Proving Congruence SSS, SAS

Transcription

1 Proving ongruence SSS, SS Use the SSS Postulate to test for triangle congruence. Use the SS Postulate to test for triangle congruence. Vocabulary included angle do land surveyors use congruent triangles? and surveyors mark and establish property boundaries. To check a measurement, they mark out a right triangle and then mark a second triangle that is congruent to the first. SSS POSTUT Is it always necessary to show that all of the corresponding parts of two triangles are congruent to prove that the triangles are congruent? In this lesson, we will explore two other methods to prove that triangles are congruent. ongruent Triangles Using Sides 1 raw a triangle and label the vertices,, and Z. 2 Use a straightedge to draw any line and select a point. Use a compass to construct S on such that S Z. 3 Using as the center, draw an arc with radius equal to. Z S S 4 Using S as the center, draw an arc with radius equal to Z. 5 et T be the point of intersection of the two arcs. raw T and S T to form ST. 6 ut out ST and place it over Z. How does ST compare to Z? T S S 200 hapter 4 ongruent Triangles Paul onklin/photodit If the corresponding sides of two triangles are congruent, then the triangles are congruent. This is the Side-Side-Side Postulate, and is written as SSS.

2 Postulate 4.1 Side-Side-Side ongruence If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Z bbreviation: SSS Z xample 1 Use SSS in Proofs IN IOO The tail of an orca whale can be viewed as two triangles that share a common side. Write a two-column proof to prove that if and. iven: ; Prove: Proof: Statements easons 1. ; 1. iven eflexive Property SSS xample 2 SSS on the oordinate Plane OOINT OT etermine whether TZ for (2, 5), Z(1, 1), T(5, 2), ( 3, 0), ( 7, 1), and ( 4, 4). xplain. Use the istance ormula to show that the corresponding sides are congruent. y Z O T x T (2 ) 5 2 (5 2) 2 [ 4 ( 7)] 2 (4 1) or or 18 TZ (5 ) 1 2 (2 1) 2 [ 7 ( 3)] 2 (1 0) or or 17 Z (2 ) 1 2 (5 1) 2 [ 4 ( 3)] 2 (4 0) or or 17 T, TZ, and Z. y definition of congruent segments, all corresponding segments are congruent. Therefore, TZ by SSS. SS POSTUT Suppose you are given the measures of two sides and the angle they form, called the included angle. These conditions describe a unique triangle. Two triangles in which corresponding sides and the included pairs of angles are congruent provide another way to show that triangles are congruent. esson 4-4 Proving ongruence SSS, SS 201 effrey ich/pictor International/PictureQuest

3 Postulate 4.2 Side-ngle-Side ongruence If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. bbreviation: SS ou can also construct congruent triangles given two sides and the included angle. ongruent Triangles using Two Sides and the Included ngle 1 raw a triangle and label its vertices,, and. 2 Select a point on line m. Use a compass to construct on m such that. 3 onstruct an angle congruent to using as a side of the angle and point as the vertex. 4 onstruct such that. raw to complete. m m m 5 ut out and place it over. How does compare to? Study Tip low Proofs low proofs can be written vertically or horizontally. xample 3 Write a flow proof. iven: Prove: Use SS in Proofs is the midpoint of. is the midpoint of. low Proof: is the midpoint of. iven is the midpoint of. iven idpoint Theorem idpoint Theorem SS Vertical s are. 202 hapter 4 ongruent Triangles

4 xample 4 Identify ongruent Triangles etermine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible. a. b. ach pair of corresponding sides are congruent. The triangles are congruent by the SSS Postulate. The triangles have three pairs of corresponding angles congruent. This does not match the SSS Postulate or the SS Postulate. It is not possible to prove the triangles congruent. oncept heck 1. OPN N raw a triangle and label the vertices. Name two sides and the included angle. 2. IN TH O armelita and onathan are trying to determine whether is congruent to. armelita π π by SS Who is correct and why? onathan ongruence cannot be determined uided Practice etermine whether NP given the coordinates of the vertices. xplain. 3. ( 4, 3), ( 2, 1), ( 2, 3), (4, 3), N(2, 1), P(2, 3) 4. ( 2, 2), ( 4, 6), ( 3, 1), (2, 2), N(4, 6), P(3, 1) 5. Write a flow proof. iven: and bisect each other. Prove: 6. Write a two-column proof. iven:, Prove: xercise 5 xercise 6 etermine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible esson 4-4 Proving ongruence SSS, SS 203

