Name Class Date Reteaching 3-1 Lines and Angles Not all lines and planes intersect. • Planes that do not intersect are parallel planes. • Lines that are in the same plane and do not intersect are parallel. • The symbol || shows that lines or planes are parallel: “Line AB is parallel to line CD.” means • Lines that are not in the same plane and do not intersect are skew. Parallel planes: plane ABDC || plane EFHG plane BFHD || plane AEGC plane CDHG || plane ABFE Examples of parallel lines: Examples of skew lines: is skew to , and . Exercises In Exercises 1–3, use the figure at the right. 1. Shade one set of parallel planes. 2. Trace one set of parallel lines with a solid line. 3. Trace one set of skew lines with a dashed line. In Exercises 4–7, use the diagram to name each of the following. 4. a line that is parallel to 5. a line that is skew to 6. a plane that is parallel to NRTP 7. three lines that are parallel to In Exercises 8–11, describe the statement as true or false. If false, explain. 8. plane HIKJ 10. and plane IEGK 9. are skew lines. 11. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 9 Name Class 3-1 Date Reteaching (continued) Lines and Angles The diagram shows lines a and b intersected by line x. Line x is a transversal. A transversal is a line that intersects two or more lines found in the same plane. The angles formed are either interior angles or exterior angles. Interior Angles between the lines cut by the transversal 3, 4, 5, and 6 in diagram above Exterior Angles outside the lines cut by the transversal 1, 2, 7, and 8 in diagram above Four types of special angle pairs are also formed. Angle Pair Definition Examples alternate interior inside angles on opposite sides of the transversal, not a linear pair 3 and 6 4 and 5 alternate exterior outside angles on opposite sides of the transversal, not a linear pair 1 and 8 2 and 7 Same-side interior inside angles on the same side of the transversal 3 and 5 4 and 6 Corresponding in matching positions above or below the transversal, but on the same side of the transversal 1 and 5 3 and 7 2 and 6 4 and 8 Exercises Use the diagram at the right to answer Exercises 12–15. 12. Name all pairs of corresponding angles. 13. Name all pairs of alternate interior angles. 14. Name all pairs of same-side interior angles. 15. Name all pairs of alternate exterior angles. Use the diagram at the right to answer Exercises 16 and 17. Decide whether the angles are alternate interior angles, same-side interior angles, corresponding, or alternate exterior angles. 16. 1 and 5 17. 4 and 6 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 10 Name Class Date Reteaching 3-2 Properties of Parallel Lines When a transversal intersects parallel lines, special congruent and supplementary angle pairs are formed. Congruent angles formed by a transversal intersecting parallel lines: • corresponding angles (Postulate 3-1) 1 5 2 6 4 7 3 8 • alternate interior angles (Theorem 3-1) 4 6 3 5 • alternate exterior angles (Theorem 3-3) 1 8 2 7 Supplementary angles formed by a transversal intersecting parallel lines: • same-side interior angles (Theorem 3-2) m4 + m5 = 180 m3 + m6 = 180 Identify all the numbered angles congruent to the given angle. Explain. 1. 2. 3. 4. Supply the missing reasons in the two-column proof. Given: g || h, i || j Prove: 1 is supplementary to 16. Statements Reasons 1) 1 3 1) 2) g || h; i || j 2) Given 3) 3 11 3) 4) 11 and 16 are supplementary. 4) 5) 1 and 16 are supplementary. 5) Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 19 Name 3-2 Class Date Reteaching (continued) Properties of Parallel Lines You can use the special angle pairs formed by parallel lines and a transversal to find missing angle measures. If m1 = 100, what are the measures of 2 through 8? m2 = 180 100 m2 = 80 m4 = 180 100 m4 = 80 Vertical angles: m1 = m3 m3 = 100 Alternate exterior angles: m1 = m7 m7 = 100 Alternate interior angles: m3 = m5 m5 = 100 Corresponding angles: m2 = m6 m6 = 80 Same-side interior angles: m3 + m8 = 180 m8 = 80 Supplementary angles: What are the measures of the angles in the figure? (2x + 10) + (3x 5) = 180 Same-Side Interior Angles Theorem Combine like terms. 