct Version

Semester Test Topic Review. Correct Version Semester Test Topic Review Correct Version List of Questions Questions to answer: What does the perpendicular bisector theorem say? What is true about the slopes of parallel lines? What is true about the

More information

Lesson 2-5: Proving Angles Congruent

Lesson 2-5: Proving Angles Congruent Lesson -5: Proving Angles Congruent Geometric Proofs Yesterday we discovered that solving an algebraic expression is essentially doing a proof, provided you justify each step you take. Today we are going

More information

Lesson 19: The Graph of a Linear Equation in Two Variables is a Line

Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Lesson 19: The Graph of a Linear Equation in Two Variables is a Line Classwork Exercises Theorem: The graph of a linear equation y = mx + b is a non-vertical line with slope m and passing through (0, b),

More information

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with

Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through

More information

You MUST know the big 3 formulas!

You MUST know the big 3 formulas! Name 3-13 Review Geometry Period Date Unit 3 Lines and angles Review 3-1 Writing equations of lines. Determining slope and y intercept given an equation y = mx + b Writing the equation of a line given

More information

Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets

Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of the way along each median

More information

Unit 8 Chapter 3 Properties of Angles and Triangles

Unit 8 Chapter 3 Properties of Angles and Triangles Unit 8 Chapter 3 Properties of Angles and Triangles May 16 7:01 PM Types of lines 1) Parallel Lines lines that do not (and will not) cross each other are labeled using matching arrowheads are always the

More information

CK-12 Geometry: Properties of Parallel Lines

CK-12 Geometry: Properties of Parallel Lines CK-12 Geometry: Properties of Parallel Lines Learning Objectives Use the Corresponding Angles Postulate. Use the Alternate Interior Angles Theorem. Use the Alternate Exterior Angles Theorem. Use Same Side

More information

3.4 Warm Up. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0

3.4 Warm Up. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0 3.4 Warm Up 1. Find the values of x and y. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0 3. m = -1, x = 5, and y = -4 3.3 Proofs with

More information

SHELBY COUNTY SCHOOLS: GEOMETRY 1ST NINE WEEKS OCTOBER 2015

SHELBY COUNTY SCHOOLS: GEOMETRY 1ST NINE WEEKS OCTOBER 2015 SHELBY COUNTY SCHOOLS: GEOMETRY 1ST NINE WEEKS OCTOBER 2015 Created to be taken with the ACT Quality Core Reference Sheet: Geometry. 1 P a g e 1. Which of the following is another way to name 1? A. A B.

More information

VOCABULARY. Chapters 1, 2, 3, 4, 5, 9, and 8. WORD IMAGE DEFINITION An angle with measure between 0 and A triangle with three acute angles.

VOCABULARY. Chapters 1, 2, 3, 4, 5, 9, and 8. WORD IMAGE DEFINITION An angle with measure between 0 and A triangle with three acute angles. Acute VOCABULARY Chapters 1, 2, 3, 4, 5, 9, and 8 WORD IMAGE DEFINITION Acute angle An angle with measure between 0 and 90 56 60 70 50 A with three acute. Adjacent Alternate interior Altitude of a Angle

More information

Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review

Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -

More information

Segment Addition Postulate: If B is BETWEEN A and C, then AB + BC = AC. If AB + BC = AC, then B is BETWEEN A and C.

Segment Addition Postulate: If B is BETWEEN A and C, then AB + BC = AC. If AB + BC = AC, then B is BETWEEN A and C. Ruler Postulate: The points on a line can be matched one to one with the REAL numbers. The REAL number that corresponds to a point is the COORDINATE of the point. The DISTANCE between points A and B, written

More information

Angles formed by Parallel Lines

Angles formed by Parallel Lines Worksheet Answers 1. a = 60, b = 120, c = 120 2. a = 90, b = 90, c = 50 3. a = 77, b = 52, c = 77, d = 51 4. a = 60, b = 120, c = 120, d= 115, e = 65, f =115, g = 125, h =55, I =125 5. a = 90, b = 163,

More information

Use the figure to name each of the following:

Use the figure to name each of the following: Name: Period Date Pre-AP Geometry Fall 2016 Semester Exam REVIEW *Chapter 1.1 Points Lines Planes Use the figure to name each of the following: 1) three non-collinear points 2) one line in three different

More information

Part I. Use Figure 1 to complete the sentence or phrase. 1) Ll and L are vertical angles.

