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Name Lesson 3-1 Date Class Ready to Go On? Skills Intervention Lines and Angles Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary parallel lines perpendicular lines skew lines transversal corresponding angles alternate interior angles alternate exterior angles parallel planes same-side interior angles Identifying Types of Lines and Planes Identify each of the following. O P A. Skew segments do not lie in the same ; and do not they are not L . B. Perpendicular segments intersect at a D E Name two segments in the figure that are skew. N M A C B angle. Name a pair of perpendicular segments in the figure. C. Parallel lines are and do not . Name a pair of parallel segments in the figure. Classifying Pairs of Angles Give an example of each angle pair. A. Corresponding angles lie on the same of the In the figure, /1 and / r side of the transversal the other two lines. are same-side interior angles. C. Alternate exterior angles lie on sides of the transversal and are are alternate exterior angles. the other two lines. In the figure, /5 and / D. Alternate interior angles lie on sides of the transversal and are the other two lines. In the figure, /3 and / Copyright © by Holt, Rinehart and Winston. All rights reserved. s are corresponding angles. B. Same-side interior angles lie on the and are 4 1 7 of the other transversal and are on the same two lines. In the figure, /3 and / 8 5 t 3 6 2 29 are alternate interior angles. Holt McDougal Geometry Name Lesson 3-1 Date Class Ready to Go On? Skills Intervention Lines and Angles Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary parallel lines perpendicular lines skew lines transversal corresponding angles alternate interior angles alternate exterior angles parallel planes same-side interior angles Identifying Types of Lines and Planes Identify each of the following. O P plane A. Skew segments do not lie in the same they are not parallel _ ; intersect and do not . _ Name two segments in the figure that are skew. coplanar parallel segments in the figure. A intersect . Name a pair of _ Sample answer: MN and BC _ and do not A. Corresponding angles lie on the same two lines. In the figure, /3 and / 8 B. Same-side interior angles lie on the between side side transversal and are on the same In the figure, /1 and / of the 8 4 1 7 of the other 5 t 3 6 2 s r are corresponding angles. same side of the transversal the other two lines. 2 are same-side interior angles. C. Alternate exterior angles lie on opposite the other two lines. In the figure, /5 and / D. Alternate interior angles lie on opposite sides of the transversal and are 7 1 29 outside are alternate exterior angles. sides of the transversal and are the other two lines. In the figure, /3 and / Copyright © by Holt, Rinehart and Winston. All rights reserved. B right Classifying Pairs of Angles Give an example of each angle pair. and are C angle. Name a pair of perpendicular _ _ Sample answer: AB and LA B. Perpendicular segments intersect at a C. Parallel lines are N M D E Sample answer: MN and AB segments in the figure. L between are alternate interior angles. Holt McDougal Geometry Name Lesson 3-2 Date Class Ready To Go On? Skills Intervention Angles Formed by Parallel Lines and Transversals Using the Corresponding Angles Postulate Find m/RST. Since two lines in the figure are R (5x + 27)° and cut (8x + 6)° S by a transversal, the pairs of corresponding T . are Write an equation relating the measures of the given angles. (5x 1 27)8 5 Solve the equation. x 5 the value of x