Table of Contents. Unit 1: Similarity, Congruence, and Proofs. Answer Key...AK-1. Introduction... v

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2 These materials may not be reproduced for any purpose. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored, or recorded in any form without written permission from the publisher ISN opyright 2013 J. Weston Walch, Publisher Portland, ME Printed in the United States of merica WLH EDUTION

3 Table of ontents Introduction... v Unit 1: Similarity, ongruence, and Proofs Lesson 1: Investigating Properties of Dilations... U1-1 Lesson 2: onstructing Lines, Segments, and ngles... U1-25 Lesson 3: onstructing Polygons... U1-79 Lesson 4: Exploring ongruence... U1-125 Lesson 5: ongruent Triangles... U1-162 Lesson 6: Defining and pplying Similarity... U1-188 Lesson 7: Proving Similarity... U1-216 Lesson 8: Proving Theorems bout Lines and ngles... U1-272 Lesson 9: Proving Theorems bout Triangles... U1-317 Lesson 10: Proving Theorems bout Parallelograms... U1-409 nswer Key...K-1 iii Table of ontents

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5 Introduction Welcome to the GPS nalytic Geometry Student Resource ook. This book will help you learn how to use algebra, geometry, data analysis, and probability to solve problems. Each lesson builds on what you have already learned. s you participate in classroom activities and use this book, you will master important concepts that will help to prepare you for the EOT and for other mathematics assessments and courses. This book is your resource as you work your way through the nalytic Geometry course. It includes explanations of the concepts you will learn in class; math vocabulary and definitions; formulas and rules; and exercises so you can practice the math you are learning. Most of your assignments will come from your teacher, but this book will allow you to review what was covered in class, including terms, formulas, and procedures. In Unit 1: Similarity, ongruence, and Proofs, you will learn about dilations, and you will construct lines, segments, angles, polygons, and triangles. You will explore congruence and then define, apply, and prove similarity. Finally, you will prove theorems about lines, angles, triangles, and parallelograms. In Unit 2: Right Triangle Trigonometry, you will begin by exploring trigonometric ratios. Then you will go on to apply trigonometric ratios. In Unit 3: ircles and Volume, you will be introduced to circles and their angles and tangents. Then you will learn about inscribed polygons and circumscribed triangles by constructing them and proving properties of inscribed quadrilaterals. You will construct tangent lines and find arc lengths and areas of sectors. Finally, you will explain and apply area and volume formulas. In Unit 4: Extending the Number System, you will start working with the number system and rational exponents. Then you will perform operations with complex numbers and polynomials. In Unit 5: Quadratic Functions, you will begin by identifying and interpreting structures in expressions. You will use this information as you learn to create and solve quadratic equations in one variable, including taking the square root of both sides, factoring, completing the square, applying the quadratic formula, and solving quadratic inequalities. You will move on to solving quadratic equations in two or more variables, and solving systems v Introduction

6 of equations. You will learn to analyze quadratic functions and to build and transform them. Finally, you will solve problems by fitting quadratic functions to data. In Unit 6: Modeling Geometry, you will study the links between the two math disciplines, geometry and algebra, as you derive equations of a circle and a parabola. You will use coordinates to prove geometric theorems about circles and parabolas and solve systems of linear equations and circles. In Unit 7: pplications of Probability, you will explore the idea of events, including independent events, and conditional probability. Each lesson is made up of short sections that explain important concepts, including some completed examples. Each of these sections is followed by a few problems to help you practice what you have learned. The Words to Know section at the beginning of each lesson includes important terms introduced in that lesson. s you move through your nalytic Geometry course, you will become a more confident and skilled mathematician. We hope this book will serve as a useful resource as you learn. vi Introduction

7 UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations ommon ore Georgia Performance Standards M9 12.G.SRT.1a M9 12.G.SRT.1b Essential Questions 1. How are the preimage and image similar in dilations? 2. How are the preimage and image different in dilations? 3. When are dilations used in the real world? WORDS TO KNOW center of dilation collinear points compression congruency transformation corresponding sides dilation a point through which a dilation takes place; all the points of a dilated figure are stretched or compressed through this point points that lie on the same line a transformation in which a figure becomes smaller; compressions may be horizontal (affecting only horizontal lengths), vertical (affecting only vertical lengths), or both a transformation in which a geometric figure moves but keeps the same size and shape; a dilation where the scale factor is equal to 1 sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on. a transformation in which a figure is either enlarged or reduced by a scale factor in relation to a center point enlargement a dilation of a figure where the scale factor is greater than 1 non-rigid motion a transformation done to a figure that changes the figure s shape and/or size reduction a dilation where the scale factor is between 0 and 1 U1-1 Lesson 1: Investigating Properties of Dilations

8 rigid motion scale factor stretch a transformation done to a figure that maintains the figure s shape and size or its segment lengths and angle measures a multiple of the lengths of the sides from one figure to the transformed figure. If the scale factor is larger than 1, then the figure is enlarged. If the scale factor is between 0 and 1, then the figure is reduced. a transformation in which a figure becomes larger; stretches may be horizontal (affecting only horizontal lengths), vertical (affecting only vertical lengths), or both U1-2 Unit 1: Similarity, ongruence, and Proofs