Part I. Use Figure 1 to complete the sentence or phrase. 1) Ll and L are vertical angles. Geometry Chapter3Review2()\~ Name _ Please show all work for full credit. Period -- Date ------ Part. Use Figure to complete the sentence or phrase. ) Ll and L are vertical angles. 2) L2 and L are corresponding

More information

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ACCELERATED GEOMETRY (June 2014)

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ACCELERATED GEOMETRY (June 2014) UNIT: Chapter 1 Essentials of Geometry UNIT : How do we describe and measure geometric figures? Identify Points, Lines, and Planes (1.1) How do you name geometric figures? Undefined Terms Point Line Plane

More information

15. K is the midpoint of segment JL, JL = 4x - 2, and JK = 7. Find x, the length of KL, and JL. 8. two lines that do not intersect

15. K is the midpoint of segment JL, JL = 4x - 2, and JK = 7. Find x, the length of KL, and JL. 8. two lines that do not intersect Name: Period Date Pre-AP Geometry Fall Semester Exam REVIEW *Chapter 1.1 Points Lines Planes Use the figure to name each of the following: 1. three non-collinear points 2. one line in three different ways

More information

Introduction to Geometry

Introduction to Geometry Introduction to Geometry Objective A: Problems involving lines and angles Three basic concepts of Geometry are: Points are a single place represented by a dot A Lines are a collection of points that continue

More information

Geometry Ch 7 Quadrilaterals January 06, 2016

Geometry Ch 7 Quadrilaterals January 06, 2016 Theorem 17: Equal corresponding angles mean that lines are parallel. Corollary 1: Equal alternate interior angles mean that lines are parallel. Corollary 2: Supplementary interior angles on the same side

More information

Geometry Note-Sheet Overview

Geometry Note-Sheet Overview Geometry Note-Sheet Overview 1. Logic a. A mathematical sentence is a sentence that states a fact or contains a complete idea. Open sentence it is blue x+3 Contains variables Cannot assign a truth variable

More information

Name Date Class. Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent.

Name Date Class. Vertical angles are opposite angles formed by the intersection of two lines. Vertical angles are congruent. SKILL 43 Angle Relationships Example 1 Adjacent angles are pairs of angles that share a common vertex and a common side. Vertical angles are opposite angles formed by the intersection of two lines. Vertical

More information

Integrated Math, Part C Chapter 1 SUPPLEMENTARY AND COMPLIMENTARY ANGLES

Integrated Math, Part C Chapter 1 SUPPLEMENTARY AND COMPLIMENTARY ANGLES Integrated Math, Part C Chapter SUPPLEMENTARY AND COMPLIMENTARY ANGLES Key Concepts: By the end of this lesson, you should understand:! Complements! Supplements! Adjacent Angles! Linear Pairs! Vertical

More information

Videos, Constructions, Definitions, Postulates, Theorems, and Properties

Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording

More information

Geometry Midterm Review 2019

Geometry Midterm Review 2019 Geometry Midterm Review 2019 Name To prepare for the midterm: Look over past work, including HW, Quizzes, tests, etc Do this packet Unit 0 Pre Requisite Skills I Can: Solve equations including equations

More information

Geometry Quarter 1 Test - Study Guide.

Geometry Quarter 1 Test - Study Guide. Name: Geometry Quarter 1 Test - Study Guide. 1. Find the distance between the points ( 3, 3) and ( 15, 8). 2. Point S is between points R and T. P is the midpoint of. RT = 20 and PS = 4. Draw a sketch

More information

M2 GEOMETRY REVIEW FOR MIDTERM EXAM

M2 GEOMETRY REVIEW FOR MIDTERM EXAM M2 GEOMETRY REVIEW FOR MIDTERM EXAM #1-11: True or false? If false, replace the underlined word or phrase to make a true sentence. 1. Two lines are perpendicular if they intersect to form a right angle.