5x + 5 = 180 Subtract 5 from each side. 5x = 175 Divide each side by 5. x = 35 Find the measure of these angles by substitution. 2x + 10 = 2(35) + 10 = 80 3x 5 = 3(35) 5 = 100 2x 20 = 2(35) 20 = 50 To find m1, use the Same-Side Interior Angles Theorem: 50 + m1 = 180, so m1 = 130 Exercises Find the value of x. Then find the measure of each labeled angle. 5. 6. 7. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 20 Name 3-3 Class Date Reteaching Proving Lines Parallel Special angle pairs result when a set of parallel lines is intersected by a transversal. The converses of the theorems and postulates in Lesson 3-2 can be used to prove that lines are parallel. Postulate 3-2: Converse of Corresponding Angles Postulate If 1 5, then a || b. Theorem 3-4: Converse of the Alternate Interior Angles Theorem If 3 6, then a || b. Theorem 3-5: Converse of the Same-Side Interior Angles Theorem If 3 is supplementary to 5, then a || b. Theorem 3-6: Converse of the Alternate Exterior Angles Theorem If 2 7, then a || b. For what value of x is b || c? The given angles are alternate exterior angles. If they are congruent, then b || c. 2x 22 = 118 2x = 140 x = 70 Exercises Which lines or line segments are parallel? Justify your answers. 1. 2. 3. Find the value of x for which g || h. Then find the measure of each labeled angle. 4. 5. 6. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 29 Name Class 3-3 Date Reteaching (continued) Proving Lines Parallel A flow proof is a way of writing a proof and a type of graphic organizer. Statements appear in boxes with the reasons written below. Arrows show the logical connection between the statements. Write a flow proof for Theorem 3-1: If a transversal intersects two parallel lines, then alternate interior angles are congruent. Given: || m Prove: 2 3 Exercises Complete a flow proof for each. 7. Complete the flow proof for Theorem 3-2 using the following steps. Then write the reasons for each step. a. 2 and 3 are supplementary. b. 1 3 c. || m d. 1 and 2 are supplementary. Theorem 3-2: If a transversal intersects two parallel lines, then same side interior angles are supplementary. Given: || m Prove: 2 and 3 are supplementary. 8. Write a flow proof for the following: Given: 2 3 Prove: a || b Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 30 Name 3-4 Class Date Reteaching Parallel and Perpendicular Lines You can use angle pairs to prove that lines are parallel. The postulates and theorems you learned are the basis for other theorems about parallel and perpendicular lines. Theorem 3-7: Transitive Property of Parallel Lines 0 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. Lines a, b, and c can be in different planes. Theorem 3-8: If two lines are perpendicular to the same line, then those two lines are parallel to each other. This is only true if all the lines are in the same plane. If a d and b d, then a || b. Theorem 3-9: Perpendicular Transversal Theorem If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other line. This is only true if all the lines are in the same plane. If a || b and c, and a d, then b d, and c d. Exercises 1. Complete this paragraph proof of Theorem 3-7. Given: d || e, e || f Prove: d || f Proof: Because it is given that d || e, then 1 is supplementary to 2 by the Theorem. Because it is given that e || f, then 2 3 by the Postulate. So, by substitution, 1 is supplementary to . By the Theorem, d || f . 2. Write a paragraph proof of Theorem 3-8. Given: t n, t o Prove: n || o Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 39 Name Class 3-4 Date Reteaching (continued) Parallel and Perpendicular Lines A carpenter is building a cabinet. A decorative door will be set into an outer frame. a. If the lines on the door are perpendicular to the top of the outer frame, what must be true about the lines? b. The outer frame is made of four separate pieces of molding. Each piece has angled corners as shown. When the pieces are fitted together, will each set of sides be parallel? Explain. c. According to Theorem 3-8, lines that are perpendicular to the same line are parallel to each other. So, since each line is perpendicular to the top of the outer frame, all the lines are parallel. Know Need The angles for the top and bottom pieces