9 Recommended Resources IXL Learning. Transformations: Dilations: Find the oordinates. This interactive website gives a series of problems and scores them immediately. If the user submits a wrong answer, a description and process for arriving at the correct answer are provided. These problems start with a graphed figure. Users are asked to input the coordinates of the dilated figure given a center and scale factor. IXL Learning. Transformations: Dilations: Graph the Image. This interactive website gives a series of problems and scores them immediately. If the user submits a wrong answer, a description and process for arriving at the correct answer are provided. These problems start with a graphed figure. Users are asked to draw a dilation of the figure on the screen using a point that can be dragged, given a center and scale factor. IXL Learning. Transformations: Dilations: Scale Factor and lassification. This interactive website gives a series of problems and scores them immediately. If the user submits a wrong answer, a description and process for arriving at the correct answer are provided. These problems start with a graphed preimage and image. Users are required to choose whether the figure is an enlargement or a reduction. Other problems ask users to enter the scale factor. Math Is Fun. Resizing. This website gives a brief explanation of the properties of dilations and how to perform them. The site also contains an interactive applet with which users can select a shape, a center point, and a scale factor. The computer then generates the dilated image. fter users explore the applet, they may answer eight multiple-choice questions in order to check understanding. U1-3 Lesson 1: Investigating Properties of Dilations

10 Lesson 1.1.1: Investigating Properties of Parallelism and the enter Introduction Think about resizing a window on your computer screen. You can stretch it vertically, horizontally, or at the corner so that it stretches both horizontally and vertically at the same time. These are non-rigid motions. Non-rigid motions are transformations done to a figure that change the figure s shape and/or size. These are in contrast to rigid motions, which are transformations to a figure that maintain the figure s shape and size, or its segment lengths and angle measures. Specifically, we are going to study non-rigid motions of dilations. Dilations are transformations in which a figure is either enlarged or reduced by a scale factor in relation to a center point. Key oncepts Dilations require a center of dilation and a scale factor. The center of dilation is the point about which all points are stretched or compressed. The scale factor of a figure is a multiple of the lengths of the sides from one figure to the transformed figure. Side lengths are changed according to the scale factor, k. The scale factor can be found by finding the distances of the sides of the preimage in relation to the image. Use a ratio of corresponding sides to find the scale factor: length of image side = scalefactor length of preimage side The scale factor, k, takes a point P and moves it along a line in relation to the center so that k P= P. U1-4 Unit 1: Similarity, ongruence, and Proofs

11 P k P = P P P is under a dilation of scale factor k through center. If the scale factor is greater than 1, the figure is stretched or made larger and is called an enlargement. ( transformation in which a figure becomes larger is also called a stretch.) If the scale factor is between 0 and 1, the figure is compressed or made smaller and is called a reduction. ( transformation in which a figure becomes smaller is also called a compression.) If the scale factor is equal to 1, the preimage and image are congruent. This is called a congruency transformation. ngle measures are preserved in dilations. The orientation is also preserved. The sides of the preimage are parallel to the corresponding sides of the image. The corresponding sides are the sides of two figures that lie in the same position relative to the figures. In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on. The notation of a dilation in the coordinate plane is given by D k (x, y) = (kx, ky). The scale factor is multiplied by each coordinate in the ordered pair. U1-5 Lesson 1: Investigating Properties of Dilations

12 The center of dilation is usually the origin, (0, 0). If a segment of the figure being dilated passes through the center of dilation, then the image segment will lie on the same line as the preimage segment. ll other segments of the image will be parallel to the corresponding preimage segments. The corresponding points in the preimage and image are collinear points, meaning they lie on the same line, with the center of dilation. V' T'U'V' is TUV under a dilation of scale factor k about center. V T' U' T U Properties of Dilations 1. Shape, orientation, and angles are preserved. 2. ll sides change by a single scale factor, k. 3. The corresponding preimage and image sides are parallel. 4. The corresponding points of the figure are collinear with the center of dilation. U1-6 Unit 1: Similarity, ongruence, and Proofs

13 Guided Practice Example 1 Is the following transformation a dilation? Justify your answer using the properties of dilations. y D ( 4, 4) D ( 2, 2) F (2, 2) E E (4, 2) (2, 1) F (4, 4) x 1. Verify that shape, orientation, and angles have been preserved from the preimage to the image. oth figures are triangles in the same orientation. D D E E F F The angle measures have been preserved. U1-7 Lesson 1: Investigating Properties of Dilations

14 2. Verify that the corresponding sides are parallel. m y (2 1) 1 1 = = ( 2 2) = 4 = and m DE x 4 therefore, DE DE. y (4 2) 2 1 = = = = x ( 4 4) 8 4 ; DE y inspection, EF EF because both lines are vertical; therefore, they have the same slope and are parallel. y [2 ( 2)] 4 = = y [4 ( 4)] 8 m = = 1 and m = = = = 1 DF x ( 2 2) 4 x ( 4 4) 8 therefore, DF DF. In fact, these two segments, DF and DF, lie on the same line. ll corresponding sides are parallel. DF ; 3. Verify that the distances of the corresponding sides have changed by a common scale factor, k. We could calculate the distances of each side, but that would take a lot of time. Instead, examine the coordinates and determine if the coordinates of the vertices have changed by a common scale factor. The notation of a dilation in the coordinate plane is given by D k (x, y) = (kx, ky). Divide the coordinates of each vertex to determine if there is a common scale factor. D( 2,2) D ( 4,4) x x 4 yd 4 = 2; 2 2 = y = 2 = D D D E(2,1) E (4,2) x x 4 ye 2 = = 2; = = 2 2 y 1 E E E F(2, 2) F (4, 4) x x 4 yf 4 = = 2; = 2 2 y 2 = F F F Each vertex s preimage coordinate is multiplied by 2 to create the corresponding image vertex. Therefore, the common scale factor is k = 2. U1-8 Unit 1: Similarity, ongruence, and Proofs

