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Name Class Date Practice Form G Congruence in Right Triangles 1. Developing Proof Complete the paragraph proof. S Given: RT # SU , RU ≅ RS. R Prove: △RUT ≅ △RST Proof: It is given that RT # SU . So, jRTS and jRTU are right angles because perpendicular lines form right angles. RT ≅ RT by the Reflexive Property of Congruence. It is given that RU ≅ RS. So, △RUT ≅ △RST by HL . 2. Look at Exercise 1. If m∠RST = 46, what is m∠RUT ? T U 46 3. Write a flow proof. Use the information from the diagram to prove that A B △ABD ≅ △CDB. ∠A and ∠C are right angles. Given AB > CD Answers may vary. Sample: △ABD and △CDB are right Definition of right . D C △ABD > △CBD HL Theorem Given BD > DB Reflexive Property of Congruence 4. Look at Exercise 3. Can you prove that △ABD ≅ △CDB without using the Hypotenuse-Leg Theorem? Explain. Yes; answers may vary. Sample: You know that AB @ CD from the diagram and DB @ BD by the Reflexive Property of Congruence. Because the triangles are right triangles, the sides are related by the Pythagorean Theorem. If we let the legs = x and the hypotenuses = y, then the length of the other leg will be 2y 2 − x 2 on both triangles. So, by SSS △ABD @ △CDB. Construct a triangle congruent to each triangle by the Hypotenuse-Leg Theorem. 5. 6. Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. Name Class Date Practice (continued) Form G Congruence in Right Triangles Algebra For what values of x or x and y are the triangles congruent by HL? 17 7. 17 3 4.25 8. x x 3 4.25 9. 4x ! 1 13 2x ! y 3; 6 2; 1 10. x!3 2x ! 1 8x " 2y x!y 3x " 3y 11. Write a paragraph proof. A Given: AD bisects EB, AB ≅ DE; ∠ECD, ∠ACB are right angles. Prove: △ACB ≅ △DCE △ACB and △DCE are right triangles E B C because each contains a right angle (definition of a right triangle). It is given that AB @ DE, so the hypotenuses of these right triangles are congruent. Because AD bisects EB, point C is the midpoint of EB. EC @ BC (definition of a midpoint), so the triangles have a pair of congruent legs. By HL, △ACB @ △DCE. D What additional information would prove each pair of triangles congruent by the Hypotenuse-Leg Theorem? 12. D A R Q E S jA and jQ are right angles. 14. L X M N LN @ XZ Y Z B 13. A D C jB and jD are right angles. 15. T R S 16. Reasoning Are the triangles congruent? Explain. No; they are both right triangles, and one pair of legs is congruent, but the hypotenuse of one triangle is congruent to a leg of the other triangle. TR @ TV V 7 Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 25 25 7 Name Class Date Practice Form K Congruence in Right Triangles 1. Developing Proof Complete the proof. W V Given: ∠WVZ and ∠VWX are right angles. WZ ≅ VX Y Prove: △WVZ ≅ △VWX To prove that right triangles △WVZ and △VWX are congruent, you Z must prove that the hypotenuses are congruent and that one ? is congruent. X leg Statements Reasons 1) ? jWVZ and jVWX are right ⦞. 1) Given 2) ? WZ @ VX 2) Given 3) ? WV @ VW 3) Reflexive Property of Congruence 4) ? △WVZ @ △VWX 4) HL Theorem 2. Look at Exercise 1. If m∠X = 54, what is m∠Z? 54 3. Look at Exercise 1. If m∠X = 54, what is m∠VWZ? 36 4. Study Exercise 1. Can you prove that △WVZ and △VWX are congruent without using the HL Theorem? Explain. Yes; answers may vary. Sample: by using the Pythagorean Theorem to find another pair of congruent sides Algebra For what values of x and y are the triangles congruent by HL? x"4 5. x 14; 7 6. 2y y!5 y!3 y!x 2y " x x!7 2; 7 7. Reasoning The LL Theorem says that two right triangles are congruent if both pairs of legs are congruent. What theorem or postulate could be used to prove that the LL Theorem is true? Explain. Answers may vary. Sample: The SAS Postulate could be used to prove the LL Theorem because the two right triangles would have two pairs of congruent legs and the included right angles would also be congruent. Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. Name Class Date Practice (continued) Form K Congruence in Right Triangles What additional information would prove each pair of triangles congruent by the Hypotenuse-Leg Theorem? 8. A 9. B T N C D L M jM and jS are right angles. F E AC @ DF 10. R S M J L 11. W X Z Y O K N JL @ MO or KL @ NO WZ @ YX or WX @ YZ Coordinate Geometry Use the figure at the right for Exercises 12–14. A(!3, 5) y 4 12. Complete the paragraph proof that shows that AC and CD are perpendicular. □ □ The slope of AC is − 2 . The slope of CD is 0.5 . □ 2 x !4 2 !2 The product of the two slopes is − 1 . Therefore the line segments are B(1, 3) !2 ? . perpendicular D(!3, !5) 13. How do you know that AB = CD? The Distance Formula shows that both line segments have length 2 25. 14. Complete the paragraph proof below that shows that △ACD ≅ △DBA. right angle ∠ACD is a ? from Exercise 12. You can also use the product of slopes to show that ∠ABD is a right angle. △ACD and △ABD share the same hypotenuse. You can use the ? to show that AB = CD. Therefore, by the ? , △ACD ≅ △DBA. Distance Formula HL Theorem Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved. 4 C(1, !3)