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