Proving Congruence ASA, AAS

Transcription

1 roving ongruence, Vocabulary included side Use the ostulate to test for triangle congruence. Use the heorem to test for triangle congruence. are congruent triangles used in construction? he ank of hina ower in ong ong has triangular trusses for structural support. hese trusses form congruent triangles. In this lesson, we will explore two additional methods of proving triangles congruent. OU uppose you were given the measures of two angles of a triangle and the side between them, the included side. o these measures form a unique triangle? ongruent riangles Using wo ngles and Included ide 1 raw a triangle and label its vertices,, and. 2 raw any line m and select a point. onstruct such that. 3 onstruct an angle congruent to at using as a side of the angle. 4 onstruct an angle congruent to at using as a side of the angle. abel the point where the new sides of the angles meet. m m m 5 ut out and place it over. ow does compare to? his construction leads to the ngle-ide-ngle ostulate, written as. tudy ip eading ath he included side refers to the side that each of the angles share. ostulate 4.3 ngle-ide-ngle ongruence If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. bbreviation: W W esson 4-5 roving ongruence, 207 ylvain randadam/hoto esearchers

2 xample 1 Write a paragraph proof. iven: roof: Use in roofs bisects and. ince bisects and, and. by the eflexive roperty. y,. O uppose you are given the measures of two angles and a nonincluded side. Is this information sufficient to prove two triangles congruent? ngle-ngle-ide ongruence odel 1. raw a triangle on a piece of patty paper. abel the vertices,, and. 2. opy,, and on another piece of patty paper and cut them out. 3. ssemble them to form a triangle in which the side is not the included side of the angles. nalyze 1. lace the original over the assembled figure. ow do the two triangles compare? 2. ake a conjecture about two triangles with two angles and the nonincluded side of one triangle congruent to two angles and the nonincluded side of the other triangle. his activity leads to the ngle-ngle-ide heorem, written as. heorem 4.5 ngle-ngle-ide ongruence If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. bbreviation: xample: roof heorem 4.5 iven:,, roof: tatements easons 1.,, 1. iven hird ngle heorem hapter 4 ongruent riangles

3 tudy ip Overlapping riangles When triangles overlap, it is a good idea to draw each triangle separately and label the congruent parts. xample 2 Write a flow proof. iven: Flow roof: Use in roofs iven iven eflexive roperty ou have learned several methods for proving triangle congruence. he oncept ummary lists ways to help you determine which method to use. efinition of ongruent riangles ethods to rove riangle ongruence ll corresponding parts of one triangle are congruent to the corresponding parts of the other triangle. he three sides of one triangle must be congruent to the three sides of the other triangle. wo sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle. wo angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle. wo angles and a nonincluded side of one triangle must be congruent to two angles and side of the other triangle. rchitect bout 28% of architects are self-employed. rchitects design a variety of buildings including offices, retail spaces, and schools. Online esearch For information about a career as an architect, visit: com/careers xample 3 etermine if riangles re ongruent IU his glass chapel was designed by Frank loyd Wright s son, loyd Wright. uppose the redwood supports, U and V, measure 3 feet, 1.6 feet, and m U and m V are 31. etermine whether U V. ustify your answer. xplore lan olve We are given three measurements of each triangle. We need to determine whether the two triangles are congruent. ince m U m V, U V. ikewise, U V so U V, and so. heck each possibility using the five methods you know. We are given information about side-side-angle (). his is not a method to prove two triangles congruent. V (continued on the next page) esson 4-5 roving ongruence, 209 U (l)ennis aconald/hotodit, (r)ichael ewman/hotodit

4 xamine Use a compass, protractor, and ruler to draw a triangle with the given measurements. For simplicity of measurement, we will use centimeters instead of feet, so the measurements of the construction and those of the support beams will be proportional. raw a segment 3.0 centimeters long. t one end, draw an angle of 31. xtend the line longer than 3.0 centimeters. 3.0 cm t the other end of the segment, draw an arc with a radius of 1.6 centimeters such that it intersects the line cm otice that there are two possible segments that could determine the triangle. ince the given measurements do not lead to a unique triangle, we cannot show that the triangles are congruent. oncept heck uided ractice 1. Find a counterexample to show why (ngle-ngle-ngle) cannot be used to prove triangle congruence. 2. O raw a triangle and label the vertices. ame two angles and the included side. 3. xplain why is a theorem, not a postulate. Write a flow proof. 4. iven:, 5. iven: W Z, Z W ZW W Z Write a paragraph proof. 6. iven: Q bisects ;. 7. iven:, Q Q Q pplication 8. U uppose and each measure 7 feet, and each measure 5.5 feet, and m m 49. etermine whether. ustify your answer. 210 hapter 4 ongruent riangles

5 ractice and pply For xercises 9, 11, 14, , 12, 13, 19, ee xamples xtra ractice ee page Write a flow proof. 9. iven: F, F 10. iven:, bisects. F 11. iven: V, V Q 12. iven: F,, F V F V iven: Q, Q 14. iven: Z is the midpoint of. 2 3 Q F Q Z Z Z Q Write a paragraph proof. 15. iven: O O, 16. iven: bisects,, O O O 17. iven: F, 18. iven: F F esson 4-5 roving ongruence, 211

6 Write a two-column proof. 19. iven: 20. iven: I I I I I For xercises 21 and 22, use the following information. eth is planning a garden. he wants the triangular sections, F and F, to be congruent. F is the midpoint of, and 16 feet. 21. uppose and each measure 4 feet and the measure of F is 29. etermine whether F F. ustify your answer. 22. uppose F is the midpoint of, and. etermine whether F F. ustify your answer. F ites he largest kite ever flown was 210 feet long and 72 feet wide. ource: uinness ook of World ecords I For xercises 23 and 24, use the following information. ustin is building a kite. uppose is 2 feet, is 2.7 feet, and the measure of is If is the midpoint of and, determine whether. ustify your answer. 24. If and, determine whether. ustify your answer. omplete each congruence statement and the postulate or theorem that applies. 25. If I V and 2 5, then I? by?. 26. If I V and I V, then I? by?. 27. If I V and bisect each other, then V? by?. 28. If I V and 1 6, then V? by?. I V 29. II II iko wants to estimate the distance between herself and a duck. he adjusts the visor of her cap so that it is in line with her line of sight to the duck. he keeps her neck stiff and turns her body to establish a line of sight to a point on the ground. hen she paces out the distance to the new point. Is the distance from the duck the same as the distance she just paced out? xplain your reasoning. 212 hapter 4 ongruent riangles ourtesy eter ynn ites

7 30. WII I nswer the question that was posed at the beginning of the lesson. ow are congruent triangles used in construction? Include the following in your answer: explain how to determine whether the triangles are congruent, and why it is important that triangles used for structural support are congruent. tandardized est ractice 31. In, and are angle bisectors and m 76. What is the measure of? aintain our kills ixed eview 32. For a positive integer x, 1 percent of x percent of 10,000 equals x. 10x. 100x. 1000x. Write a flow proof. (esson 4-4) 33. iven:, 34. iven: Z W Z bisects W. W WZ Z Z Verify that each of the following preserves congruence and name the congruence transformation. (esson 4-3) 35. y 36. y ' O ' ' ' x ' ' O x etting eady for the ext esson Write each statement in if-then form. (esson 2-3) 37. appy people rarely correct their faults. 38. champion is afraid of losing. QUII I lassify each triangle according to its sides. (o review classification by sides, see esson 4-1.) esson 4-5 roving ongruence, 213