are 35°. The angles for the sides are 55° . Plan Determine whether each set of sides will be parallel. Draw the pieces as fitted together to determine the measures of the new angles formed. Use this to decide if each set of sides will be parallel. The new angle is the sum of the angles that come together. Since 35 + 55 = 90, the pieces form right angles. Two lines that are perpendicular to the same line are parallel. So, each set of sides is parallel. Exercises 3. An artist is building a mosaic. The mosaic consists of the repeating pattern shown at the right. What must be true of a and b to ensure that the sides of the mosaic are parallel? 4. Error Analysis A student says that according to Theorem 3-9, if and student’s error. , then . Explain the Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 40 Name Class Date Reteaching 3-5 Parallel Lines and Triangles Triangle Angle-Sum Theorem: The measures of the angles in a triangle add up to 180. In the diagram at the right, ∆ACD is a right triangle. What are m1 and m2? Step 1 Angle Addition Postulate m1 + mDAB = 90 Substitution Property m1 + 30 = 90 Subtraction Property of Equality m1 = 60 Step 2 Triangle Angle-Sum Theorem m1 + m2 + mABC = 180 Substitution Property 60 + m2 + 60 = 180 Addition Property of Equality m2 + 120 = 180 Subtraction Property of Equality m2 = 60 Exercises Find m1. 1. 2. 3. 4. 5. 6. Algebra Find the value of each variable. 7. 8. 9. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 49 Name Class Date Reteaching (continued) 3-5 Parallel Lines and Triangles In the diagram at the right, 1 is an exterior angle of the triangle. An exterior angle is an angle formed by one side of a polygon and an extension of an adjacent side. For each exterior angle of a triangle, the two interior angles that are not next to it are its remote interior angles. In the diagram, 2 and 3 are remote interior angles to 1. The Exterior Angle Theorem states that the measure of an exterior angle is equal to the sum of its remote interior angles. So, m1 = m2 + m3. What are the measures of the unknown angles? Triangle Angle-Sum Theorem mABD + mBDA + mBAD = 180 Substitution Property 45 + m1 + 31 = 180 Subtraction Property of Equality m1 = 104 Exterior Angle Theorem mABD + mBAD = m2 Substitution Property 45 + 31 = m2 Subtraction Property of Equality 76 = m2 Exercises What are the exterior angle and the remote interior angles for each triangle? 10. 11. 12. exterior: exterior: exterior: interior: interior: interior: Find the measure of the exterior angle. 13. 14. 15. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 50 Name 3-6 Class Date Reteaching Constructing Parallel and Perpendicular Lines Parallel Postulate Through a point not on a line, there is exactly one line parallel to the given line. Given: Point D not on Construct: Step 1 Draw parallel to . Step 2 With the compass tip on B, draw an arc that intersects between B and D. Label this intersection point F. Continue the arc to intersect at point G. Step 3 Without changing the compass setting, place the compass tip on D and draw an arc that intersects above B and D. Label this intersection point H. Step 4 Place the compass tip on F and open or close the compass so it reaches G. Draw a short arc at G. Step 5 Without changing the compass setting, place the compass tip on H and draw an arc that intersects the first arc drawn from H. Label this intersection point J. Step 6 Draw , which is the required line parallel to . Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 59 Name Class 3-6 Date Reteaching (continued) Constructing Parallel and Perpendicular Lines Exercises Construct a line parallel to line m and through point Y. 1. 2. 3. Perpendicular Postulate Through a point not on a line, there is exactly one line perpendicular to the given line. Given: Point D not on Construct: a line perpendicular to through D Step 1 Construct an arc centered at D that intersects Label those points G and H. at two points. Step 2 Construct two arcs of equal length centered at points G and H. Step 3 Construct the line through point D and the intersection of the arcs from Step 2. Step 1 Step 2 Step 3 Construct a line perpendicular to line n and through point X. 4. 5. 6. