2.7 Angles and Intersecting Lines

Transcription

1 Investigating g Geometry TIVITY Use before Lesson.7.7 ngles and Intersecting Lines M T R I LS graphing calculator or computer Q U S T I O N What is the relationship between the measures of the angles formed by intersecting lines? You can use geometry drawing software to investigate the measures of angles formed when lines intersect. X P L O R Measure linear pairs formed by intersecting lines STP raw two intersecting lines raw and label ]. raw and label ] so that it intersects ]. raw and label the point of intersection. STP STP Type a name: XPLOR ancel OK Measure angles Measure,, and. Move point to change the angles. Save Save as XPLOR by choosing Save from the F menu and typing the name. R W O N L U S I O N S Use your observations to complete these exercises. escribe the relationship between and.. escribe the relationship between and.. What do you notice about and?. In xplore, what happens when you move to a different position? o the angle relationships stay the same? Make a conjecture about two angles supplementary to the same angle. 5. o you think your conjecture will be true for supplementary angles that are not adjacent? xplain. hapter Reasoning and Proof

2 X P L O R Measure complementary angles classzone.com Keystrokes STP raw two perpendicular lines raw and label ]. raw point on ]. raw and label ] ]. raw and label point on ] so that is between and as shown in Step. STP STP G G F F raw another line raw and label ] G so that G is in the interior of. raw point F on ] G as shown. Save as XPLOR. Measure angles Measure F, F, G, and G. Move point G to change the angles. X P L O R Measure vertical angles formed by intersecting lines STP raw two intersecting lines raw and label ]. raw and label ] so that it intersects ]. raw and label the point of intersection. STP Measure angles Measure,,, and. Move point to change the angles. Save as XPLOR. R W O N L U S I O N S Use your observations to complete these exercises 6. In xplore, does the angle relationship stay the same as you move G? 7. In xplore, make a conjecture about the relationship between G and G. Write your conjecture in if-then form. 8. In xplore, the intersecting lines form two pairs of vertical angles. Make a conjecture about the relationship between any two vertical angles. Write your conjecture in if-then form. 9. Name the pairs of vertical angles in xplore. Use this drawing to test your conjecture from xercise 8..7 Prove ngle Pair Relationships

3 Prove ngle Pair.7 Relationships efore You identified relationships between pairs of angles. Now You will use properties of special pairs of angles. Why? So you can describe angles found in a home, as in x.. Key Vocabulary complementary angles, p. 5 supplementary angles, p. 5 linear pair, p. 7 vertical angles, p. 7 Sometimes, a new theorem describes a relationship that is useful in writing proofs. For example, using the Right ngles ongruence Theorem will reduce the number of steps you need to include in a proof involving right angles. THORM THORM. Right ngles ongruence Theorem ll right angles are congruent. For Your Notebook Proof: below WRIT PROOFS When you prove a theorem, write the hypothesis of the theorem as the GIVN statement. The conclusion is what you must PROV. P RO O F Right ngles ongruence Theorem GIVN c and are right angles. PROV c > STTMNTS. and are right angles.. m 5 908, m m 5 m. > RSONS. Given. efinition of right angle. Transitive Property of quality. efinition of congruent angles X M P L Use right angle congruence VOI RRORS The given information in xample is about perpendicular lines. You must then use deductive reasoning to show the angles are right angles. Write a proof. GIVN c } }, } } PROV c > STTMNTS. } }, } }. and are right angles.. > RSONS. Given. efinition of perpendicular lines. Right ngles ongruence Theorem hapter Reasoning and Proof

4 THORMS For Your Notebook THORM. ongruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If and are supplementary and and are supplementary, then >. Proof: xample, below; x. 6, p. 9 THORM.5 ongruent omplements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If and 5 are complementary and 6 and 5 are complementary, then > Proof: x. 7, p. 9; x., p. 0 To prove Theorem., you must prove two cases: one with angles supplementary to the same angle and one with angles supplementary to congruent angles. The proof of Theorem.5 also requires two cases. X M P L Prove a case of ongruent Supplements Theorem Prove that two angles supplementary to the same angle are congruent. GIVN c and are supplements. and are supplements. PROV c > STTMNTS. and are supplements. and are supplements.. m m m m m m 5 m m. m 5 m 5. > RSONS. Given. efinition of supplementary angles. Transitive Property of quality. Subtraction Property of quality 5. efinition of congruent angles at classzone.com GUI PRTI for xamples and. How many steps do you save in the proof in xample by using the Right ngles ongruence Theorem?. raw a diagram and write GIVN and PROV statements for a proof of each case of the ongruent omplements Theorem..7 Prove ngle Pair Relationships 5

