Identify relationships between lines and identify angles formed by transversals

Size: px

Start display at page:

Download "Identify relationships between lines and identify angles formed by transversals"

Olivia Cross
5 years ago
Views:

1 NAME ~ ~ Practice with Exa.mples For use with pages DATE Identify relationships between lines and identify angles formed by transversals VOCABULARY Two lines are parallel lines if they are coplanar and do not intersect. Lines that do not intersect and are not coplanar are called skew lines. I Two planes that do not intersect are called parallel planes. A transversal is a line that intersects two or more coplanar lines at different points. When two lines are cut by a transversal, two angles are corresponding angles if they occupy corresponding positions. When two lines are cut by a transversal, two angles are alternate exterior angles if they lie outside the two lines on opposite sides of the transversal. I When two lines are cut by a transversal, two angles are alternate interior angles if they lie between the two lines on opposite sides of-the transversal., When two lines are cut by a transversal, two angles are consecutive interior angles (or same side interior angles) if they lie between the two lines on the same side of the transversal.. Postulate 13 Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Postulate 14 Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular I to the given line. G1IfI) Identifyi~g Relationships in Space Think of each segment in the diagram as part of a line. Which of the lines ~ppear to fit the description? ~ ~ a. parallel to AB b. skew to AB H c. parallel to Be d. Are planes ABE and CDE parallel? A E c B Practice Workbook vvith Examples

2 LESSON 3.1 NAME ~ _ For use with pages DATE SOLUTION ~ ~ a. Only CD is parallel to AB. ~ ~ ~ b. ED and EC are skew to AB. ~ ~ c. Only AD is parallel to Be. d. No, the two planes are not parallel. At the very least, we can see that the two planes intersect at point E..~~f!!.~~~I!.~.!l!.~.~lf.i!.'!!l!.~'!..! :. Think of each segment in the diagram as part of a line. Fill in the blank with parallel, skew, or perpendicular. ~ ~ 1. DE and CFare B ~~ ~ A 2. AD, BE, and CF are o 3. Plane ABC and plane DEF are ~ _ ~ ~ 4. BE and AB are Think of each segment in the diagram as part of a line. There may be more than one right answer. 5. Name a line perpendicular to lid. B c.~ 'fl I II, Iii 6. Name a plane parallel to DCH. G. ~ 7. Name a lme parallel to Be.. ~ 8. Name a line skew to FG. H All right;; reserved. Practice Workbook with Examples

3 LESSON 3.1 NAME' For use with pages _ DATE List all pairs of angles that fit the description. a. corresponding b. alternate exterior c. alternate interior d. consecutive interior a. Ll and L3 L2 and L4 L8 and L6 L7 and L5 b. Ll and L5 L8 and L4 c. L2 and L6 L7 and L3 d. L2 and L3 L7 and L6 Exercises for Example 2 e e ~ ~. ~ " ".. e! e Complete the statement with corresponding, alternate interior, alternate exterior, or consecutive interior. 9. L4 and L8 are angles. 10. L2 and L6 are angles Ll and L8 are angles. 12. L8 and L2 are angles: 13. L4 and L5 are ~ angles. 14. L5 and L 1 are angles. Geometll'Y Practice Workbook vvith Examples Copyright McDougal Littell lnc, All rights reserved,

4 NAME _ DATE For use with pages Write different types of proofs and prove results about perpendicular lines VOCABULARY :.,,. i, I Ii 'Q I ii, I I;,, A flow proof uses arrows to show the flow of the logical argument. Theorem 3.1 If two lines intersect to form a linear pair of congruent I angles, then the lines are perpendicular. Theorem 3.2 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Theorem 3.3 If two lines are perpendicular, then they intersect to form four right angles. Comparing Types of Proofs Write a two-column proof of Theorem 3.1 (a flow proof is provided in Example 2 on page 137 of ~he text). Given: Ll =: L2, Ll and L2 are a linear pair. Prove: g.lh 9 2 h,, [': " I 1. Ll =: L2 2. mll = ml2 Statements 3. L 1 and L2 are a linear pair 4. L 1 and L2 are supplementary 5. ml1 + ml2 = ml1 + mll = (mll) = ml1 = L 1 is a right L 10. g.l h 1. Given Reasons 2. Definition of congruent angles 3. Given 4. Linear Pair Postulate 5. Definition of supplementary angles 6. Substitution property of equality 7. Distributive property 8. Division property of equality 9. Definition of right angle 10. Definition of perpendicular lines (,! Copyright McDougal Littell lnc. All rights reserved, Practice Workbook vvith Examples