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 60 Name Class 3-7 Date Reteaching Equations of Lines in the Coordinate Plane To find the slope m of a line, divide the change in the y values by the change in the rise change in y x values from one point to another. Slope is or . run change in x What is the slope of the line through the points (5, 3) and (4, 9)? Slope change in y 93 6 2 = = change in x 4 5 9 3 Exercises Find the slope of the line passing through the given points. 1. (5, 2), (1, 8) 2. (1, 8), (2, 4) 3. (2, 3), (2, 4) If you know two points on a line, or if you know one point and the slope of a line, then you can find the equation of the line. Write an equation of the line that contains the points J(4, 5) and K(2, 1). Graph the line. If you know two points on a line, first find the slope using m m y2 y1 . x2 x1 1 5 6 1 2 4 6 Now you know two points and the slope of the line. Select one of the points to substitute for (x1, y1). Then write the equation using the point-slope form y y1 = m(x x1). y 1 = 1(x (2)) Substitute. y 1 = 1(x + 2) Simplify within parentheses. You may leave your equation in this form or further simplify to find the slope-intercept form. y 1 = x 2 y = x 1 Answer: Either y 1 = 1(x + 2) or y = x 1 is acceptable. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 69 Name Class 3-7 Date Reteaching (continued) Equations of Lines in the Coordinate Plane Exercises Graph each line. 4. y 1 x4 2 1 3 5. y 4 ( x 3) 6. y 3 = 6(x 3) If you know two points on a line, or if you know one point and the slope of a line, then you can find the equation of the line using the formula y y1 = m(x x1). Use the given information to write an equation for each line. 7. slope 1, y-intercept 6 8. slope 4 , y-intercept 3 5 9. 10. 11. passes through (7, 4) and (2, 2) 12. passes through (3, 5) and (6, 1) Graph each line. 13. y = 4 14. x = 24 15. y = 22 Write each equation in slope-intercept form. 16. y 7 = 2(x 1) 1 3 17. y 2 ( x 5) Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 70 3 2 18. y 5 ( x 3) Name 3-8 Class Date Reteaching Slopes of Parallel and Perpendicular Lines Remember that parallel lines are lines that are in the same plane that do not intersect, and perpendicular lines intersect at right angles. Write an equation for the line that contains G(4, 3) and 1 is parallel to : – x 2 y = 6 . Write another equation 2 for the line that contains G and is perpendicular to . Graph the three lines. Step 1 Rewrite in slope-intercept form: y 1 x 3. 4 Step 2 Use point-slope form to write an equation for each line. 1 Parallel line: m Perpendicular line: m = 4 4 1 y (3) ( x 4) y (3) = 4(x 4) 4 1 y x4 y = 4x + 13 4 Exercises In Exercises 1 and 2, are lines m1 and m2 parallel? Explain. 1. 2. Find the slope of a line (a) parallel to and (b) perpendicular to each line. 1 4 x+3 5. y 3 = 4(x + 1) 6. 4x 2y = 8 5 5 Write an equation for the line parallel to the given line that contains point C. 1 1 7. y x 4 ; C(4, 3) 8. y x 3 ; C(3, 1) 2 2 1 9. y x 2 ; C(4, 3) 10. y = 2x 4; C(3, 3) 3 3. y = 3x + 4 4. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 79 Name Class 3-8 Date Reteaching (continued) Slopes of Parallel and Perpendicular Lines Write an equation for the line perpendicular to the given line that contains P. 11. P(5, 3); y = 4x 12. P(2, 5); 3x + 4y = 1 13. P(2, 6); 2x y = 3 14. P(2, 0); 2x 3y 5= 9 Given points J(1, 4), K(2, 3), L(5, 4), and M(0, 3), are and parallel, perpendicular, or neither? 1 7 – 3 5 Their slopes are not equal, so they are not parallel. neither 1 7 –1 3 5 The product of their solpes is not –1, so they are not perpendicular. Exercises Tell whether and are parallel, perpendicular, or neither. 16. J(4, 5), K(5, 1), L(6, 0), M(4, 3) 15. J(2, 0), K(1, 3), L(0, 4), M(1, 5) 17. : 6y + x = 7 18. : 16 = –5y x 20. 1 : y x2 5 1 : y 5x 2 : 3x + 2y = 5 19. : 4x + 5y = 22 21. : 2y 1 x 2 2 : x + 2y = 1 22. : 2x + 8y = 8 23. Right Triangle Verify that ∆ABC is a right triangle for A(0, 4), B(3, 2), and C(1, 4). Graph the triangle and explain your reasoning. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 79 80 : 2x y = 1 : y = 1 :x=0 Name Class 3-1 Date Standardized Test Prep Lines and Angles Multiple Choice For Exercises 1–7, choose the correct letter. D For Exercises 1–3, use the figure at the right. F 1. Which line segment is parallel to GE? DH FG KI HI H 2. Which two line segments are skew? DE and GE EI and GK GK and DH HI and DF E G I J K 3. Which line segment is parallel to plane FGKJ? FD HI GE For Exercises 4–7, use the figure at the right. 1 4. Which is a pair of alternate interior angles? /3 and /6 /6 and /5 /2 and /7 /4 and /6 KI 2 3 t 4 5 6 7 u 8 5. Which angle corresponds to /7? /1 /3 /4 /6 /5 and /8 /1 and /8 6. Which pair of angles are alternate exterior angles? /1 and /5 /3 and /6 7. Which pair of angles are same-side interior angles? /1 and /5 /3 and /6 /4 and /8 Short Response 8. Describe the parallel planes, parallel lines, and skew lines in a cube. Draw a sketch to illustrate your answer. Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 /3 and /5 Name Class 3-2 Date Standardized Test Prep Properties of Parallel Lines Multiple Choice For Exercises 1–6, choose the correct letter. For Exercises 1–4, use the figure at the right. 1 1. Which angle is congruent to /1? /2 /6 /5 /7 2 3 f 4 6 g 8 7 z 5 2. Which angle is not supplementary to /6? /2 /4 /5 /8 3. Which can be used to prove directly that /1 > /8? Alternate Interior Angles Theorem Corresponding Angles Postulate Same-Side Interior Angles Theorem Alternate Exterior Angles Theorem 4. If m/5 5 42, what is m/4? 42 48 128 138 For Exercises 5 and 6, use the figure at the right. 4x 5. What is the value of x? 10 30 25 120 (x 30) (5x 25) 1 6. What is the measure of /1? 45 60 120 125 Short Response r 7. Write a two-column proof of the Alternate Exterior Angles Theorem s (Theorem 3-2). Given: r 6 s 1 Prove: /1 > /8 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 17 6 8 d Name Class 3-3 Date Standardized Test Prep Proving Lines Parallel Multiple Choice For Exercises 1–6, choose the correct letter. e (7x 3) 1. For what value of x is d 6 e? 20 25 (2x 3) d 35 37 q For Exercises 2 and 3, use the figure below right. /8 is supplementary to /12. /1 > /6 /10 is supplementary to /11. /5 > /13 3. Which statement proves that x 6 y? /2 is supplementary to /3. /6 > /9 /14 is supplementary to /15. /12 > /13 For Exercises 4–6, use the figure at the right. 58 122 x 3 4 1 2 7 8 5 6 y 9 10 11 12 13 14 15 16 m , 4. If / 6 m, what is m/1? 22 b a 2. Which statement proves that a 6 b? 130 1 (5x 12) (2x 14) 2 5. For what value of x is / 6 m? 22 54 58 122 122 130 6. If / 6 m, what is m/2? 22 58 Short Response 7. Write a flow proof. 1 Given: /2 and /3 are supplementary. c 2 Prove: c 6 d 3 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 27 d Name Class 3-4 Date Standardized Test Prep Parallel and Perpendicular Lines Multiple Choice For Exercises 1–5, choose the correct letter. 1. Which can be used to prove d ' t ? s Transitive Property of Parallel Lines Transitive Property of Congruence t Perpendicular Transversal Theorem d Converse of the Corresponding Angles Postulate a 2. A carpenter is building a frame. Which values of a and b will ensure that the sides of the finished frame are parallel? a 5 40 and b 5 60 a 5 30 and b 5 60 a 5 45 and b 5 50 a 5 40 and b 5 40 a b b b b a a For Exercises 3 and 4, use the map at the right. 3. If Adam Ct. is perpendicular to Bertha Dr. and Charles St., Ad am what must be true? Ct Dr a t. sS Adam Ct. ' Edward Rd. Bertha Dr. 6 Charles St. Adam Ct. 6 Dana La. Dana La. ' Charles St. Rd ar na rd wa Da La Ed is parallel to Edward Rd. What must be true? Ch 4. Adam Ct. is perpendicular to Charles St. and Charles St. . le Dana La. ' Charles St. rth Bertha Dr. 6 Charles St. Be Adam Ct. 6 Dana La. . . Adam Ct. ' Edward Rd. . 5. If a ' b, b ' c, c 6 d, and d ' e, which is not true? a'e a6d a6c b6d Short Response 6. Write a paragraph proof. Given: a 6 b, b 6 c, and d ' c a Prove: a ' d b c d Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 37 Name Class Date Standardized Test Prep 3-5 Parallel Lines and Triangles Gridded Response 14 1. What is the value of w? 30 w 19 2. What is the value of z? 22 x° y° z° 104 3. What is the value of s? ( s 1 s 3) 7 ( 4. What is the value of e? 1 s 3) 7 60 e 69 5. What is the value of t on the truss of the bridge? t 55 Answers 1. 