into the expression To find m/RST, . Find the measure of /RST. Finding Angle Measures Find each angle measure. D A. m/DEC Since the two labeled angles are on sides (24x – 13)° E F C the other of the transversal and are (20x + 7)° angles. two lines, they are . Since the lines in the figure are parallel, the labeled angles are Write an equation relating the measures of the angles. Solve the equation. x 5 the value of x into the expression To find m/DEC, . Find the measure of /DEC. B. m/DEF so the sum of their m/DEC and m/DEF form a measures is 8. from Subtract m/DEF 5 to find m/DEF. 2 107 5 Copyright © by Holt, Rinehart and Winston. All rights reserved. 30 Holt McDougal Geometry Name Lesson 3-2 Date Class Ready To Go On? Skills Intervention Angles Formed by Parallel Lines and Transversals Using the Corresponding Angles Postulate Find m/RST. parallel Since two lines in the figure are are and cut (8x + 6)° S angles by a transversal, the pairs of corresponding congruent R (5x + 27)° T . Write an equation relating the measures of the given angles. (5x 1 27)8 5 (8x 1 6)8 7 Solve the equation. x 5 To find m/RST, substitute the value of x into the expression (8x 1 6) or (5x 1 27) . 628 Find the measure of /RST. Finding Angle Measures Find each angle measure. D A. m/DEC opposite Since the two labeled angles are on inside of the transversal and are (24x – 13)° E angles. Since the lines in the figure are parallel, the labeled angles are Write an equation relating the measures of the angles. congruent . 20x 1 7 5 24x 2 13 5 Solve the equation. x 5 To find m/DEC, F C the other alternate interior two lines, they are sides (20x + 7)° substitute the value of x into the expression 24x 2 13 or 20x 1 7 . 1078 Find the measure of /DEC. B. m/DEF m/DEC and m/DEF form a measures is Subtract m/DEF 5 linear pair so the sum of their 180 8. 107 180 from 2 107 5 Copyright © by Holt, Rinehart and Winston. All rights reserved. 180 to find m/DEF. 738 30 Holt McDougal Geometry Name Lesson 3-3 Date Class Ready To Go On? Skills Intervention Proving Lines Parallel Using the Converse of the Corresponding Angles Postulate Use the given information to show that p i q. 14 5 2 3 8 76 Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9 p Substitute the value of x into each expression. m/2 5 12( ) 2 25 5 ; m/8 5 9( q )125 Does m/2 5 m/8? Since m/2 m/8, by the Property of Congruence. angles formed by two coplanar lines cut by a Since the , p i q. transversal are 14 5 2 3 8 76 Determining Whether Lines are Parallel Using the given information and the diagram to show that p i q. p A. /1 /7 What type of angles are /1 and /7? If two coplanar lines are cut by a transversal so that a pair of q . Since /1 /7, angles are congruent, then the two lines are i . B. m/3 5 m/5 What type of angles are /3 and /5? Since m/3 5 m/5, . Since , p i q by the Converse of . the R Proving Lines Parallel _ _ Write a paragraph proof to show that RS i QT . Given: m/R 5 1318, m/Q 5 498 _ _ Prove: RS i QT Q 49° S T angles by Since m/R 5 1318 and m/Q 5 498, /R and /Q are angles. Since /R and /Q lie on the same side the definition of angles. of two coplanar lines cut by a transversal, they are Angles Theorem, when same-side angles By the Converse of the are 131° _ _ , then the two lines are parallel, so RS i QT . Copyright © by Holt, Rinehart and Winston. All rights reserved. 