Prove Triangles Congruent by ASA and AAS

Transcription

1 4.5 rove Triangles ongruent by S and S efore ou used the SSS, SS, and H congruence methods. Now ou will use two more methods to prove congruences. hy So you can recognize congruent triangles in bikes, as in xs ey ocabulary flow proof Suppose you tear two angles out of a piece of paper and place them at a fixed distance on a ruler. an you form more than one triangle with a given length and two given angle measures as shown below In a polygon, the side connecting the vertices of two angles is the included side. two angle measures and the length of the included side, you can make only one triangle. So, all triangles with those measurements are THOMS or our Notebook OSTT 21 ngle-side-ngle (S) ongruence ostulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are If ngle >, then Side } > }, and ngle >, n > n. THOM 4.6 ngle-ngle-side (S) ongruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are If ngle >, then ngle >, and Side } > }, n > n. roof: xample 2, p rove Triangles ongruent by S and S 249

2 M 1 Identify congruent triangles an the triangles be proven congruent with the information given in the diagram If so, state the postulate or theorem you would use. a. b. c. Solution OI OS ou need at least one pair of congruent corresponding sides to prove two triangles a. The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are The triangles are congruent by the S ongruence Theorem. b. There is not enough information to prove the triangles are congruent, because no sides are known to be c. Two pairs of angles and their included sides are The triangles are congruent by the S ongruence ostulate. O OOS ou have written two-column proofs and paragraph proofs. flow proof uses arrows to show the flow of a logical argument. ach reason is written below the statement it justifies. M 2 rove the S ongruence Theorem rove the ngle-ngle-side ongruence Theorem. GIN c >, >, } > } O c n > n > > Third Thm. > } > } n > n S ongruence ost. at classzone.com GI TI for xamples 1 and 2 1. In the diagram at the right, what postulate or theorem can you use to prove that nst > nt xplain. 2. ewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof. S T 250 hapter 4 ongruent Triangles

3 M 3 rite a flow proof In the diagram, } } and >. rite a flow proof to show n > n. Solution GIN c } }, > O c n > n > and are supplements. and are supplements. ef. of supplementary angles } } > ongruent Supps. Thm. } > } eflexive rop. m 5 m ef. of lines n > n S ongruence ost. > ll right are >. M 4 Standardized Test ractice I TOS The forestry service uses fire tower lookouts to watch for forest fires. hen the lookouts spot a fire, they measure the angle of their view and radio a dispatcher. The dispatcher then uses the angles to locate the fire. How many lookouts are needed to locate a fire Not enough information The locations of tower, tower, and the fire form a triangle. The dispatcher knows the distance from tower to tower and the measures of and. So, he knows the measures of two angles and an included side of the triangle. y the S ongruence ostulate, all triangles with these measures are So, the triangle formed is unique and the fire location is given by the third vertex. Two lookouts are needed to locate the fire. c The correct answer is. GI TI for xamples 3 and 4 3. In xample 3, suppose > is also given. hat theorem or postulate besides S can you use to prove that n > n 4. HT I In xample 4, suppose a fire occurs directly between tower and tower. ould towers and be used to locate the fire xplain. 4.5 rove Triangles ongruent by S and S 251

4 ONT SMM or our Notebook Triangle ongruence ostulates and Theorems ou have learned five methods for proving that triangles are SSS SS H (right ns only) S S ll three sides are Two sides and the included angle are The hypotenuse and one of the legs are Two angles and the included side are Two angles and a (nonincluded) side are In the xercises, you will prove three additional theorems about the congruence of right triangles: ngle-eg, eg-eg, and Hypotenuse-ngle. 4.5 ISS SI TI HOMO 5 O-OT SOTIONS on p. S1 for xs. 5, 9, and 27 5 STNI TST TI xs. 2, 7, 21, and O Name one advantage of using a flow proof rather than a two-column proof. 2. ITING ou know that a pair of triangles has two pairs of congruent corresponding angles. hat other information do you need to show that the triangles are congruent M 1 on p. 250 for xs. 3 7 INTI ONGNT TINGS Is it possible to prove that the triangles are congruent If so, state the postulate or theorem you would use. 3. n, nqs 4. n, nj 5. nq, ns S S J 6. O NSIS escribe the error in concluding that n > n. y, n > n. 252 hapter 4 ongruent Triangles