15 4. Verify that corresponding vertices are collinear with the center of dilation,. y D ( 4, 4) D ( 2, 2) F (2, 2) E (2, 1) E (4, 2) F (4, 4) x straight line can be drawn connecting the center with the corresponding vertices. This means that the corresponding vertices are collinear with the center of dilation. 5. Draw conclusions. The transformation is a dilation because the shape, orientation, and angle measures have been preserved. dditionally, the size has changed by a scale factor of 2. ll corresponding sides are parallel, and the corresponding vertices are collinear with the center of dilation. U1-9 Lesson 1: Investigating Properties of Dilations

16 Example 2 Is the following transformation a dilation? Justify your answer using the properties of dilations T (0, 5) U (6, 5) V (6, 0) U (9, 5) y V (9, 0) x 1. Verify that shape, orientation, and angles have been preserved from the preimage to the image. The preimage and image are both rectangles with the same orientation. The angle measures have been preserved since all angles are right angles. U1-10 Unit 1: Similarity, ongruence, and Proofs

17 2. Verify that the corresponding sides are parallel. TU is on the same line as TU ; therefore, V is on the same line as V ; therefore, TU TU. V V. y inspection, UV and U V are vertical; therefore, UV UV. T remains unchanged from the preimage to the image. ll corresponding sides are parallel. 3. Verify that the distances of the corresponding sides have changed by a common scale factor, k. Since the segments of the figure are on a coordinate plane and are either horizontal or vertical, find the distance by counting. In TUV : In TU V : TU = V = 9 TU = V = 6 UV = T = 5 UV = T = 5 The formula for calculating the scale factor is: length of imageside scalefactor= length of preimage side Start with the horizontal sides of the rectangle. TU 6 2 V 6 2 = = = = TU 9 3 V 9 3 oth corresponding horizontal sides have a scale factor of 2 3. Next, calculate the scale factor of the vertical sides. UV 5 = = UV 5 1 T 5 T = 5 = 1 oth corresponding vertical sides have a scale factor of 1. U1-11 Lesson 1: Investigating Properties of Dilations

18 4. Draw conclusions. The vertical corresponding sides have a scale factor that is not consistent with the scale factor of 2 for the horizontal sides. Since all 3 corresponding sides do not have the same common scale factor, the transformation is NOT a dilation. Example 3 The following transformation represents a dilation. What is the scale factor? Does this indicate enlargement, reduction, or congruence? ' ' ' Determine the scale factor. Start with the ratio of one set of corresponding sides. length of imageside scalefactor= length of preimage side = = 10 4 The scale factor appears to be 1 4. U1-12 Unit 1: Similarity, ongruence, and Proofs

19 2. Verify that the other sides maintain the same scale factor = = and = =. Therefore, = = = and the scale factor, k, is Determine the type of dilation that has occurred. If k > 1, then the dilation is an enlargement. If 0 < k < 1, then the dilation is a reduction. If k = 1, then the dilation is a congruency transformation. 1 Since k =, k is between 0 and 1, or 0 < k < 1. 4 The dilation is a reduction. U1-13 Lesson 1: Investigating Properties of Dilations

20 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations Practice 1.1.1: Investigating Properties of Parallelism and the enter Determine whether each of the following transformations represents a dilation. Justify your answer using the properties of dilations. 1. ompare ST to S T. y S(0, 6) S (0, 3) T T -2 (4, 0) (8, 0) x 2. ompare quadrilateral STUV to quadrilateral S'T'U'V'. U1-14 Unit 1: Similarity, ongruence, and Proofs S(0, 6) S (0, 3) V T (4, 0) T (8, 0) ( 8, 0) V -1 ( 4, 0) y U (0, 2) U (0, 6) x continued

21 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations 3. ompare PQR to PQ R. y P (0, 7) Q (8, 7) Q (10, 7) R R (8, 0) (10, 0) x 4. ompare TUV to T UV. y T (0, 8) T(0, 5) U (3, 5) U (4.8, 8) V V (3, 0) (4.8, 0) x continued U1-15 Lesson 1: Investigating Properties of Dilations

22 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations For problems 5 and 6, the following transformations represent dilations. Determine the scale factor and whether the dilation is an enlargement, a reduction, or a congruency transformation. 5. O 7.8 O M N N 4.5 M 6. P Q Q 2 5 R R P Use the given information in each problem that follows to answer the questions. 7. triangle with vertices D, E, and F has side lengths as follows: DE = 12.2, EF = 7.6, and FD = 8.4. If the image is dilated through the center (0, 0) and now has side lengths of DE =, EF =, and FD =, what is the scale factor? Does this dilation indicate an enlargement, a reduction, or a congruency transformation? Explain. U1-16 Unit 1: Similarity, ongruence, and Proofs continued

23 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations 8. company makes triangular wedges used to install laminate flooring. ustomers have complained that the wedge is too small. The company s designers propose dilating the wedge. The drawing below shows the side views of the original wedge and the dilated wedge. What is the scale factor? Does the scale factor represent an enlargement, a reduction, or a congruency transformation? Explain. y Q (0, 2.5) Q (0, 3.125) R R (6, 0) (7.5, 0) x continued U1-17 Lesson 1: Investigating Properties of Dilations