More information

Chapter 2: Introduction to Proof. Assumptions from Diagrams

Chapter 2: Introduction to Proof. Assumptions from Diagrams Chapter 2: Introduction to Proof Name: 2.6 Beginning Proofs Objectives: Prove a conjecture through the use of a two-column proof Structure statements and reasons to form a logical argument Interpret geometric

More information

Identify parallel lines, skew lines and perpendicular lines.

Identify parallel lines, skew lines and perpendicular lines. Learning Objectives Identify parallel lines, skew lines and perpendicular lines. Parallel Lines and Planes Parallel lines are coplanar (they lie in the same plane) and never intersect. Below is an example

More information

Hartmann HONORS Geometry Chapter 3 Formative Assessment * Required

Hartmann HONORS Geometry Chapter 3 Formative Assessment * Required Hartmann HONORS Geometry Chapter 3 Formative Assessment * Required 1. First Name * 2. Last Name * Vocabulary Match the definition to the vocabulary word. 3. Non coplanar lines that do not intersect. *

More information

Instructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code:

Instructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code: 306 Instructional Unit Area 1. Areas of Squares and The students will be -Find the amount of carpet 2.4.11 E Rectangles able to determine the needed to cover various plane 2. Areas of Parallelograms and

More information

GEOMETRY. Background Knowledge/Prior Skills. Knows ab = a b. b =

GEOMETRY. Background Knowledge/Prior Skills. Knows ab = a b. b = GEOMETRY Numbers and Operations Standard: 1 Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers, and number

More information

Unit 2 Language Of Geometry

Unit 2 Language Of Geometry Unit 2 Language Of Geometry Unit 2 Review Part 1 Name: Date: Hour: Lesson 1.2 1. Name the intersection of planes FGED and BCDE 2. Name another point on plane GFB 3. Shade plane GFB 4. Name the intersection

More information

4 Triangles and Congruence

4 Triangles and Congruence www.ck12.org CHAPTER 4 Triangles and Congruence Chapter Outline 4.1 TRIANGLE SUMS 4.2 CONGRUENT FIGURES 4.3 TRIANGLE CONGRUENCE USING SSS AND SAS 4.4 TRIANGLE CONGRUENCE USING ASA, AAS, AND HL 4.5 ISOSCELES

More information

Geometry Rules. Triangles:

Geometry Rules. Triangles: Triangles: Geometry Rules 1. Types of Triangles: By Sides: Scalene - no congruent sides Isosceles - 2 congruent sides Equilateral - 3 congruent sides By Angles: Acute - all acute angles Right - one right

More information

Geometry Review for Semester 1 Final Exam

Geometry Review for Semester 1 Final Exam Name Class Test Date POINTS, LINES & PLANES: Geometry Review for Semester 1 Final Exam Use the diagram at the right for Exercises 1 3. Note that in this diagram ST plane at T. The point S is not contained

More information

*Chapter 1.1 Points Lines Planes. Use the figure to name each of the following:

*Chapter 1.1 Points Lines Planes. Use the figure to name each of the following: Name: Period Date Pre- AP Geometry Fall 2015 Semester Exam REVIEW *Chapter 1.1 Points Lines Planes Use the figure to name each of the following: 1) three non-collinear points 2) one line in three different

More information

Identify relationships between lines and identify angles formed by transversals

Identify relationships between lines and identify angles formed by transversals NAME ~ ~------------------ Practice with Exa.mples For use with pages 129-134 DATE Identify relationships between lines and identify angles formed by transversals VOCABULARY Two lines are parallel lines

More information

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1

Geometry Vocabulary Math Fundamentals Reference Sheet Page 1 Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and

More information

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never

Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never 1stSemesterReviewTrueFalse.nb 1 Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never Classify each statement as TRUE or FALSE. 1. Three given points are always coplanar. 2. A