5 INTRSTING LINS When two lines intersect, pairs of vertical angles and linear pairs are formed. The relationship that you used in Lesson.5 for linear pairs is formally stated below as the Linear Pair Postulate. This postulate is used in the proof of the Vertical ngles ongruence Theorem. POSTULT For Your Notebook POSTULT Linear Pair Postulate If two angles form a linear pair, then they are supplementary. and form a linear pair, so and are supplementary and m m THORM For Your Notebook THORM.6 Vertical ngles ongruence Theorem Vertical angles are congruent. Proof: xample, below >, > X M P L Prove the Vertical ngles ongruence Theorem Prove vertical angles are congruent. GIVN c 5 and 7 are vertical angles. PROV c 5 > US IGRM You can use information labeled in a diagram in your proof. STTMNTS. 5 and 7 are vertical angles.. 5 and 6 are a linear pair. 6 and 7 are a linear pair.. 5 and 6 are supplementary. 6 and 7 are supplementary.. 5 > 7 RSONS. Given. efinition of linear pair, as shown in the diagram. Linear Pair Postulate. ongruent Supplements Theorem GUI PRTI for xample In xercises 5, use the diagram.. If m 5 8, find m, m, and m.. If m 5 678, find m, m, and m. 5. If m 5 78, find m, m, and m. 6. Which previously proven theorem is used in xample as a reason? 6 hapter Reasoning and Proof

6 X M P L Standardized Test Practice LIMINT HOIS Look for angle pair relationships in the diagram. The angles in the diagram are supplementary, not complementary or congruent, so eliminate choices and. Which equation can be used to find x? (x ) 5 90 (x ) x x 5 T P 8 (x )8 P R S Solution ecause TPQ and QPR form a linear pair, the sum of their measures is 808. c The correct answer is. GUI PRTI for xample Use the diagram in xample. 7. Solve for x. 8. Find m TPS..7 XRISS SKILL PRTI HOMWORK KY 5 WORK-OUT SOLUTIONS on p. WS for xs. 5,, and 9 5 STNRIZ TST PRTI xs., 7, 6, 0, and 5. VOULRY opy and complete: If two lines intersect at a point, then the? angles formed by the intersecting lines are congruent.. WRITING escribe the relationship between the angle measures of complementary angles, supplementary angles, vertical angles, and linear pairs. XMPLS and on pp. 5 for xs. 7 INTIFY NGLS Identify the pair(s) of congruent angles in the figures below. xplain how you know they are congruent.. N P P. is supplementary to. is supplementary to F. M 508 S 508 R F 5. F G H J K 58 8 M L W X Y Z 6. G L M K H J.7 Prove ngle Pair Relationships 7

7 7. SHORT RSPONS The x-axis and y-axis in a coordinate plane are perpendicular to each other. The axes form four angles. re the four angles congruent right angles? xplain. XMPL on p. 6 for xs. 8 FINING NGL MSURS In xercises 8, use the diagram at the right. 8. If m 5 58, find m, m, and m. 9. If m 5 688, find m, m, and m. 0. If m 5 78, find m, m, and m.. If m 5 68, find m, m, and m. XMPL on p. 7 for xs.. LGR Find the values of x and y.. (8x 7)8 x8 (7y )8 5y8 (7y )8 (6y 8)8 (6x 6)8 (9x )8. (0x )8 (8y 8)8 6y8 6(x )8 5. RROR NLYSIS escribe the error in stating that > and >. > > 6. MULTIPL HOI In a figure, and are complementary angles and m 5 x8. Which expression can be used to find m? (x 90)8 (80 x)8 (80 x)8 (90 x)8 FINING NGL MSURS In xercises 7, copy and complete the statement given that m FH 5 m HG 5 m HF If m 5 08, then m 6 5?. 8. If m HF 5 58, then m 5?. 9. If m , then m 5?. 0. If m HF 5 8, then m HG 5?.. If m 5 8, then m 5?. 7 H 6 G F NLYZING STTMNTS Two lines that are not perpendicular intersect such that and are a linear pair, and are a linear pair, and and are vertical angles. Tell whether the statement is true.. >. >. > 5. > 6. > 7. m m LGR Find the measure of each angle in the diagram. 8. 0y8 (y )8 (x )8 (7x )8 9. (5x 5)8 (5y 5)8 (7y 9)8 (6x 50)8 8 5 WORK-OUT SOLUTIONS on p. WS 5 STNRIZ TST PRTI

8 0. OPN-N MTH In the diagram, m Y and ] XY bisects. Give two more true statements about the diagram. X Y RWING ONLUSIONS In xercises, use the given statement to name two congruent angles. Then give a reason that justifies your conclusion.. In triangle GF, ] GH bisects GF.. is a supplement of 6, and 9 is a supplement of 6.. } is perpendicular to }, and } and } intersect at.. 5 is complementary to, and is complementary to. 5. HLLNG Sketch two intersecting lines j and k. Sketch another pair of linesl and m that intersect at the same point as j and k and that bisect the angles formed by j and k. Linel is perpendicular to line m. xplain why this is true. PROLM SOLVING XMPL on p. 5 for x PROVING THORM. Prove the second case of the ongruent Supplements Theorem where two angles are supplementary to congruent angles. GIVN c and are supplements. and are supplements. > PROV c > 7. PROVING THORM.5 opy and complete the proof of the first case of the ongruent omplements Theorem where two angles are complementary to the same angles. GIVN c and are complements. and are complements. PROV c > STTMNTS. and are complements. and are complements.. m m m m ?.? 5. > RSONS.?.?. Transitive Property of quality. Subtraction Property of quality 5.?.7 Prove ngle Pair Relationships 9