2. a 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 3. a 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 4. a 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 5. a 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 47 a 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Name Class Date Standardized Test Prep 3-6 Constructing Parallel and Perpendicular Lines Multiple Choice For Exercises 1–3, choose the correct letter. 1. Which diagram below shows the first step in parallel line construction on a point outside a line? N N N N P P P 2. What kind of line is being constructed in this series of diagrams? Q Q parallel Q perpendicular congruent similar 3. Which diagram below shows line m parallel to line n? P n m C o n B m m A m n Short Response 4. Error Analysis Explain the error in the construction at the right. P The student is attempting to draw a perpendicular line through a point not on the line. B A Q Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 57 n Name Class Date Standardized Test Prep 3-7 Equations of Lines in the Coordinate Plane Multiple Choice For Exercises 1–4, choose the correct letter. 1. What is the slope of the line passing through the points (2, 7) and (21, 3)? 3 2 4 7 4 3 2. What is the correct equation of the line 6 shown at the right? 3 y 5 2x 1 3 y 5 23 x 1 3 3 y 5 22 x 2 3 y 5 2 23 x 2 3 4 2 the line is 22. What is the equation of the line? y 5 25 x 1 2 y 5 2 25 x 2 2 2 O 2 4 6 4 (4, 3) 6 5 y 5 22x 2 2 (0, 3) 2 3. The x-intercept of a line is 25 and the y-intercept of 5 y x 6 4 y 5 22x 2 5 1 3 5 4. What is the slope-intercept form of the equation y 2 7 5 22 (x 1 4)? 5 y 2 2 5 2 2 (x 1 2) y 5 2 47 x 1 2 5 5 y 1 7 5 2x 1 2 y 5 22x 2 3 Short Response y 4 5. Error Analysis A student has attempted to graph an equation that contains the point (1, 24) and has a slope of 13 . a. What is the correct equation in slope-intercept form? b. What is the student’s error on the graph? 2 x 4 2 O 2 4 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 67 4 (2, 1) (1, 4) Name Class 3-8 Date Standardized Test Prep Slopes of Parallel and Perpendicular Lines Multiple Choice For Exercises 1–4 choose the correct letter. 1. Which pair of slopes could represent perpendicular lines? 1 7, 7 3 1 2 2, 4 2 4, 43 1 1 3, 3 2. The lines shown in the figure at the right are (2, 7) parallel. perpendicular. y 8 6 (8, 5) 4 neither parallel nor perpendicular. both parallel and perpendicular. 3. Two lines are perpendicular when 10 8 4 2 the product of their slopes is 21. x O 2 (1, 2) 2 the product of their slopes is greater than 0. (5, 4) 4 4 they have the same slope. 6 their slopes are undefined. 5 1 4. Which is the equation for the line perpendicular to y 5 23x 1 113 and containing P(22, 3)? 3 y 2 2 5 25 (x 2 3) 5 y 5 23x 1 413 3 3 y 5 25x 1 415 y 5 5x 1 415 Extended Response 5. Graph the vertices of ABCD where 8 A(21, 3), B(26, 22), C(21, 27), and D(4, 22). a. Explain how you know the opposite sides of ABCD are parallel. 4 b. Explain how you know the adjacent 2 y 6 x sides of ABCD are perpendicular. 8 6 4 c. What is the length of each side, to the nearest inch, if each grid space is equal to 2 in.? 2 O 2 4 6 d. What kind of figure is ABCD? 8 Prentice Hall Geometry • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 77 2 4 6 8 Chapter 3 Test Name:_______________________ Date:_______________________ ______________________________________________________________________________ 1 Find the measure of the third angle of a triangle given the measure of two angles are 63º and 22º. A B C D 2 Find the measure of the third angle of a triangle given the measure of two angles are 86º and xº. A B C D 3 5º 41º 85º 95º (86 – x)º (90 – x)º (94 – x)º (180 – x)º Find the value of k. A B C D 107 121 25 59 ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 1 of 12 Chapter 3 Test ______________________________________________________________________________ 4 The horizontal lines are parallel. Find the value of x. A B C D 5 6 27 28 29 The horizontal lines are parallel. Find the measure of A B C D 36º 54º 116º 144º A B C D 50 60 70 120 Find ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 2 of 12 Chapter 3 Test ______________________________________________________________________________ 