31 Holt McDougal Geometry Name Lesson 3-3 Date Class Ready To Go On? Skills Intervention Proving Lines Parallel Using the Converse of the Corresponding Angles Postulate Use the given information to show that p i q. 14 5 2 3 8 76 Given: m/2 5 (12x 2 25)8 and m/8 5 (9x 1 2)8; x 5 9 p Substitute the value of x into each expression. m/2 5 12( 9 ) 2 25 5 Does m/2 5 m/8? Since m/2 Since the 5 838 ; m/8 5 9( 9 838 Yes m/8, /2 corresponding /8 by the Property of Congruence. angles formed by two coplanar lines cut by a congruent transversal are )125 q , p i q. 14 5 2 3 8 76 Determining Whether Lines are Parallel Using the given information and the diagram to show that p i q. p A. /1 /7 q Alternate exterior angles What type of angles are /1 and /7? If two coplanar lines are cut by a transversal so that a pair of angles are congruent, then the two lines are parallel alternate exterior . Since /1 /7, p i q . B. m/3 5 m/5 What type of angles are /3 and /5? /5 . Since /3 Alternate Interior Angles Theorem . Since m/3 5 m/5, the /3 Alternate interior angles /5 , p i q by the Converse of R Proving Lines Parallel _ _ Write a paragraph proof to show that RS i QT . Given: m/R 5 1318, m/Q 5 498 _ _ Prove: RS i QT the definition of supplementary are supplementary S T angles by angles. Since /R and /Q lie on the same side of two coplanar lines cut by a transversal, they are By the Converse of the 49° Q Since m/R 5 1318 and m/Q 5 498, /R and /Q are 131° Same-Side same-side interior angles. Angles Theorem, when same-side angles _ _ supplementary , then the two lines are parallel, so RS i QT . Copyright © by Holt, Rinehart and Winston. All rights reserved. 31 Holt McDougal Geometry Name Lesson 3-4 Date Class Ready To Go On? Skills Intervention Perpendicular Lines Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary perpendicular bisector distance from a point to a line Proving Properties of Lines Write a two-column proof. Given: m/1 5 m/2, b i c Prove: d ' b d b 12 Plan your proof: c Step 1: Write the given information in the two-column proof. Step 2: Since it is given that m/1 5 m/2, you know that /1 is . to /2 by the definition of Put this information in the two-column proof. Step 3: If two intersecting lines form a linear pair of . So you know that d the lines are angles, then c. Put this information in Step 3 of the two-column proof. Step 4: It is given that b i c. In Step 3, you proved that d c. You can conclude Theorem. that d ' b because of the Complete Step 4 of the two-column proof. Statements Reasons 1. 1. Given 2. 2. 3. 3. 4. d ' b 4. Copyright © by Holt, Rinehart and Winston. All rights reserved. 32 Holt McDougal Geometry Name Lesson 3-4 Date Class Ready To Go On? Skills Intervention Perpendicular Lines Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary perpendicular bisector distance from a point to a line Proving Properties of Lines Write a two-column proof. Given: m/1 5 m/2, b i c Prove: d ' b d b 12 Plan your proof: c Step 1: Write the given information in the two-column proof. congruent Step 2: Since it is given that m/1 5 m/2, you know that /1 is to /2 by the definition of congruent angles . Put this information in the two-column proof. Step 3: If two intersecting lines form a linear pair of the lines are perpendicular congruent ' . So you know that d angles, then c. Put this information in Step 3 of the two-column proof. Step 4: It is given that b i c. In Step 3, you proved that d that d ' b because of the ' Perpendicular Transversal c. You can conclude Theorem. Complete Step 4 of the two-column proof. Statements Reasons 1. m/1 5 m/2, b i c 1. Given 2. /1 > /2 2. 3. d'c 3. Two 4. d ' b Copyright © by Holt, Rinehart and Winston. All rights reserved. Def. of > /s intersecting lines form a linear pair of > /s, lines are '. 4. 