24 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations 9. The laminate flooring designers proposed another wedge design, shown below. Does this design represent a dilation? Explain. y S (0, 2.5) T (6, 0) T (7.5, 0) x 10. neighborhood committee is planning a new community pool. The committee has proposed a design for the pool. The design consists of two rectangles. The inner rectangle is the pool, and has been dilated about (0, 0) to create the concrete walkway that will border the pool. The vertices of the pool are P ( 2, 4), Q (2, 4), R (2, 4), and S ( 2, 4). The vertices of the outside edge of the concrete walkway are P' ( 3, 6), Q' (3, 6), R' (3, 6), and S' ( 3, 6). What is the scale factor? Does this represent an enlargement, a reduction, or a congruency transformation? Explain. U1-18 Unit 1: Similarity, ongruence, and Proofs

25 Lesson 1.1.2: Investigating Scale Factors Introduction figure is dilated if the preimage can be mapped to the image using a scale factor through a center point, usually the origin. You have been determining if figures have been dilated, but how do you create a dilation? If the dilation is centered about the origin, use the scale factor and multiply each coordinate in the figure by that scale factor. If a distance is given, multiply the distance by the scale factor. Key oncepts The notation is as follows: Dk ( x, y) = ( kxky, ). Multiply each coordinate of the figure by the scale factor when the center is at (0, 0). y k D = D D (x, y) D (kx, ky) x The lengths of each side in a figure also are multiplied by the scale factor. If you know the lengths of the preimage figure and the scale factor, you can calculate the lengths of the image by multiplying the preimage lengths by the scale factor. Remember that the dilation is an enlargement if k > 1, a reduction if 0 < k < 1, and a congruency transformation if k = 1. U1-19 Lesson 1: Investigating Properties of Dilations

26 Guided Practice Example 1 If has a length of 3 units and is dilated by a scale factor of 2.25, what is the length of? Does this represent an enlargement or a reduction? 1. To determine the length of, multiply the scale factor by the length of the segment. = 3; k = 2.25 '' = k '' = = 6.75 is 6.75 units long. 2. Determine the type of dilation. Since the scale factor is greater than 1, the dilation is an enlargement. Example 2 triangle has vertices G (2, 3), H ( 6, 2), and J (0, 4). If the triangle is dilated by a scale factor of 0.5 through center (0, 0), what are the image vertices? Draw the preimage and image on the coordinate plane. 1. Start with one vertex and multiply each coordinate by the scale factor, k. D k = (kx, ky) G' = D 0.5 [G (2, 3)] = D 0.5 (0.5 2, 0.5 3) = (1, 1.5) 2. Repeat the process with another vertex. Multiply each coordinate of the vertex by the scale factor. H' = D 0.5 [H ( 6, 2)] = D 0.5 (0.5 6, 0.5 2) = ( 3, 1) U1-20 Unit 1: Similarity, ongruence, and Proofs

27 3. Repeat the process for the last vertex. Multiply each coordinate of the vertex by the scale factor. J' = D 0.5 [ J (0, 4)] = D 0.5 (0.5 0, 0.5 4) = (0, 2) 4. List the image vertices. G' (1, 1.5) H' ( 3, 1) J' (0, 2) 5. Draw the preimage and image on the coordinate plane. y H ( 6, 2) H ( 3, 1) G (1, 1.5) -2-3 G (2, 3) J (0, 4) J (0, 2) x U1-21 Lesson 1: Investigating Properties of Dilations

28 Example 3 What are the side lengths of D EF with a scale factor of 2.5 given the preimage and image below and the information that DE = 1, EF = 9.2, and FD = 8.6? D E D 1 E F F 1. hoose a side to start with and multiply the scale factor (k) by that side length. DE = 1; k = 2.5 D'E' = k DE D'E' = = hoose a second side and multiply the scale factor by that side length. EF = 9.2; k = 2.5 E'F' = k EF E'F' = = 23 U1-22 Unit 1: Similarity, ongruence, and Proofs

29 3. hoose the last side and multiply the scale factor by that side length. FD = 8.6; k = 2.5 F'D' = k FD F'D' = = Label the figure with the side lengths. D 2.5 E D 1 E F F U1-23 Lesson 1: Investigating Properties of Dilations

30 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 1: Investigating Properties of Dilations Practice 1.1.2: Investigating Scale Factors Determine the lengths of the dilated segments given the preimage length and the scale factor. 1. is 13.5 units long and the segment is dilated by a scale factor of k = FG is 19 units long and the segment is dilated by a scale factor of k = QR is 5.8 units long and the segment is dilated by a scale factor of 80%. 4. VW is 27 5 units long and the segment is dilated by a scale factor of 1 9. Determine the image vertices of the dilations given a center and scale factor. 5. TUV has the following vertices: T ( 9, 3), U ( 6, 6), and V ( 2, 3). What are the vertices under a dilation with a center at (0, 0) and a scale factor of 1 3? 6. DE has the following vertices: ( 1, 0), D ( 5, 6), and E (3, 4). What are the vertices under a dilation with a center at (0, 0) and a scale factor of 2? 7. NOP has the following vertices: N ( 6, 2), O (3, 5), and P (4, 8). What are the vertices under a dilation with a center at (0, 0) and a scale factor of 160%? 8. EFG has the following vertices: E (4, 9), F (5, 3), and G (9, 10). What are the vertices under a dilation with a center at (0, 0) and a scale factor of 30%? 9. IJK has the following vertices: I (6, 5), J (2, 2), and K ( 3, 4). What are the final vertices after 2 successive dilations with a center at (0, 0) and a scale factor of 3 4? What is the scale factor from IJK to I JK? 10. Jenelle is sophomore class president. She s creating a poster for spirit week. First, she drew the design on graph paper. Then she projected the design onto a wall where she d taped a giant sheet of poster paper. The projector increased the image by 960%. If the original poster design is 7.5 inches by 10 inches, what are the dimensions of the full-size poster? U1-24 Unit 1: Similarity, ongruence, and Proofs