5 , L~SSON NAME ~~ DATE -,,--_ For use with pages , Exercises/or Example 1 :. 1. Write a two-column proof of Theorem 3.2. Note that you are asked to complete a paragraph proof of this theorem in Practice and Applications Exercise 17 on page Write a paragraph proof of Theorem 3.3. Note that you are asked to complete a flow proof related to this theorem in Practice and Applications Exercise 18 on page 139 and a two-column proof related to this theorem in Exercise 19. mti!d Application of the Theorems.tl Find the value of x. a. b. m n $OUJTION. a.x = 90 because, by Theorem 3.3, since k and e are perpendicular, all four angles formed are right angles. By definition of a right angle, x is 90~ b. By Theorem 3.3, since m and n are perpendicular, all four angles formed are right angles. By Theorem 3.2, the 62 angle and the X O angle are complementary. Thus x + 62 = 90, so x = 28.. Practice Workbook with Examples

6 LESSON 3.2 NAME ~ _ For use with pages DATE Exercises for Example 2 '.. ~ '! '.' " Find the value of x Practice Workbook with Examples

7 /. NAME~ ~ ~ ~ _ For use with pages DATE Prove and use results about parallel lines and transversals and use properties of parallel lines to solve problems IVOCABULARY Postulate 15 Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Theorem 3.4 If two parallel lines are cut by a transversal, then the pairs.of alternate.interior angles are congruent. Theorem 3.5 If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. Theorem 3.6 If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem 3.7 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Given that ml 1 = 32, find each measure. Tell which postulate or theorem you use. a. ml2 b. ml3 c. ml4 d. mls ( \--.:. SOLUTmON a. ml2 = 32 b. ml3 = 32 c. ml4 = ml3= d. mls = 32 Corresponding Angles Postulate Alternate Exterior Angles Theorem Linear Pair Postulate Vertical Angles Theorem Practice Workbook with Examples

8 - '! LESSON NAME ~ _ DATE For use with pages 143~149.~~.~r.~~~I!.~.!.f!.~.~l!.l!.'!!l!.~~.!. Find each measure given that ml6 = ml7 3. ml9 5. mlll 7. ml13 2. ml8 4. mlio 6. ntl12!!..sing Properties of Parallel Lines Use properties of parallel lines to find the value of x. (x - 8) ~ 55 x = 63 Alternate Exterior Angles Theorem Add, Exercises for Example 2... Use properties of parallel lines to find the value of x Practice Workbook with Examples

10 NAME ~ _ DATE Practice lii.1ith Examples For use with pages '. Prove that two lines are parallel and use properties of parallel lines to solve problems. 'VOCABULARV ~-! I! I, i.il'> I Postulate 16 Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Theorem 3.8 Alternate Interior Angles Converse If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Theorem 3.9 Consecutive Interior Angles Converse If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. Theorem 3.10 Alternate Exterior Angles Converse If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Proving that Two Lines are Parallel Prove that lines j and k are parallel. ~--~..----~ j ~~ k Given: ml 1 = 53 0 ml2 = Prove: j II k Statements 1. mll = Given Reasons ml2 = ml3 + ml2 = ml = ml3= ml3 = mli 6. L3 = Ll 7. -j II k 2. Linear Pair Postulate 3. Substitution prop. of equality. 4. Subtraction prop. of equality 5. Substitution prop. of equality 6. Def. of congruent angles 7. Corresponding Angles Converse. Practice Workbook vvith Examples

11 i!.~ LESSON 3.4 NAME ~ ~ ~-- DATE -'- _ For use with pages i {~.r:!.~~~l!.f!.!.l!.~. ~1!.i!.'!!.I?~i!..! ; :, Prove the statement from the given information. 1. Prove: e 1/ m 2. Prove: n II 0 Identifying Parallel Lines Determine which rays are. parallel a. Is PN parallel to SR? b. Is PO parallel to SQ? ( SOLUTION a. Decide whether PN II SR. mlnps = = 140 mlrsp = = LNPS and LRSP are congruent alternate interior angles, so PN II SR b. Decide whether PO 1/ SQ. mlops = 101 mlpsq = 98 LOPS and LPSQ are alternate interior angles, but they are not ---7 ~. congruent, so PO and S!,L are not parallel. Practice Workbook with Examples

12 LESSON 3.4 NAME ~ DATE For use with pages ~~~!.t?~~~.~.!.l!!..~l!.;!.t!!l!.~f!..?. Find the value of x that makes a II b a b 5. a --~:7-- b ~ _ Co'pyright McDougal Littell Inc. Practice Workbook With ExampleS