32 Perpendicular Transversal Theorem Holt McDougal Geometry Name Section 3A Date Class Ready to Go On? Quiz E 3-1 Lines and Angles D Identify each of the following. F 1. a pair of parallel segments A 2. a pair of perpendicular segments B 3. a pair of skew segments C 4. a pair of parallel planes Give an example of each angle pair. 1 2 8 7 5. same-side interior angles 3 4 6 5 6. alternate exterior angles 7. corresponding angles 8. alternate interior angles 3-2 Angles Formed by Parallel Lines and Transversals Find each angle measure. 9. 10. 58° (9x – 8)° (7x + 6)° x° 11. (11x + 4)° (15x – 40)° Copyright © by Holt, Rinehart and Winston. All rights reserved. 33 Holt McDougal Geometry Name Section 3A Date Class Ready to Go On? Quiz E 3-1 Lines and Angles Identify each of the following. Sample answers given for 1–3. 1. a pair of parallel segments 2. a pair of perpendicular segments _ _ DE and BA _ _ DE and EF 3. a pair of skew segments and BC EA F A B DEF and BAC Give an example of each angle pair. 5. same-side interior angles 6. alternate exterior angles C 4. a pair of parallel planes _ _ D 1 2 8 7 /2 and /3 or /7 and /6 3 4 6 5 /1 and /5 or /8 and /4 7. corresponding angles /7 and /5; /2 and /4; /1 and /3; /8 and /6 8. alternate interior angles /2 and /6 or /7 and /3 3-2 Angles Formed by Parallel Lines and Transversals Find each angle measure. 9. 10. 58° (9x – 8)° (7x + 6)° x° 588 558 11. (11x + 4)° (15x – 40)° 1258 Copyright © by Holt, Rinehart and Winston. All rights reserved. 33 Holt McDougal Geometry Name Section 3A Date Class Ready to Go On? Quiz continued 3-3 Proving Lines Parallel Use the given information and the theorems and postulates you have learned to show that a i b. 12. m/3 1 m/6 5 1808 a b 1 2 4 3 7 8 6 5 13. /1 /7 14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16 15. m/7 5 m/3 _ _ 16. Write a paragraph proof to show that DC i AB . D a 1 2 A4 3 C 72° b 7 8 108° 6 5 B t 1 3-4 Perpendicular Lines m 17. Complete the two-column proof below. Given: t ' m, m/1 5 m/2 Prove: n ' t 2 Statements Reasons 1. t ' m, m/1 5 m/2 1. Given 2. /1 /2 2. 3. 3. C onverse of the Alternate Exterior Angles Theorem 4. n ' t 4. Copyright © by Holt, Rinehart and Winston. All rights reserved. n 34 Holt McDougal Geometry Name Date Class Ready to Go On? Quiz continued Section 3A 3-3 Proving Lines Parallel Use the given information and the theorems and postulates you have learned to show that a i b. 12. m/3 1 m/6 5 1808 Interior Converse of the Same-Side Angles Theorem 14. m/4 5 (7x 2 1)8, m/8 5 (5x 1 31)8, x 5 16 1 2 4 3 7 8 6 5 m/4 5 m/8 5 1118, /4 > /1; Alternate Exterior Angles Theorem 15. m/7 5 m/3 Alternate /7 > /3 by the def. of >/s; Interior Angles Theorem _ _ 16. Write a paragraph proof to show that DC i AB . D Sample answer: 1088 1 728 5 1808; so /C and b Converse of the Corresponding Angles Postulate 13. /1 /7 so a /B are supplementary by the definition of a supplementary angles. Since /C and /B are on the same side of the transversal and between the other two lines; they are same-side interior angles. When same-side interior angles are supplementary the lines are parallel. 1 2 A4 3 C 72° b 7 8 108° 6 5 B t 1 3-4 Perpendicular Lines m 17. Complete the two-column proof below. Given: t ' m, m/1 5 m/2 Prove: n ' t 2 Statements Reasons 1. t ' m, m/1 5 m/2 1. Given 2. /1 /2 2. 3. min 4. n ' t Copyright © by Holt, Rinehart and Winston. All rights reserved. n Def. > /s 3. C onverse of the Alternate Exterior Angles Theorem 4. 