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Name LESSON 7-3 Date Class Practice A Triangle Similarity: AA, SSS, SAS Fill in the blanks to complete each postulate or theorem. to the three sides of 1. If the three sides of one triangle are another triangle, then the triangles are similar. 2. If two sides of one triangle are proportional to two sides of another triangle and their . included angles are congruent, then the triangles are to two angles of another 3. If two angles of one triangle are triangle, then the triangles are similar. Name two pairs of congruent angles in Exercises 4 and 5 to show that the triangles are similar by the Angle-Angle (AA) Similarity Postulate. 4. ! 5. 8 37° 4 $ 37° : " 9 # 5 6 & % Substitute side lengths into the ratios in Exercise 6. If the ratios are equal, the triangles are similar by the Side-Side-Side (SSS) Similarity Theorem. 6. 12 ' 6 12 ( 9 ) + * 24 18 , GH ___ HI ___ JK GI ___ KL JL Name one pair of congruent angles and substitute side lengths into the ratios in Exercise 7. If the ratios are equal and the congruent angles are in between the proportional sides, the triangles are similar by the Side-Angle-Side (SAS) Similarity Theorem. 7. 0 5 8 1 20 4 10 16 2 congruent angles: Copyright © by Holt, Rinehart and Winston. All rights reserved. 3 PQ ___ ST 19 QR ___ TU Holt Geometry LESSON 7-3 Practice A LESSON 7-3 Triangle Similarity: AA, SSS, SAS Fill in the blanks to complete each postulate or theorem. proportional 1. If the three sides of one triangle are another triangle, then the triangles are similar. Practice B Triangle Similarity: AA, SSS, SAS For Exercises 1 and 2, explain why the triangles are similar and write a similarity statement. to the three sides of 1. # similar included angles are congruent, then the triangles are 3. If two angles of one triangle are triangle, then the triangles are similar. congruent Name two pairs of congruent angles in Exercises 4 and 5 to show that the triangles are similar by the Angle-Angle (AA) Similarity Postulate. 4. ! 8 5. 37° 4 $ % " to two angles of another 37° 7 8 . 4 2. $ ! 2. If two sides of one triangle are proportional to two sides of another triangle and their 5 9 6 Possible answer: ACB and Possible answer: Every equilateral ECD are congruent vertical triangle is also equiangular, so angles. mB mD 100°, each angle in both triangles so B D. Thus, ABC measures 60°. Thus, TUV EDC by AA . WXY by AA . : " 9 # 5 B Y; C Z 6 & % For Exercises 3 and 4, verify that the triangles are similar. Explain why. F V; D T 3. JLK and JMN 4. PQR and UTS + 6. 12 ' 6 12 ( 9 ) + * * , 12 0 _1_ 9 ___ HI ___ 2 5 20 8 _1_ 18 KL 12 ___ GI ___ 2 _1_ 24 JL 2 5 2 8 ___ 10 PQ ___ ST _4_ 5 16 ___ 20 QR ___ TU Holt Geometry TS $ 1.25 ' Holt Geometry Review for Mastery Triangle Similarity: AA, SSS, and SAS " $ 0 4 5 2 4 78° E Side-Side-Side (SSS) Similarity 3 3 ; 3 3 3 C 15 18 12 10 F A 14.4 E ABC DEF D B If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Side-Angle-Side (SAS) Similarity D 12 B If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. C 15 57° 57° 12 10 18 F A E ABC DEF 3 1 _ _ _ Possible answer: Draw diagonals HK, HJ, QS, _ and QT. G and P are right angles, so they are Explain how you know the triangles are similar, and write a similarity statement. ) ( * ' + 4 3 1. 0 2 R K 2. U 49° H 24 18 J Q 92° 1 S GH _3_, so GHK PQT by SAS. It is given that GK ___ congruent. ___ PT PQ 2 IJ _3_, so HIJ QRS by SAS. Because GHK HI ___ I R. ___ QR RS 2 HJ _3_ HK _3_ and GHK PQT. Because HIJ QRS, ___ PQT, ___ QT 2 QS 2 and IHJ RQS. It is given that H Q. So by the Angle Addition T 92° 16 39° 27 L G V Q T by the Def. of . By the Sum Thm., mS 39 and HJG LJK by the Vert. GJ __ HJ ___ 2 . GHJ Thm. ___ LJ KJ 3 mU 49, so S V and KLJ by SAS . R U. QRS TUV by AA . HJ _3_, so KHJ TQS by SAS. HK ___ Postulate, KHJ TQS. ___ QT QS 2 JK ___ HK _3_. All the corresponding angles are Because KHJ TQS, ___ ST QT 2 congruent; all the corresponding sides are proportional. Thus, GHIJK 3. Verify that ABC MNP. BC ____ CA __ 4 ; ABC AB ____ ____ MN NP PM A 12 5 C P 12 8 B 15 10 N M 15 MNP by SSS . PQRST by the definition of similar polygons. Copyright © by Holt, Rinehart and Winston. All rights reserved. F A 6 5. Use triangle similarity to prove that GHIJK PQRST. 21 57° 57° ABC DEF 3 7 2 78° C If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Angle-Angle (AA) Similarity # also each have a right angle, so ABC ADB by AA . They have a similarity ratio of _2_. ABC and BDC share C. They also each have 1 a right angle, so ABC BDC by AA . They have a similarity ratio 2 3 to 1. By the Transitive Property of Similarity, ADB BDC. of ____ 3 3 to 1. They have a similarity ratio of ___ 3 6 2 3 AD 1 and DC 3. Find the perimeter of ABC. 3. Use the similarity ratios you found in Exercise 1 and the answer to Exercise 2 to find the perimeters of ADB and BDC. D B 1. Prove similarity relationships between triangles in the figure. Give a similarity ratio for each relationship you find. Possible answer: ABC and ADB share A. They & 20 Copyright © by Holt, Rinehart and Winston. All rights reserved. 7-3 ! 5 US 5 3.25 # LESSON Use the figure for Exercises 1–3. Copyright © by Holt, Rinehart and Winston. All rights reserved. UT right angles, so they are congruent. Thus, CDG CEF by AA . Triangle Similarity: AA, SSS, SAS 3 Thus, PQR UTS by SSS . Possible answer: C C by the Reflexive Property. CGD and F are _4_ 5 Practice C 4. Find ST. Thus, JLK JMN by SAS . QR ___ PQ ___ PR _3_. ___ DE 9.75 19 13 ___ Possible answer: 5. DE 3 Q T Possible answer: It is given that JL _4_. KL ___ JMN L. ___ MN JM 3 For Exercise 5, explain why the triangles are similar and find the stated length. 4 Copyright © by Holt, Rinehart and Winston. All rights reserved. 2. 3.5 6 , % 10 16 congruent angles: 7-3 8 12 . Name one pair of congruent angles and substitute side lengths into the ratios in Exercise 7. If the ratios are equal and the congruent angles are in between the proportional sides, the triangles are similar by the Side-Angle-Side (SAS) Similarity Theorem. LESSON 4 3 3 24 6 ___ JK 1 2 18 GH ___ 7. 24 1.8 3.6 16 - 1 2.1 0 Substitute side lengths into the ratios in Exercise 6. If the ratios are equal, the triangles are similar by the Side-Side-Side (SSS) Similarity Theorem. Holt Geometry Copyright © by Holt, Rinehart and Winston. All rights reserved. 55 001_062_Go08an_CRF_c07.indd 22 22 Holt Geometry Holt Geometry 4/12/07 4:32:37 PM