5 7. MTI HOI hich postulate or theorem can you use to prove that n > nhj H J S SS S Not enough information M 2 on p. 250 for xs OING OO State the third congruence that is needed to prove that ngh > nmn using the given postulate or theorem. 8. GIN c } GH > } MN, G > M, > se the S ongruence Theorem. 9. GIN c } G > } M, G > M, > se the S ongruence ostulate. G M 10. GIN c } H > } N, H > N, > se the SS ongruence ostulate. H N OING TINGS xplain how you can prove that the indicated triangles are congruent using the given postulate or theorem. 11. n > n by SS 12. n > n by S 13. n > n by S TMINING ONGN Tell whether you can use the given information to determine whether n > n. xplain your reasoning. 14. >, } > }, } > } 15. >, >, > 16. >, >, } > } 17. } > }, } > }, } > } INTI ONGNT TINGS Is it possible to prove that the triangles are congruent If so, state the postulate(s) or theorem(s) you would use. 18. n, n 19. nt, nt 20. nqm, nn T M N 21. TN SONS se the graph at the right. a. Show that >. xplain your reasoning. b. Show that >. xplain your reasoning. c. Show that n > n. xplain your reasoning. y (2, 5) (6, 6) 22. HNG se a coordinate plane. a. Graph the lines y 5 2x 1 5, y 5 2x 2 3, and x 5 0 in the same coordinate plane. (0, 1) b. onsider the equation y 5 mx 1 1. or what values of m will the graph of the equation form two triangles if added to your graph or what values of m will those triangles be congruent xplain. 2 1 (4, 2) x 4.5 rove Triangles ongruent by S and S 253

6 OM SOING ONGN IN IS xplain why the triangles are M O OO opy and complete the flow proof. on p. 251 for x. 25 GIN c } i }, } > } O c n > n > > n > n } > } M 4 on p. 251 for x SHOT SONS ou are making a map for an orienteering race. articipants start at a large oak tree, find a boulder 250 yards due east of the oak tree, and then find a maple tree that is 508 west of north of the boulder and 358 east of north of the oak tree. Sketch a map. an you locate the maple tree xplain. 27. IN In the airplane at the right, and are right angles, } > }, and >. hat postulate or theorem allows you to conclude that n > n IGHT TINGS In esson 4.4, you learned the Hypotenuse-eg Theorem for right triangles. In xercises 28 30, write a paragraph proof for these other theorems about right triangles. 28. eg-eg () Theorem If the legs of two right triangles are congruent, then the triangles are 29. ngle-eg () Theorem If an angle and a leg of a right triangle are congruent to an angle and a leg of a second right triangle, then the triangles are 30. Hypotenuse-ngle (H) Theorem If an angle and the hypotenuse of a right triangle are congruent to an angle and the hypotenuse of a second right triangle, then the triangles are O-OT SOTIONS on p. S1 5 STNI TST TI

7 31. OO rite a two-column proof. GIN c } > } J, J > J, > O c n > nj 32. OO rite a flow proof. GIN c } > }, > O c n > n J 33. OO rite a proof. 34. OO rite a proof. GIN c NM > M, > N GIN c is the midpoint of } and }. O c nnm > nm O c n > n N M 35. HNG rite a proof. GIN c n > n, is the midpoint of }, is the midpoint of }. O c n > n > n MI I ind the value of x that makes m i n. (p. 161) 36. x8 518 m n x8 m n m (x 1 16)8 n I repare for esson 4.6 in xs rite an equation of the line that passes through point and is parallel to the line with the given equation. (p. 180) 39. (0, 3), y 5 x (22, 4), y 5 22x 1 3 ecide which method, SSS, SS, or H, can be used to prove that the triangles are (pp. 234, 240) 41. nhj > nj 42. nt > nt 43. n > nq H J T T TI for esson 4.5, p ONIN QI at classzone.com 255