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Name ________________________________________ Date __________________ Class__________________ LESSON 5-5 Reading Strategies Use a Contradiction Indirect proofs can be written by following these steps: 1. Identify the conjecture to be proven. 2. Assume the opposite of the conclusion is true. 3. Use direct reasoning to show the assumption leads to a contradiction. 4. Conclude that since the assumption is false, the original conjecture must be true. Find the two statements in each set that contradict each other or are opposites. 1. XY � AB AB ⊥ XY AB ≅ XZ _________________________________________________________________________________________ 2. In ΔABC, m∠A > m∠B. In ΔABC, m∠C = 60°. In ΔABC, m∠A = 50° and m∠B = 70°. _________________________________________________________________________________________ 3. �PQR is a right triangle. ∠P is an acute angle. ∠Q is an obtuse angle. _________________________________________________________________________________________ Write True or False. Explain your answer. 4. Two supplementary angles can both be obtuse. _________________________________________________________________________________________ _________________________________________________________________________________________ 5. A scalene triangle can have two congruent sides. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 5-42 Holt McDougal Geometry 5. ∠V, ∠S, ∠T 9. x > 1 6. JK , KH , HJ 11. 4 < z < 26 7. No; 3 + 5 = 8, which is not greater than the length of the third side. 10. 1; 15 12. Diagram 1 Practice B 8. Yes; the sum of each pair of lengths is greater than the length of the third side. Challenge 1. m∠K < m∠M 2. AB < DE 3. QR > ST 4. 7 < x < 58 5. 1. largest angle: ∠Z; smallest angle: ∠X 5 17 <x< 3 2 6. −2 < x < 10.5 2. longest side: KL ; shortest side: JL 7. x > 4 3. CD, BC, BD, AB, AD 8. Possible answer: The legs of a compass and the length spanned by it form a triangle, but the lengths of the legs cannot change. Therefore any two settings of the compass are subject to the Hinge Theorem. To draw a largerdiameter circle, the measure of the hinge angle must be made larger. To draw a smaller-diameter circle, the measure of the hinge angle must be made smaller. 4. PT , PQ, QT , QR, RT , ST , RS 5. YZ, XY , XZ, WX , WZ, VW , VZ 6, 7. Proofs may vary. Problem Solving 1. Fairbanks to Nome 2. towers K and L 3. targets 2 and 3 Practice C 4. targets 1 and 4 5. C 6. J 7. A 1. The length of BD increases, and the length of AC decreases. 2. between zero and (a + b) Reading Srategies 1. XY || AB, AB ⊥ XY 3. 10 < x < 58 2. In UABC, m∠A > m∠B. In UABC, m∠A = 50° and m∠B = 70° 3. UPQR is a right triangle. ∠Q is an obtuse angle. 4. False; supplementary angles have measures that add up to be 180°, so both angles cannot be obtuse because their sum would be greater than 180°. 5. False; a scalene triangle has three unequal sides by definition. 1 (y + z) 3 5. Yes, BD can be longer than DC 2 < x < 188 5 6. No, DC cannot be longer than BC ; possible answer: the inequalities lead to the contradiction that x must be both less 4 and greater than 2. than 3 7. AB, CD, DE, FA, EF , BC 8. DE 5-6 INEQUALITIES IN TWO TRIANGLES Reteach Practice A 1. third side 2. included angle 3. AB > DE 4. m∠I < m∠L 1. TV < XY 2. m∠G > m∠L 3. AB > AD 4. m∠FHE < m∠HFG 5. PS < PQ 6. m∠UTV > m∠WTV 7. 28 > 2x − 2 4. 6 < x < 8. x < 15 5. −2 < x < 7 6. 3 < x < 21 7. 3 < x < 57 8. −0.6 < x < 7 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A55 Holt McDougal Geometry