7 Line r is parallel to line t. Find m 5. A B C D 8 146 134 24 34 Construct a rectangle with side lengths a and b. A B C D ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 3 of 12 Chapter 3 Test ______________________________________________________________________________ 9 Construct a quadrilateral with one pair of parallel opposite sides, each side of length 2a. A B C D 10 The measures of the three angles of a triangle are given. Find the value of x and state whether the triangle is acute, obtuse, or right. x + 10, x – 20, x + 25 A B C D x x x x = 25, acute = 55, acute = 25, right = 55, right ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 4 of 12 Chapter 3 Test ______________________________________________________________________________ 11 What is the relationship between A B C D 12 13 ? and are parallel. and are skew. and are neither parallel nor skew. There is not enough information. What is the relationship between A B C D and and ? and are parallel. and are skew. and are neither parallel nor skew. There is not enough information. Determine whether two parallel lines that go along the sideline of a field and the center bar of the top-goal post of the soccer net are parallel lines, skew lines, neither, or there is not enough information. A B C D parallel lines skew lines neither not enough information ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 5 of 12 Chapter 3 Test ______________________________________________________________________________ 14 Write an equation in slope-intercept form of the line with slope 5 and y-intercept –4. A B C D 15 Write an equation in point-slope form of the line through point J(–8, 10) with slope –5. A B C D 16 Write an equation in point-slope form of the line through points (7, 6) and (9, 2). Use (7, 6) as the point (x1, y1). A B C D 17 (y + 6) = –2(x + 7) (y + 6) = 2(x + 7) (y –6) = –2(x – 7) (y – 6) = 2(x – 7) Find the value of x for which || m. A B C D 24 26 12.75 14 ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 6 of 12 Chapter 3 Test ______________________________________________________________________________ 18 If and A B C D 19 90 105 75 not enough information Determine whether and are parallel, perpendicular, or neither. A(–1, –4), B(2, 11), C(1, 1), D(4, 10) A B C 20 , what is parallel perpendicular neither Determine whether and are parallel, perpendicular, or neither. A(2, 8), B(–1, –2), C(3, 7), D(0, –3) A B C parallel perpendicular neither ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 7 of 12 Chapter 3 Test ______________________________________________________________________________ 21 Graph y = 5x – 12 and y – 5x = 19. Are the lines parallel, perpendicular, or neither? A The lines are neither parallel nor perpendicular. B The lines are parallel. ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 8 of 12 Chapter 3 Test ______________________________________________________________________________ C The lines are parallel. D The lines are neither parallel nor perpendicular. ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 9 of 12 Chapter 3 Test ______________________________________________________________________________ 22 Graph y = –2x + 5 and . Are the lines parallel, perpendicular, or neither? A The lines are neither parallel nor perpendicular. B The lines are parallel. C The lines are perpendicular. D The lines are neither parallel nor perpendicular. ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 10 of 12 Chapter 3 Test ______________________________________________________________________________ 23 Which is a correct two-column proof? Given: Prove: and are supplementary. A B C D 24 none of these To construct a line parallel to a given line m through a point not on m, you need to know how to construct __?__ angles. A B C D right acute congruent obtuse ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 11 of 12 Chapter 3 Test ______________________________________________________________________________ 25 Which diagram shows plane PQR and plane QRS intersecting only in ? A B C D ______________________________________________________________________________________________ Copyright © 2005 - 2006 by Pearson Education Page 12 of 12