34 Perpendicular Transversal Theorem Holt McDougal Geometry Name Section 3A Date Class Ready to Go On? Enrichment Finding Angle Measures Use the figure at the right and the given information to answer the questions below. s i t, s i r, l i m, n i m r s m/1 5 (7x)8 m/2 5 (4x 1 18)8 m/3 5 (11a 1 10b)8 m/4 5 (6a 1 18b)8 m m/5 5 (3y )8 n 6 t 1 35 2 4 7 m/6 5 (5a 1 2)8 m/7 5 (28b 2 5)8 1. Find the value of x. 2. Find m/1. 3. Find m/2. 4. How are /1 and /3 related? 6. What is m/4? 5. What is m/3? 7. What is the value of a? 9. Find m/5. 8. What is the value of b? 10. Find the value of y. 11. Is n i m? Explain your answer. 12. Write a paragraph proof to show that s i r. Copyright © by Holt, Rinehart and Winston. All rights reserved. 35 Holt McDougal Geometry Name Section 3A Date Class Ready to Go On? Enrichment Finding Angle Measures Use the figure at the right and the given information to answer the questions below. s i t, s i r, l i m, n i m r s m/1 5 (7x)8 m/2 5 (4x 1 18)8 m/3 5 (11a 1 10b)8 m/4 5 (6a 1 18b)8 m m/5 5 (3y )8 n 6 t 1 35 2 4 7 m/6 5 (5a 1 2)8 m/7 5 (28b 2 5)8 1. Find the value of x. 6 2. Find m/1. 3. Find m/2. 428 4. How are /1 and /3 related? Same-side interior angles 6. What is m/4? 5. What is m/3? 8 9. Find m/5. 1388 7. What is the value of a? 1388 428 8. What is the value of b? 5 10. Find the value of y. 428 14 11. Is n i m? Explain your answer. Sample answer: No, because m/7 5 28(5) 2 5 5 1358, and m/4 m/7. 12. Write a paragraph proof to show that s i r. Sample answer: m/6 5 5(8) 1 2 5 428; and m/6 5 m/1. By the definition Corresponding of congruent angles; /6 > /1. By the Converse of the Angles Postulate s i r Copyright © by Holt, Rinehart and Winston. All rights reserved. 35 Holt McDougal Geometry Name Date Class Ready to Go On? Skills Intervention Lesson 3-5 Slopes of Lines Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary rise run slope y Finding the Slope of a Line Use the slope formula to determine the slope of each line. ‹__› B A.AB y 22 What is the slope formula? m 5 _________ 2 x 1 2 –4 –2 O What are the coordinates of A? Substitute the coordinates of A and B into the slope formula ‹__› to find the slope of AB . of B? C –4 A 2 4 x 6 D 32 m 5 _________ 5 2 _____ 26 __ ‹ › B.BD What are the coordinates of D? Substitute the coordinates of B and D into the slope formula to find the slope of BD . 23 2 2 5 ______ m 5 __________ 21 The slope is ‹__› ‹__› . What kind of line is BD ? Determining Whether Lines are Parallel, Perpendicular, or Neither ‹___› ‹___› LM passes through L(4, 2) and M(0, 24), and XY passes through X(22, 5) and Y(2, 21). Use slopes to determine whether the lines are parallel, perpendicular, or neither. Graph the coordinates and draw each line on the grid at the right. Find the slope of each line by substituting the coordinates into the slope formula. 22 y 1 ‹__› y_______ 24 2 Slope of LM 5 x 2 x 5 __________ 5 2 1 02 4 2 –4 –2 O –2 2 4 –4 __ ‹ › 25 ___________ 5 Slope of XY 5 _____ 5 _____ 2 Are they parallel? Do the lines have the same slope? Are the lines perpendicular? Is the product of the slopes 21? The lines are neither Copyright © by Holt, Rinehart and Winston. All rights reserved. nor . 36 Holt McDougal Geometry Name Date Class Ready to Go On? Skills Intervention Lesson 3-5 Slopes of Lines Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary rise run slope y Finding the Slope of a Line Use the slope formula to determine the slope of each line. ‹__› B A.AB y 22 y 1 What is the slope formula? m 5 _________ x 2 2 x 1 (6, 1) 2 (1, 3) What are the coordinates of A? Substitute the coordinates of A and B into the slope formula ‹__› to find the slope of AB . of B? –4 –2 O C –4 A 2 4 x 6 D 2 32 1 m 5 _________ 5 2 _____ 1 5 26 __ ‹ › B.BD (1, 23) What are the coordinates of D? Substitute the coordinates of B and D into the slope formula to find the slope of BD . 