31 UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 2: onstructing Lines, Segments, and ngles ommon ore Georgia Performance Standard M9 12.G.O.12 Essential Questions 1. What is the difference between sketching geometric figures, drawing geometric figures, and constructing geometric figures? 2. What tools are used with geometric constructions and why? 3. How can you justify a construction was done correctly? WORDS TO KNOW altitude angle bisect compass congruent construct construction drawing endpoint the perpendicular line from a vertex of a figure to its opposite side; height two rays or line segments sharing a common endpoint; the symbol used is to cut in half an instrument for creating circles or transferring measurements that consists of two pointed branches joined at the top by a pivot having the same shape, size, or angle to create a precise geometric representation using a straightedge along with either patty paper (tracing paper), a compass, or a reflecting device a precise representation of a figure using a straightedge and a compass, patty paper and a straightedge, or a reflecting device and a straightedge a precise representation of a figure, created with measurement tools such as a protractor and a ruler either of two points that mark the ends of a line segment; a point that marks the end of a ray U1-25 Lesson 2: onstructing Lines, Segments, and ngles

32 equidistant line median of a triangle midpoint midsegment parallel lines perpendicular bisector perpendicular lines ray segment sketch straightedge the same distance from a reference point the set of points between two points P and Q and the infinite number of points that continue beyond those points the segment joining the vertex to the midpoint of the opposite side a point on a line segment that divides the segment into two equal parts a line segment joining the midpoints of two sides of a figure lines that either do not share any points and never intersect, or share all points a perpendicular line constructed through the midpoint of a segment two lines that intersect at a right angle (90 ) a line with only one endpoint a part of a line that is noted by two endpoints a quickly done representation of a figure; a rough approximation of a figure a bar or strip of wood, plastic, or metal having at least one long edge of reliable straightness U1-26 Unit 1: Similarity, ongruence, and Proofs

33 Recommended Resources Math Open Reference. isecting an angle. This site provides step-by-step instructions for bisecting an angle with a compass and straightedge. Math Open Reference. opying a line segment. This site provides step-by-step instructions for copying a line segment with a compass and straightedge. Math Open Reference. opying an angle. This site provides step-by-step instructions for copying an angle with a compass and straightedge. Math Open Reference. Perpendicular at a point on a line. This site provides step-by-step instructions for constructing a perpendicular line at a point on the given line. Math Open Reference. Perpendicular bisector of a line segment. This site provides step-by-step instructions for constructing a perpendicular bisector of a line segment using a compass and straightedge. Math Open Reference. Perpendicular to a line from an external point. This site provides step-by-step instructions for constructing a perpendicular line from a point not on the given line. U1-27 Lesson 2: onstructing Lines, Segments, and ngles

34 Lesson 1.2.1: opying Segments and ngles Introduction Two basic instruments used in geometry are the straightedge and the compass. straightedge is a bar or strip of wood, plastic, or metal that has at least one long edge of reliable straightness, similar to a ruler, but without any measurement markings. compass is an instrument for creating circles or transferring measurements. It consists of two pointed branches joined at the top by a pivot. It is believed that during early geometry, all geometric figures were created using just a straightedge and a compass. Though technology and computers abound today to help us make sense of geometry problems, the straightedge and compass are still widely used to construct figures, or create precise geometric representations. onstructions allow you to draw accurate segments and angles, segment and angle bisectors, and parallel and perpendicular lines. Key oncepts geometric figure precisely created using only a straightedge and compass is called a construction. straightedge can be used with patty paper (tracing paper) or a reflecting device to create precise representations. onstructions are different from drawings or sketches. drawing is a precise representation of a figure, created with measurement tools such as a protractor and a ruler. sketch is a quickly done representation of a figure or a rough approximation of a figure. When constructing figures, it is very important not to erase your markings. Markings show that your figure was constructed and not measured and drawn. n endpoint is either of two points that mark the ends of a line, or the point that marks the end of a ray. line segment is a part of a line that is noted by two endpoints. n angle is formed when two rays or line segments share a common endpoint. constructed figure and the original figure are congruent; they have the same shape, size, or angle. Follow the steps outlined below and on the next page to copy a segment and an angle. U1-28 Unit 1: Similarity, ongruence, and Proofs

35 opying a Segment Using a ompass 1. To copy, first make an endpoint on your paper. Label the endpoint. 2. Put the sharp point of your compass on endpoint. Open the compass until the pencil end touches endpoint. 3. Without changing your compass setting, put the sharp point of your compass on endpoint. Make a large arc. 4. Use your straightedge to connect endpoint to any point on your arc. 5. Label the point of intersection of the arc and your segment D. Do not erase any of your markings. is congruent to D. opying a Segment Using Patty Paper 1. To copy, place your sheet of patty paper over the segment. 2. Mark the endpoints of the segment on the patty paper. Label the endpoints and D. 3. Use your straightedge to connect points and D. is congruent to D. U1-29 Lesson 2: onstructing Lines, Segments, and ngles