More information

Writing Linear Equations

Writing Linear Equations Writing Linear Equations Name: SHOW ALL WORK!!!!! For full credit, show all work on all problems! Write the slope-intercept form of the equation of each line. 1. 3x 2y = 16 2. 13x 11y = 12 3. 4x y = 1

More information

Parallel Lines cut by a Transversal Notes, Page 1

Parallel Lines cut by a Transversal Notes, Page 1 Angle Relationships Review 2 When two lines intersect, they form four angles with one point in 1 3 common. 4 Angles that are opposite one another are VERTIAL ANGLES. Some people say instead that VERTIAL

More information

Find the coordinates of the midpoint of the segment with the given endpoints. Use the midpoint formula.

Find the coordinates of the midpoint of the segment with the given endpoints. Use the midpoint formula. Concepts Geometry 1 st Semester Review Packet Use the figure to the left for the following questions. 1) Give two other names for AB. 2) Name three points that are collinear. 3) Name a point not coplanar

More information

Period: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means

Period: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means : Analytic Proofs of Theorems Previously Proved by Synthetic Means Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of

More information

UNIT 6: Connecting Algebra & Geometry through Coordinates

UNIT 6: Connecting Algebra & Geometry through Coordinates TASK: Vocabulary UNIT 6: Connecting Algebra & Geometry through Coordinates Learning Target: I can identify, define and sketch all the vocabulary for UNIT 6. Materials Needed: 4 pieces of white computer

More information

What could be the name of the plane represented by the top of the box?

What could be the name of the plane represented by the top of the box? hapter 02 Test Name: ate: 1 Use the figure below. What could be the name of the plane represented by the top of the box? E F I 2 Use the figure below. re points,, and E collinear or noncollinear? noncollinear

More information

CCM Unit 10 Angle Relationships

CCM Unit 10 Angle Relationships CCM6+7+ Unit 10 Angle Relationships ~ Page 1 CCM6+7+ 2015-16 Unit 10 Angle Relationships Name Teacher Projected Test Date Main Concepts Page(s) Unit 10 Vocabulary 2-6 Measuring Angles with Protractors

More information

Reteaching Transversals and Angle Relationships

Reteaching Transversals and Angle Relationships Name Date Class Transversals and Angle Relationships INV Transversals A transversal is a line that intersects two or more coplanar lines at different points. Line a is the transversal in the picture to

More information

POTENTIAL REASONS: Definition of Congruence:

POTENTIAL REASONS: Definition of Congruence: Sec 1.6 CC Geometry Triangle Proofs Name: POTENTIAL REASONS: Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point

More information

Geometry Agenda. Week 1.6 Objective Grade. Lines and Angles. Practice. Angles Formed by Parallel Lines Practice Proving Lines Parallel.

Geometry Agenda. Week 1.6 Objective Grade. Lines and Angles. Practice. Angles Formed by Parallel Lines Practice Proving Lines Parallel. Name Period Geometry Agenda Week 1.6 Objective Grade Monday September 26, Tuesday September 27, Wednesday September 28, Thursday September 29, Friday September 30, Lines and Angles Practice Angles Formed

More information

theorems & postulates & stuff (mr. ko)

theorems & postulates & stuff (mr. ko) theorems & postulates & stuff (mr. ko) postulates 1 ruler postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of

More information

CP Math 3 Page 1 of 34. Common Core Math 3 Notes - Unit 2 Day 1 Introduction to Proofs. Properties of Congruence. Reflexive. Symmetric If A B, then B

CP Math 3 Page 1 of 34. Common Core Math 3 Notes - Unit 2 Day 1 Introduction to Proofs. Properties of Congruence. Reflexive. Symmetric If A B, then B CP Math 3 Page 1 of 34 Common Core Math 3 Notes - Unit 2 Day 1 Introduction to Proofs Properties of Congruence Reflexive A A Symmetric If A B, then B A Transitive If A B and B C then A C Properties of