23 2 3 2 6 5 ______ m 5 __________ 1 21 0 The slope is undefined ‹__› ‹__› . What kind of line is BD ? Vertical Determining Whether Lines are Parallel, Perpendicular, or Neither ‹___› ‹___› LM passes through L(4, 2) and M(0, 24), and XY passes through X(22, 5) and Y(2, 21). Use slopes to determine whether the lines are parallel, perpendicular, or neither. 4 2 Graph the coordinates and draw each line on the grid at the right. Find the slope of each line by substituting the coordinates into the slope formula. 6 or __ 3 __ 22 y 1 ‹__› y_______ 24 2 2 __________ 4 2 Slope of LM 5 x 2 x 5 5 1 2 4 02 –4 –2 O –2 2 4 –4 __ ‹ › 21 2 5 26 _____ 23 ___________ 5 Slope of XY 5 _____ 5 4 2 2 2 22 Do the lines have the same slope? Is the product of the slopes 21? The lines are neither Copyright © by Holt, Rinehart and Winston. All rights reserved. parallel No No nor Are they parallel? No Are the lines perpendicular? No perpendicular . 36 Holt McDougal Geometry Name Lesson 3-6 Date Class Ready to Go On? Skills Intervention Lines in the Coordinate Plane Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary point-slope form slope-intercept form Writing Equations of Lines A. Write the equation of the line with slope 3 through (21, 4) in point-slope form. What is point-slope form? y 2 y 15 Substitute for m, for x 1and for y 1: B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form. y 22 2 2 24 2 5 ______ 5 ___________ Use the slope formula to find the slope. m 5 _________ 5 ______ 2 x 1 2 (26) What is slope-intercept form? y 5 ___ Substitute 22 for m, 26 for x, and 2 for y, and then simplify to find b. 3 b 5 5 _____ (26) 1 b Write the equation in slope-intercept form. Graphing Lines 1 Graph the line. y 1 2 5 2 __ (x 2 3) 2 The equation is given in is . The line goes through the point point and then rise and run a line connecting the two points. 4 form. The slope of the line 2 . Plot the –4 –2 O –2 to find another point. Draw 2 4 –4 Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. 3 x 1 4 and 3x 2 2y 5 6 y 5 __ 2 and the y-intercept is . The slope of the first line is Solve the second equation for y to rewrite it in slope-intercept form. The slope of the second line is and the y-intercepts are The slopes lines are are and the y-intercept is . . The lines . Copyright © by Holt, Rinehart and Winston. All rights reserved. 37 Holt McDougal Geometry Name Date Class Ready to Go On? Skills Intervention Lesson 3-6 Lines in the Coordinate Plane Find these vocabulary words in the lesson and the Multilingual Glossary. Vocabulary point-slope form slope-intercept form Writing Equations of Lines A. Write the equation of the line with slope 3 through (21, 4) in point-slope form. m(x 2 x 1) What is point-slope form? y 2 y 15 Substitute 3 for m, 21 for x 1and 4 for y 1: y 2 4 5 3(x 1 1) B. Write the equation of the line through points (26, 2) and (3, 24) in slope-intercept form. y 22 y 1 2 6 2 2 24 2 2 5 ______ 5 ___________ Use the slope formula to find the slope. m 5 _________ 5 ______ x 2 2 x 1 9 3 3 2 (26) What is slope-intercept form? y 5 mx 1 b ___ Substitute 22 for m, 26 for x, and 2 for y, and then simplify to find b. 3 2 22 3 y 5 ___ 22 x 1 (22) 22 Write the equation in slope-intercept form. 3 5 _____ (26) 1 b b 5 Graphing Lines 1 Graph the line. y 1 2 5 2 __ (x 2 3) 2 The equation is given in point-slope Sample answers: form. The slope of the line 1 2 __ is 2 . The line goes through the point 21 point and then rise and run a line connecting the two points. 