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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
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2. In ΔABC, m∠A > m∠B.
In ΔABC, m∠C = 60°.
In ΔABC, m∠A = 50° and m∠B = 70°.
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3. �PQR is a right triangle.
∠P is an acute angle.
∠Q is an obtuse angle.
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Write True or False. Explain your answer.
4. Two supplementary angles can both be obtuse.
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5. A scalene triangle can have two congruent sides.
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Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
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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.
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Holt McDougal Geometry