5 pplication 9. PISION IHT The United States Navy light emonstration Squadron, the lue ngels, fly in a formation that can be viewed as two triangles with a common side. Write a two-column proof to prove that ST QT if T is the midpoint of S Q and S Q. S T Q Practice and pply or xercises , See xamples xtra Practice See page etermine whether H given the coordinates of the vertices. xplain. 10. ( 3, 2), ( 7, 4), ( 1, 9), (2, 3), (4, 7), H(9, 1) 11. ( 1, 1), ( 2, 2), ( 5, 1), (2, 1), (3, 2), H(2, 5) 12. ( 1, 1), (0, 6), (2, 3), (3, 1), (5, 3), H(8, 1) 13. (3, 9), (4, 6), (1, 5), (1, 7), (2, 4), H( 1, 3) Write a flow proof. 14. iven:,, 15. iven: Q T Q Q W Q Q WQT Prove: Prove: QWT Q Q W T Write a two-column proof. 16. iven: is isosceles. 17. iven: N QP is the midpoint of. NP QPN Prove: Prove: NP QPN Q N P 18. iven: 19. iven: H bisects. Prove: H Prove: H 204 hapter 4 ongruent Triangles laine Thompson/P/Wide World Photos

6 20. TS cat s ear is triangular in 21. S This photograph shows a shape. Write a two-column proof to flock of geese flying in formation. prove ST PN if S P N, Write a two-column proof to prove T P, S N, and that H, if H and T. is the midpoint of H. S N P T H etermine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible S or xercises 26 and 27, use the following information. baseball diamond is a square with four right angles and all sides congruent. 26. Write a two-column proof to prove that the distance from first base to third base is the same as the distance from home plate to second base. 27. Write a two-column proof to prove that the angle formed by second base, home plate, and third base is the same as the angle formed by second base, home plate, and first base. aseball The infield is a square 90 feet on each side. Source: ITI THININ evise a plan and write a two-column proof for the following. iven:,,, Prove: 29. WITIN IN TH nswer the question that was posed at the beginning of the lesson. How do land surveyors use congruent triangles? Include the following in your answer: description of three methods to prove triangles congruent, and another example of a career that uses properties of congruent triangles. esson 4-4 Proving ongruence SSS, SS 205 (tl).. & Vikki Hart/Photoisc, (tr)hase Swift/OIS, (b)index Stock

7 Standardized Test Practice 30. Which of the following statements about the figure is true? 90 a b a b 90 a b 90 a b b a 31. lassify the triangle with the measures of the angles in the ratio 3:6:7. isosceles acute obtuse right aintain our Skills ixed eview Identify the congruent triangles in each figure. (esson 4-3) N W Z P ind each measure if P Q Q. (esson 4-2) 35. m m m m m m 6 or xercises 41 43, use the graphic at the right. (esson 3-3) 41. ind the rate of change from first quarter to the second quarter. 42. ind the rate of change from the second quarter to the third quarter. 43. ompare the rate of change from the first quarter to the second, and the second quarter to the third. Which had the greater rate of change? US TO Snapshots P slides in 2001 ross domestic product in private industries, which generate 88% of P, slowed to 4.1% in 2000 from 4.8% in % irst quarter 1.3% Second quarter Source: The ureau of conomic nalysis Percentage changes for the first three quarters of 2001: 0.3% Third quarter 1.1% y Shannon eilly and Suzy Parker, US TO etting eady for the Next esson PQUISIT SI and are angle bisectors and segment bisectors. Name the indicated segments and angles. (To review bisectors of segments and angles, see essons 1-5 and 1-6.) 44. a segment congruent to 45. an angle congruent to 46. an angle congruent to 47. a segment congruent to 206 hapter 4 ongruent Triangles