2 (3, 22) . Plot the to find another point. Draw The slopes lines are the same are . parallel Copyright © by Holt, Rinehart and Winston. All rights reserved. 2 –4 –2 O –2 2 4 –4 Classifying Pairs of Lines Determine whether the lines are parallel, intersect, or coincide. 3 x 1 4 and 3x 2 2y 5 6 y 5 __ _32 2 4 . and the y-intercept is The slope of the first line is Solve the second equation for y to rewrite it in slope-intercept form. The slope of the second line is 4 _32 23 . different y 5 __ 3 x 2 3 2 and the y-intercept is and the y-intercepts are 37 . The lines Holt McDougal Geometry Name Date Class Ready to Go On? Quiz Section 3B 3-5 Slopes of Lines Use the slope formula to determine the slope of each line. ‹__› 1.AD ‹__› 2.AB y 6 D A 2 C –2 O –2 ‹__› 3.AC –4 x 2 4 6 B ‹__› 4.DB Find the slope of the line through the given points. 5. R(4, 7) and S(22, 0) 6. C(0, 24) and D(5, 9) 7. H(3, 5) and I(24, 2) 8. S(26, 1) and T(3, 26) Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither. ‹___› ‹__› ‹__› ‹___› 9.CD for A(21, 0), B(1, 5), and MN for L(23, 2), M(21, 5), 10.LM and AB C(4, 5), and D(22, 4) N(2, 3), and P(1, 25) 4 2 2 –4 –2 O –2 2 –4 –2 O –2 4 2 4 –4 –4 4 11.‹PR __ › and ‹PS __ › for P(2, 21), Q(2, 1), 12.‹GH ___ › and ‹FJ __ › for F(23, 2), G(22, 5) R(23, 1), and S(22, 22) H(2, 4), and J(2, 1) 4 2 2 –4 –2 O –2 –4 –2 O –2 4 Copyright © by Holt, Rinehart and Winston. All rights reserved. 2 4 –4 –4 4 38 Holt McDougal Geometry Name Date Class Ready to Go On? Quiz Section 3B 3-5 Slopes of Lines Use the slope formula to determine the slope of each line. ‹__› 1.AD ‹__› 2.AB y 1 2 __ 2 __ 7 2 __ 2 or __ 1 4 2 10 5 2 ___ or 2 __ 4 2 ‹__› 3.AC ‹__› 4.DB 6 D A 2 C –2 O –2 –4 x 2 4 6 B Find the slope of the line through the given points. 5. R(4, 7) and S(22, 0) 6. C(0, 24) and D(5, 9) __ 7 6 8. S(26, 1) and T(3, 26) 3 __ 7 5 7. H(3, 5) and I(24, 2) ___ 13 7 2 __ 9 Graph each pair of lines and use their slopes to determine if they are parallel, perpendicular, or neither. ‹___› ‹__› ‹__› ‹___› 9.CD for A(21, 0), B(1, 5), and MN for L(23, 2), M(21, 5), 10.LM and AB C(4, 5), and D(22, 4) N(2, 3), and P(1, 25) 4 2 2 –4 –2 O –2 2 –4 –2 O –2 4 2 4 –4 –4 4 Neither Perpendicular 11.‹PR __ › and ‹PS __ › for P(2, 21), Q(2, 1), 12.‹GH ___ › and ‹FJ __ › for F(23, 2), G(22, 5) R(23, 1), and S(22, 22) H(2, 4), and J(2, 1) 4 2 2 –4 –2 O –2 –4 –2 O –2 4 Copyright © by Holt, Rinehart and Winston. All rights reserved. 2 4 –4 –4 4 Neither 38 Parallel Holt McDougal Geometry Name Section 3B Date Class Ready to Go On? Quiz continued 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 13. the line through (23, 21) and (3, 23) in slope-intercept form 3 14. the line through (6, 22) with slope 2 __ in point-slope form 4 15. the line with y-intercept 23 through the point (2, 5) in point-slope form 16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form Graph each line. 3 (x 1 2) 18. y 2 1 5 __ 5 17. y 5 3x 2 1 19. y 5 25 4 4 4 2 2 –4 –2 2 –4 –2 O –2 4 –4 2 –4 –2 O –2 4 –4 2 4 –4 Write the equation of each line. 20. 21. 22. y y 4 2 –4 –2 O –2 y 4 2 x 2 –4 –2 O –2 2 2 –4 –2 O –2 x –4 4 2 4 x –4 Determine whether the lines are parallel, intersect, or coincide. 1 x 1 2 23. y 5 2 __ 5 24. 2x 1 3y 5 9 25. y 5 5x 2 3 2 x 2 1 y 5 5x 1 1 x 1 5y 5 10 y 5 __ 3 Copyright © by Holt, Rinehart and Winston. All rights reserved. 