36 opying an ngle Using a ompass 1. To copy, first make a point to represent the vertex on your paper. Label the vertex E. 2. From point E, draw a ray of any length. This will be one side of the constructed angle. 3. Put the sharp point of the compass on vertex of the original angle. Set the compass to any width that will cross both sides of the original angle. 4. Draw an arc across both sides of. Label where the arc intersects the angle as points and. 5. Without changing the compass setting, put the sharp point of the compass on point E. Draw a large arc that intersects the ray. Label the point of intersection as F. 6. Put the sharp point of the compass on point of the original angle and set the width of the compass so it touches point. 7. Without changing the compass setting, put the sharp point of the compass on point F and make an arc that intersects the arc in step 5. Label the point of intersection as D. 8. Draw a ray from point E to point D. Do not erase any of your markings. is congruent to E. opying an ngle Using Patty Paper 1. To copy, place your sheet of patty paper over the angle. 2. Mark the vertex of the angle. Label the vertex E. 3. Use your straightedge to trace each side of. is congruent to E. U1-30 Unit 1: Similarity, ongruence, and Proofs

37 Guided Practice Example 1 opy the following segment using only a compass and a straightedge. M N 1. Make an endpoint on your paper. Label the endpoint P. Original segment onstruction M N P 2. Put the sharp point of your compass on endpoint M. Open the compass until the pencil end touches endpoint N. Original segment onstruction M N P U1-31 Lesson 2: onstructing Lines, Segments, and ngles

38 3. Without changing your compass setting, put the sharp point of your compass on endpoint P. Make a large arc. Original segment onstruction M N P 4. Use your straightedge to connect endpoint P to any point on your arc. Original segment onstruction M N P 5. Label the point of intersection of the arc and your segment Q. Original segment onstruction M N Q Do not erase any of your markings. MN is congruent to PQ. P U1-32 Unit 1: Similarity, ongruence, and Proofs

39 Example 2 opy the following angle using only a compass and a straightedge. J 1. Make a point to represent vertex J. Label the vertex R. Original angle onstruction J R 2. From point R, draw a ray of any length. This will be one side of the constructed angle. Original angle onstruction J R U1-33 Lesson 2: onstructing Lines, Segments, and ngles

40 3. Put the sharp point of the compass on vertex J of the original angle. Set the compass to any width that will cross both sides of the original angle. Original angle onstruction J R 4. Draw an arc across both sides of J. Label where the arc intersects the angle as points K and L. Original angle onstruction K L J R 5. Without changing the compass setting, put the sharp point of the compass on point R. Draw a large arc that intersects the ray. Label the point of intersection as S. Original angle onstruction K L S J R U1-34 Unit 1: Similarity, ongruence, and Proofs

41 6. Put the sharp point of the compass on point L of the original angle and set the width of the compass so it touches point K. Original angle onstruction K L S J R 7. Without changing the compass setting, put the sharp point of the compass on point S and make an arc that intersects the arc you drew in step 5. Label the point of intersection as T. Original angle onstruction K T L S J R 8. Draw a ray from point R to point T. Original angle onstruction K T L S J Do not erase any of your markings. J is congruent to R. R U1-35 Lesson 2: onstructing Lines, Segments, and ngles

42 Example 3 Use the given line segment to construct a new line segment with length Use your straightedge to draw a long ray. Label the endpoint. 2. Put the sharp point of your compass on endpoint of the original segment. Open the compass until the pencil end touches. 3. Without changing your compass setting, put the sharp point of your compass on and make a large arc that intersects your ray. 4. Mark the point of intersection as point D. D U1-36 Unit 1: Similarity, ongruence, and Proofs

43 5. Without changing your compass setting, put the sharp point of your compass on D and make a large arc that intersects your ray. D 6. Mark the point of intersection as point E. D E Do not erase any of your markings. E = 2 U1-37 Lesson 2: onstructing Lines, Segments, and ngles

44 Example 4 Use the given angle to construct a new angle equal to Follow the steps from Example 2 to copy. Label the vertex of the copied angle G. 2. Put the sharp point of the compass on vertex of the original angle. Set the compass to any width that will cross both sides of the original angle. 3. Draw an arc across both sides of. Label where the arc intersects the angle as points and. U1-38 Unit 1: Similarity, ongruence, and Proofs

45 4. Without changing the compass setting, put the sharp point of the compass on G. Draw a large arc that intersects one side of your newly constructed angle. Label the point of intersection H. G H 5. Put the sharp point of the compass on of the original angle and set the width of the compass so it touches. 6. Without changing the compass setting, put the sharp point of the compass on point H and make an arc that intersects the arc created in step 4. Label the point of intersection as J. H G J U1-39 Lesson 2: onstructing Lines, Segments, and ngles

46 7. Draw a ray from point G to point J. H G J Do not erase any of your markings. G= + Example 5 Use the given segments to construct a new segment equal to D. D 1. Draw a ray longer than that of. Label the endpoint M. M 2. Follow the steps from Example 3 to copy onto the ray. Label the second endpoint P. M P U1-40 Unit 1: Similarity, ongruence, and Proofs

47 3. Put the sharp point of the compass on endpoint M of the ray. opy segment D onto the same ray. Label the endpoint N. M N P Do not erase any of your markings. NP = D ngles can be subtracted in the same way. U1-41 Lesson 2: onstructing Lines, Segments, and ngles