13 NAME~. ~~ ~ ~ ~ DATE Practice with. Examples' For use with pages Use properties of parallel lines and construct parallel lines =l using straightedge and compass VOCABULARY. Theorem 3.11 If two lines are parallel to the same line, then they are'. i parallel to each other., Theorem 3.12 In a plane, if two lines are perpendicular to the same I. line, then they are parallel to each other, I L-....J " Showing Lines are Parallel Explain how you would show that k II f. a. k b. n a. Because the 40 angle and angle I form a linear pair, ml 1 must equal 140, Thus L 1 and the other 140 angle are congruent. Because they are also corresponding angles, lines k and e are parallel by the Corresponding Angles Converse postulate, b. Because the three angles with measures of 6xo, 5~0, and Xo form a straight line, their sum must be 180. So 6x + 5x + x = 180, Thus 12x = 180, and therefore x = 15. We can now conclude that the angle with the 6xo measure is a right angle [6. (15) = 90]. Therefore, line n is perpendicular to line f. Since line k is also perpendicular to line n (the 90 angle is indicated), lines k and e are parallel by Theorem 3,12.!. Geovneti'Y Practice VVorkbook with Examples Copyright McDougal littell lnc.

14 LESSON CONTiNUED NAME _ DATE -For use with pages ~~~!.~~~I!.~.!.~L~l!.I!.r!!.l!.~~.!. ~. Explain how you would show kill k f 3. f.~ Naming Parallel Lines Determine which lines, if any, must be parallel. a Lines c arid d are parallel because they have congruent corresponding angles. Likewise, lines d and e are parallel because they have congruent corresponding angles. Also, lines c and e are parallel because they are both parallel to the same line, line d. Because ml 1 = 105 (L 1 and the 75 angle form a linear pair), L I and the 100 angle are not congruent. Since L 1 and the 100 angle are corresponding angles that are not congruent, lines a and b are not parallel. Copyright McDougal Littell. Inc. Geometoy Practice Workbook with Examples

15 m:::z;s'?t'f'=on efeh ft LESSON 3.5 NAME For use with pages _ DATE.~~~!.~~~l!.~.!.l!.~. ~1!.f!.I!!J!.~I!..?. Determine which lines, if any, must be parallel..~, 4. b 5. h c d 9 6. Practice Workbook with Examples

16 NAME _ DATE :-- _ For use with pages Find slopes of lines and use slope to identify parallel lines in a coordinate plane and write equations of parallel lines in a coordinate plane Postulate 17 Slopes of Parallel Lines In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. ~ finding the Slope of a Line Find the slope of the line that passes through the points (3, - 3) and (0, 9). SOLUTION Let (Xl' Yl) = (3, -3) and (x 2 ' h) = (0,9). 9 - (- 3) = -4 The slope of the line is , Exercises for Example 1 ~,. Find the slope of the line that passes through the given points. 1. (4,2) and (6,8) 2. (-3, -1) and (- 5, -11) 3. (;- 8, 12)and (0, -12) 4. (8,3) and (14,5) 5. (-7, -5) and (5,4) 6. (-18,5) and (4, 5) i i I' I Practice Workbook with Examples

17 NAME _ DATE _ For ase with pages Find slopes of lines and use slope to identify parallel lines in a coordinate plane and write equations of parallel lines in a coordinate plane VOCAiSUIi..ARY Postulate 17 Slopes of Parallel Lines In a coordinate plane, two nonverticallines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Find the slope of the line that passes through the points (3, - 3) and (0, 9). SOLUTION Let (xl' Yl) = (3, - 3) and (x2' h) = (0,9). m= Y2 - YI X 2 - Xl 9 - (- 3) = -4 The slope of the line is - 4. Exercises toit Example 1.. Find the slope of the line that passes through the given points. 1. (4,2) and (6, 8) 2. (- 3, -1) and (- 5, -11) 3. (;- 8,12) and (0, -12) 4. (8,3) and (14,5) 5. (-7, -5) and (5,4) 6. (-18,5) and (4, 5) AII.rights reserved. Practice Workbook with Examples

18 NAME For use with pages DATE Ide'ntiifvilffl Parallel lines Find the slope of each line. Is a lib? 'YV /l- / (O,5) a b / ; ~ / 1/1 -< (0,2) (-5( 0)/! i /! / J/r -1 0 x.. _c c ) i/ 1/ I / /1 Find the slope of a. Line a passes through (- 5; 0) and (0, 5) lna = 0 - (- 5) = "5 = 1 Find the slope of b. Line b passes through (- 2, 0) and (0, 2) mb = 0 - (- 2) = "2 = 1 Compare the slopes. Because a and b have the same slope, they are parallel. Exercises for Example 2.. ~ ~... " ".... e c.' e Find the slope of each line. Which lines are parallel? Y 8., I h7~ 2) 1- i(5, 1) i! 1 x' i ( 1) -( -7,! - ~) i! I.~.~. I Practice Workbook with Examples Copyright McDougal Littell Inc.