More information

Geometry Curriculum Map

Geometry Curriculum Map Geometry Curriculum Map Unit 1 st Quarter Content/Vocabulary Assessment AZ Standards Addressed Essentials of Geometry 1. What are points, lines, and planes? 1. Identify Points, Lines, and Planes 1. Observation

More information

10) the plane in two different ways Plane M or DCA (3 non-collinear points) Use the figure to name each of the following:

10) the plane in two different ways Plane M or DCA (3 non-collinear points) Use the figure to name each of the following: Name: Period Date Pre-AP Geometry Fall 2015 Semester Exam REVIEW *Chapter 1.1 Points Lines Planes Use the figure to name each of the following: 1) three non-collinear points (A, C, B) or (A, C, D) or any

More information

5 and Parallel and Perpendicular Lines

5 and Parallel and Perpendicular Lines Ch 3: Parallel and Perpendicular Lines 3 1 Properties of Parallel Lines 3 Proving Lines Parallel 3 3 Parallel and Perpendicular Lines 3 Parallel Lines and the Triangle Angles Sum Theorem 3 5 The Polgon

More information

Course: Geometry Level: Regular Date: 11/2016. Unit 1: Foundations for Geometry 13 Days 7 Days. Unit 2: Geometric Reasoning 15 Days 8 Days

Course: Geometry Level: Regular Date: 11/2016. Unit 1: Foundations for Geometry 13 Days 7 Days. Unit 2: Geometric Reasoning 15 Days 8 Days Geometry Curriculum Chambersburg Area School District Course Map Timeline 2016 Units *Note: unit numbers are for reference only and do not indicate the order in which concepts need to be taught Suggested

More information

Naming Angles. One complete rotation measures 360º. Half a rotation would then measure 180º. A quarter rotation would measure 90º.

Naming Angles. One complete rotation measures 360º. Half a rotation would then measure 180º. A quarter rotation would measure 90º. Naming Angles What s the secret for doing well in geometry? Knowing all the angles. An angle can be seen as a rotation of a line about a fixed point. In other words, if I were mark a point on a paper,

More information

Writing Equations of Parallel and Perpendicular Lines

Writing Equations of Parallel and Perpendicular Lines Writing Equations of Parallel and Perpendicular Lines The coordinate plane provides a connection between algebra and geometry. Postulates 17 and 18 establish a simple way to find lines that are parallel

More information

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled.

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled. Test Date: November 3, 2016 Format: Scored out of 100 points. 8 Multiple Choice (40) / 8 Short Response (60) Topics: Points, Angles, Linear Objects, and Planes Recognizing the steps and procedures for

More information

3 John likes to experiment with geometric. 4 Which of the following conjectures is true for

3 John likes to experiment with geometric. 4 Which of the following conjectures is true for 1 Rectangle ABCD is drawn on a coordinate plane. Each angle measures 90. The rectangle is reflected across the y axis, translated 9 units down, and then rotated 180 clockwise about the origin. What would

More information

Department: Course: Chapter 1

Department: Course: Chapter 1 Department: Course: 2016-2017 Term, Phrase, or Expression Simple Definition Chapter 1 Comprehension Support Point Line plane collinear coplanar A location in space. It does not have a size or shape The

More information

.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3)

.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3) Co-ordinate Geometry Co-ordinates Every point has two co-ordinates. (3, 2) x co-ordinate y co-ordinate Plot the following points on the plane..(3, 2) A (4, 1) D (2, 5) G (6, 3) B (3, 3) E ( 4, 4) H (6,

More information

Geometry Fundamentals Midterm Exam Review Name: (Chapter 1, 2, 3, 4, 7, 12)

Geometry Fundamentals Midterm Exam Review Name: (Chapter 1, 2, 3, 4, 7, 12) Geometry Fundamentals Midterm Exam Review Name: (Chapter 1, 2, 3, 4, 7, 12) Date: Mod: Use the figure at the right for #1-4 1. What is another name for plane P? A. plane AE B. plane A C. plane BAD D. plane

More information

GEOMETRY is the study of points in space

GEOMETRY is the study of points in space CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of

More information