48 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 2: onstructing Lines, Segments, and ngles Practice 1.2.1: opying Segments and ngles opy the following segments using a straightedge and a compass D 3. E F Use the line segments from problems 1 3 to construct the line segments described in problems 4 and D EF U1-42 Unit 1: Similarity, ongruence, and Proofs continued

49 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 2: onstructing Lines, Segments, and ngles opy the following angles using a straightedge and a compass. 6. G 7. H 8. J Use the angles from problems 6 8 to construct the angles described in problems 9 and H G 10. G+ J U1-43 Lesson 2: onstructing Lines, Segments, and ngles

50 Lesson 1.2.2: isecting Segments and ngles Introduction Segments and angles are often described with measurements. Segments have lengths that can be measured with a ruler. ngles have measures that can be determined by a protractor. It is possible to determine the midpoint of a segment. The midpoint is a point on the segment that divides it into two equal parts. When drawing the midpoint, you can measure the length of the segment and divide the length in half. When constructing the midpoint, you cannot use a ruler, but you can use a compass and a straightedge (or patty paper and a straightedge) to determine the midpoint of the segment. This procedure is called bisecting a segment. To bisect means to cut in half. It is also possible to bisect an angle, or cut an angle in half using the same construction tools. midsegment is created when two midpoints of a figure are connected. triangle has three midsegments. Key oncepts isecting a Segment segment bisector cuts a segment in half. Each half of the segment measures exactly the same length. point, line, ray, or segment can bisect a segment. point on the bisector is equidistant, or is the same distance, from either endpoint of the segment. The point where the segment is bisected is called the midpoint of the segment. isecting a Segment Using a ompass 1. To bisect, put the sharp point of your compass on endpoint. Open the compass wider than half the distance of. 2. Make a large arc intersecting. 3. Without changing your compass setting, put the sharp point of the compass on endpoint. Make a second large arc. It is important that the arcs intersect each other in two places. 4. Use your straightedge to connect the points of intersection of the arcs. 5. Label the midpoint of the segment. Do not erase any of your markings. is congruent to. U1-44 Unit 1: Similarity, ongruence, and Proofs

51 isecting a Segment Using Patty Paper 1. Use a straightedge to construct on patty paper. 2. Fold the patty paper so point meets point. e sure to crease the paper. 3. Unfold the patty paper. 4. Use your straightedge to mark the midpoint of. 5. Label the midpoint of the segment. is congruent to. isecting an ngle n angle bisector cuts an angle in half. Each half of the angle has exactly the same measure. line or ray can bisect an angle. point on the bisector is equidistant, or is the same distance, from either side of the angle. isecting an ngle Using a ompass 1. To bisect, put the sharp point of the compass on the vertex of the angle. 2. Draw a large arc that passes through each side of the angle. 3. Label where the arc intersects the angle as points and. 4. Put the sharp point of the compass on point. Open the compass wider than half the distance from to. 5. Make a large arc. 6. Without changing the compass setting, put the sharp point of the compass on. 7. Make a second large arc. It is important that the arcs intersect each other in two places. 8. Use your straightedge to create a ray connecting the points of intersection of the arcs with the vertex of the angle,. 9. Label a point, D, on the ray. Do not erase any of your markings. D is congruent to D. U1-45 Lesson 2: onstructing Lines, Segments, and ngles

52 isecting an ngle Using Patty Paper 1. Use a straightedge to construct on patty paper. 2. Fold the patty paper so the sides of line up. e sure to crease the paper. 3. Unfold the patty paper. 4. Use your straightedge to mark the crease line with a ray. 5. Label a point, D, on the ray. D is congruent to D. U1-46 Unit 1: Similarity, ongruence, and Proofs

53 Guided Practice Example 1 Use a compass and straightedge to find the midpoint of D. Label the midpoint of the segment M. D 1. opy the segment and label it D. D 2. Make a large arc intersecting D. Put the sharp point of your compass on endpoint. Open the compass wider than half the distance of D. Draw the arc. D U1-47 Lesson 2: onstructing Lines, Segments, and ngles

54 3. Make a second large arc. Without changing your compass setting, put the sharp point of the compass on endpoint D, then make the second arc. It is important that the arcs intersect each other in two places. D 4. onnect the points of intersection of the arcs. Use your straightedge to connect the points of intersection. D U1-48 Unit 1: Similarity, ongruence, and Proofs

55 5. Label the midpoint of the segment M. M D Do not erase any of your markings. M is congruent to MD. Example 2 onstruct a segment whose measure is 1 4 the length of PQ. P Q 1. opy the segment and label it PQ. P Q U1-49 Lesson 2: onstructing Lines, Segments, and ngles

56 2. Make a large arc intersecting PQ. Put the sharp point of your compass on endpoint P. Open the compass wider than half the distance of PQ. Draw the arc. P Q 3. Make a second large arc. Without changing your compass setting, put the sharp point of the compass on endpoint Q, then make the second arc. It is important that the arcs intersect each other in two places. P Q U1-50 Unit 1: Similarity, ongruence, and Proofs

57 4. onnect the points of intersection of the arcs. Use your straightedge to connect the points of intersection. Label the midpoint of the segment M. P M Q PM is congruent to MQ. PM and MQ are both 1 2 the length of PQ. 5. Find the midpoint of PM. Make a large arc intersecting PM. Put the sharp point of your compass on endpoint P. Open the compass wider than half the distance of PM. Draw the arc. P M Q U1-51 Lesson 2: onstructing Lines, Segments, and ngles