39 Holt McDougal Geometry Name Section 3B Date Class Ready to Go On? Quiz continued 3-6 Lines in the Coordinate Plane Write the equation of each line in the given form. 1 y 5 2 __ x 2 2 3 13. the line through (23, 21) and (3, 23) in slope-intercept form y 1 2 5 2__ 3 (x 2 6) 3 __ 14. the line through (6, 22) with slope 2 in point-slope form 4 4 15. the line with y-intercept 23 through the point (2, 5) in point-slope form y 2 5 5 4(x 2 2) 1 x 1 2 y 5 __ 2 16. the line with x-intercept 24 and y-intercept 2 in slope-intercept form Graph each line. 3 (x 1 2) 18. y 2 1 5 __ 5 17. y 5 3x 2 1 19. y 5 25 4 4 4 2 2 –4 –2 2 –4 –2 O –2 4 –4 2 –4 –2 O –2 4 –4 2 4 –4 Write the equation of each line. 20. 21. 22. y y 4 2 –4 –2 O –2 y 4 2 x 2 –4 –2 O –2 2 2 –4 –2 O –2 x –4 x54 4 2 4 x –4 3 x 2 3 y 5 __ 5 y 5 23 Determine whether the lines are parallel, intersect, or coincide. 1 x 1 2 23. y 5 2 __ 5 24. 2x 1 3y 5 9 25. y 5 5x 2 3 2 x 2 1 y 5 5x 1 1 x 1 5y 5 10 y 5 __ 3 Coincide Copyright © by Holt, Rinehart and Winston. All rights reserved. Intersect 39 Parallel Holt McDougal Geometry Name Section 3B Date Class Ready to Go On? Enrichment Slopes and Lengths of Segments Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and D(24, 21). 4 2 1. Sketch and label the quadrilateral using the grid at the right. _ –4 –2 O –2 _ 2 4 2 4 –4 2. Find the slopes of AC and BD . 3. How are the segments related? Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0). Use the information to answer the following questions: 4. Sketch and label the quadrilateral using the grid at the right. 4 2 –4 –2 O –2 –4 Find the length of each segment. 5. Find PQ. 6. Find QR. 7. Find RS. 8. Find PS. 9. What can you conclude about the side lengths of the quadrilateral? _ 10. What is the slope of PR ? _ 11. What is the slope of QS ? 12. What can you conclude about the diagonals of the quadrilateral? 13. Is the quadrilateral a square? Explain your answer. _ 14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle. Copyright © by Holt, Rinehart and Winston. All rights reserved. 40 Holt McDougal Geometry Name Section 3B Date Class Ready to Go On? Enrichment Slopes and Lengths of Segments 4• • Quadrilateral ABCD has vertices A(25, 3), B(21, 4), C(5, 23) and D(24, 21). •• 1. Sketch and label the quadrilateral using the grid at the right. –4 –2 O • • –2 _ _ 2. Find the slopes of AC and BD . 3. How are the segments related? 2 3 and __ 5 2 __ 5 3 4 2 •• –4 –2 O –2 5. Find PQ. 8. Find PS. 5 units 5 units 9. What can you conclude about the side lengths of the quadrilateral? _ 10. What is the slope of PR ? 2 4 • • 5 units 7. Find RS. 2 6. Find QR. 5 units •• –4 •• Find the length of each segment. •• They are perpendicular. 4. Sketch and label the quadrilateral using the grid at the right. 4 –4 Quadrilateral PQRS has vertices P(2, 3), Q(2, 22), R(22, 25), S(22, 0). Use the information to answer the following questions: 2 They are congruent. _ 11. What is the slope of QS ? 1 2 __ 2 12. What can you conclude about the diagonals of the quadrilateral? They are perpendicular. 13. Is the quadrilateral a square? Explain your answer. right _ No; the sides do not meet at angles. 14. A triangle has vertices L(2, 8), M(5, 9), and N(4, 2). Write a paragraph proof to show that triangle LMN is a right triangle. _ __1 is and the slope of Sample answer: Using the slope formula; the slope of LM 3 _ is 23. The product of the slopes is 21, so by the Perpendicular Lines Theorem, LN the segments are perpendicular. Perpendicular lines form right angles, and by definition, a right triangle is a triangle that has one right angle. Copyright © by Holt, Rinehart and Winston. All rights reserved. 40 Holt McDougal Geometry