9 PROOF Use the given information and the diagram to prove the statement. 8. GIVN c is a right angle. 9. GIVN c } JK } JM, } KL } ML, is a right angle. J > M, K > L PROV c > PROV c } JM } ML and } JK } KL J K M L 0. MULTI-STP PROLM Use the photo of the folding table. a. If m 5 x8, write expressions for the other three angle measures. b. stimate the value of x. What are the measures of the other angles? c. s the table is folded up, gets smaller. What happens to the other three angles? xplain your reasoning.. PROVING THORM.5 Write a two-column proof for the second case of Theorem.5 where two angles are complementary to congruent angles. WRITING PROOFS Write a two-column proof.. GIVN c >. GIVN c QRS and PSR are PROV c > supplementary. PROV c QRL > PSR M L R S P P N K. STIRS Use the photo and the given information to prove the statement. GIVN c is complementary to. is complementary to. PROV c > TW, and ] TX and ] TW are opposite rays. You want to show STX > VTX. 5. XTN RSPONS STV is bisected by ] a. raw a diagram. b. Identify the GIVN and PROV statements for the situation. c. Write a two-column proof. 0 5 WORK-OUT SOLUTIONS on p. WS 5 STNRIZ TST PRTI

10 6. USING IGRMS opy and complete the statement with <, >, or 5. a. m? m 7 b. m? m 6 c. m 8 m 6? d. If m 5 08, then m 5? m HLLNG In xercises 7 and 8, write a two-column proof. 7. GIVN c m WYZ 5 m TWZ GIVN c The hexagon is regular. PROV c SWZ > XYW PROV c m m X Y Z S W T MIX RVIW PRVIW Prepare for Lesson. in xs In xercises 9 5, sketch a plane. Then sketch the described situation. (p. ) 9. Three noncollinear points that lie in the plane 50. line that intersects the plane at one point 5. Two perpendicular lines that lie in the plane 5. plane perpendicular to the given plane 5. Sketch the next figure in the pattern. (p. 7) QUIZ for Lessons.6.7 Match the statement with the property that it illustrates. (p. ). If } HJ > } LM, then } LM > } HJ.. Reflexive Property of ongruence. If > and >, then >.. Symmetric Property of ongruence. XYZ > XYZ. Transitive Property of ongruence. Write a two-column proof. (p. ) GIVN c XWY is a straight angle. ZWV is a straight angle. PROV c XWV > ZWY X Z W V Y XTR PRTI for Lesson.7, p. 899 ONLIN QUIZ at classzone.com

11 MIX RVIW of Problem Solving Lessons.5.7. MULTI-STP PROLM In the diagram below, ] bisects and ] bisects. a. Prove m 5 m. b. If m 5 998, what is m? xplain.. SHORT RSPONS You are cutting a rectangular piece of fabric into strips that you will weave together to make a placemat. s shown, you cut the fabric in half lengthwise to create two congruent pieces. You then cut each of these pieces in half lengthwise. o all of the strips have the same width? xplain your reasoning. 5. XTN RSPONS formula you can use to calculate the total cost of an item including sales tax is T 5 c( s), where T is the total cost including sales tax, c is the cost not including sales tax, and s is the sales tax rate written as a decimal. a. Solve the formula for s. Give a reason for each step. b. Use your formula to find the sales tax rate on a purchase that was $6.75 with tax and $5 without tax. c. Look back at the steps you used to solve the formula for s. ould you have solved for s in a different way? xplain. 6. OPN-N In the diagram below, m G What additional information do you need to find m and m? xplain your reasoning. STT TST PRTI classzone.com G F. GRI NSWR The cross section of a concrete retaining wall is shown below. Use the given information to find the measure of in degrees. m 5 m m 5 m m m m m m XTN RSPONS xplain how the ongruent Supplements Theorem and the Transitive Property of ngle ongruence can both be used to show how angles that are supplementary to the same angle are congruent. 7. SHORT RSPONS Two lines intersect to form,,, and. The measure of is three times the measure of and m 5 m. Find all four angle measures. xplain your reasoning. 8. SHORT RSPONS Part of a spider web is shown below. If you know that and are complements and that ] and ] F are opposite rays, what can you conclude about and F? xplain your reasoning. hapter Reasoning and Proof