58 6. Make a second large arc. Without changing your compass setting, put the sharp point of the compass on endpoint M, and then draw the second arc. It is important that the arcs intersect each other in two places. P M Q 7. onnect the points of intersection of the arcs. Use your straightedge to connect the points of intersection. Label the midpoint of the smaller segment N. P N M Q Do not erase any of your markings. PN is congruent to NM. PN is 1 the length of PQ. 4 U1-52 Unit 1: Similarity, ongruence, and Proofs

59 Example 3 Use a compass and a straightedge to bisect an angle. 1. Draw an angle and label the vertex J. J 2. Make a large arc intersecting the sides of J. Put the sharp point of the compass on the vertex of the angle and swing the compass so that it passes through each side of the angle. Label where the arc intersects the angle as points L and M. L J M U1-53 Lesson 2: onstructing Lines, Segments, and ngles

60 3. Find a point that is equidistant from both sides of J. Put the sharp point of the compass on point L. Open the compass wider than half the distance from L to M. Make an arc beyond the arc you made for points L and M. L J Without changing the compass setting, put the sharp point of the compass on M. Make a second arc that crosses the arc you just made. It is important that the arcs intersect each other. Label the point of intersection N. L M N J M U1-54 Unit 1: Similarity, ongruence, and Proofs

61 4. Draw the angle bisector. Use your straightedge to create a ray connecting the point N with the vertex of the angle, J. L N J Do not erase any of your markings. LJN is congruent to NJM. M Example 4 onstruct an angle whose measure is 3 4 the measure of S. S 1. opy the angle and label the vertex S. S U1-55 Lesson 2: onstructing Lines, Segments, and ngles

62 2. Make a large arc intersecting the sides of S. Put the sharp point of the compass on the vertex of the angle and swing the compass so that it passes through each side of the angle. Label where the arc intersects the angle as points T and U. T S U 3. Find a point that is equidistant from both sides of S. Put the sharp point of the compass on point T. Open the compass wider than half the distance from T to U. Make an arc beyond the arc you made for points T and U. T S U Without changing the compass setting, put the sharp point of the compass on U. Make a second arc that crosses the arc you just made. It is important that the arcs intersect each other. Label the point of intersection W. T W S U U1-56 Unit 1: Similarity, ongruence, and Proofs

63 4. Draw the angle bisector. Use your straightedge to create a ray connecting the point W with the vertex of the angle, S. T W S U TSW is congruent to WSU. The measure of TSW is 1 2 the measure of S. The measure of WSU is 1 2 the measure of S. 5. Find a point that is equidistant from both sides of WSU. Label the intersection of the angle bisector and the initial arc as X. Put the sharp point of the compass on point X. Open the compass wider than half the distance from X to U. Make an arc. T X W S U (continued) U1-57 Lesson 2: onstructing Lines, Segments, and ngles

64 Without changing the compass setting, put the sharp point of the compass on U. Make a second arc. It is important that the arcs intersect each other. Label the point of intersection Z. T X W Z S U 6. Draw the angle bisector. Use your straightedge to create a ray connecting point Z with the vertex of the original angle, S. T X W Z S Do not erase any of your markings. U XSZ is congruent to ZSU. TSZ is 3 4 the measure of TSU. U1-58 Unit 1: Similarity, ongruence, and Proofs

65 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 2: onstructing Lines, Segments, and ngles Practice 1.2.2: isecting Segments and ngles Use a compass and straightedge to copy each segment, and then construct the bisector of each segment D 3. E F Use a compass and straightedge to construct each segment as specified. 4. onstruct a segment whose measure is 3 4 the length of in problem onstruct a segment whose measure is 1 4 the length of D in problem 2. continued U1-59 Lesson 2: onstructing Lines, Segments, and ngles

66 PRTIE UNIT 1 SIMILRITY, ONGRUENE, ND PROOFS Lesson 2: onstructing Lines, Segments, and ngles Use a compass and straightedge to copy each angle, and then construct the bisector of each angle. 6. G 7. H 8. J Use a compass and straightedge to construct each angle as specified. 9. onstruct an angle whose measure is 3 4 the measure of H in problem onstruct an angle whose measure is 1 4 the measure of J in problem 8. U1-60 Unit 1: Similarity, ongruence, and Proofs

67 Lesson 1.2.3: onstructing Perpendicular and Parallel Lines Introduction Geometry construction tools can also be used to create perpendicular and parallel lines. While performing each construction, it is important to remember that the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but in constructions this is not allowed. You can adjust the opening of your compass to verify that lengths are equal. Key oncepts Perpendicular Lines and isectors Perpendicular lines are two lines that intersect at a right angle (90 ). perpendicular line can be constructed through the midpoint of a segment. This line is called the perpendicular bisector of the line segment. It is impossible to create a perpendicular bisector of a line, since a line goes on infinitely in both directions, but similar methods can be used to construct a line perpendicular to a given line. It is possible to construct a perpendicular line through a point on the given line as well as through a point not on a given line. onstructing a Perpendicular isector of a Line Segment Using a ompass 1. To construct a perpendicular bisector of, put the sharp point of your compass on endpoint. Open the compass wider than half the distance of. 2. Make a large arc intersecting. 3. Without changing your compass setting, put the sharp point of the compass on endpoint. Make a second large arc. It is important that the arcs intersect each other. 4. Use your straightedge to connect the points of intersection of the arcs. 5. Label the new line m. Do not erase any of your markings. is perpendicular to line m. U1-61 Lesson 2: onstructing Lines, Segments, and ngles