19 LESSON NAME_.~ _ DATE For use with pages ,Writing an Equation of a Parallel Line Line k has the equation y = - x - 4. Line.e is parallel to kand passes through the point (1, 5). Write an equation of.e. SOLUTION I I: iii t I Find the slope. The slope of k is - 1. Because parallel lines have the same slope, the slope of.e is also - 1. Solve for b. Use (x, y) = (1,5) and m. = -1. y=mx+b 5=-1(1)+b 5=-I+b 6=b Write an equation. Because m = - 1 and b = 6, an equation of.e is y = - x + 6. t~.~..~~.f!!.~ ~~~~. ~l!.l!.'!!.f!.~i!..~. Write an equation of the line the passes through the. given point P and is parallel to the line with the given equation. 9. P(10, 3), y = x - 12 f I II 10. p( - 5,2), y = -x - 9 I PC-I, 2),y = '3x - 2. Practice Workbook with Examples

20 NAME ~ ~-- DATE For use with pages Use slope to identify perpendicular lines in a coordinate plane and write equations of perpendicular lines VOCABULARY Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, two nonverticallines are perpendicular if and only if the product of their slopes is - 1. Deciding Whether Lines are Perpendicular a.. Decide whether ~ ~ PQ and QR are perpendicular. b. Decide whether the lines are perpendicular. Line f: 2x - 3y = -4 Line k: 3x + 2y = 3 SOLUTiON a. Find each slope. ~ Slope of PQ = 0 - (- 4) =4 ~ Slope of OR = -- = - = Multiply slopes to see if the lines are perpendicular. l. (-3) = -~ 4 4. ~ ~ The product of the slopes is not - 1. SO, PQ and QR are not perpendicular. ~- ~,c~practice Vvorkbook with Examples Copyright McDougal Littell lnc.

21 l --1& LESSON ~ I NAME ~~ _ DATE For use with pages b. Rewrite each equation in slope-intercept form to find the slope. Line e. Line k: Y = 3x + 3 y = -"2x + "2 2 3 slope = - slope = ' Multiply the slopes to see if the lines are perpendicular. G). (-1) = - 1, so the lines are perpendicular..~~.i!.~i?~~i!.~.!.~!..~jf.'!.'!!l!/i!..!,. Decide whether lines k and e are perpendicular. 1. k passes through (3, 2) and (- 1, 5) e passes through (0,2) and (3, 6) I i 2. k has the equation 2x - 4y = - 3 e has the equation x + 2y = - 6 Practice Workbook vvith Exqmples

22 ~.' LESSON NAME _ DATE _ For use with pages Wiitifl!j the Equation of a Perpendicular Line Line k has equation y = ~x - ~. Find an equation of line f that passes through P(3, -1) and is perpendicular to k. SOU.JTION First determine the slope of e. For k and e to be perpendicular, their slopes must equal - 1. Ink' me = me = me = -- 2 the product of. 3 Then use In = -2 and (x, y) = (3,-1) to find b. y=mx+b 3-1 = --. (3) + b 2 2=b 2 S. f o : 3 7 0, an equation 0 -(.IS Y = -lx + 2.~~~!.~~~I!.~.!.l!.~.~Jf.l!.'!!P..~I!..?. Line j is perpendicular to the line with the given equation and line j passes through P. Write an equation of line j. 3. 4x + 7y = 13,P(-2,6) 4. 5x - 2y = 3,P(O, -~)5. x + 5y = 6,P(-1,2) ~..~.".'.-:-.. ~. PIrlMir.R Workbook with Examples Copyright McDougal.Littell Inc.

23 LESSON 3.7 NAME For use with pages _ DATE Writin.g, = the Equation of a,perpendicular Line Line k has equation y = ~x - rfind an equation of line f that passes' through P(3, - 1) and is perpendicular to k. SOLUTION, I First determine the slope of.e. For k and.e to be perpendicular, their slopes must equal - 1. m k rne = m = -1 3 e the product of, 3 me =-- 2, 3 Then use m = -'2 and (x, y) = (3, - 1) to find b. y =mx + b -1 = _l. (3) + b 2 S 2=b 2. f ()' 3 7 0, an equation 0 '(, IS Y = -'2x + '2.~~.~!.~~~'!.~.!.C?~. ~'!.l!.'!!.1?~'!..?. line j is perpendicular to the line with the given equation and line j passes through P. Write an equation of line j. 3. 4x + 7y = 13,P(-2,6) 4. 5x - 2y = 3,P(O, -%),5. x + 5y = 6,P(-1,2) Ge!timetry, Practice Workbook with Examples. Copyright McDougal Littell Inc. All rights reserved,

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

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