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Lesson 5 – Indirect Proof and Inequalities in One Triangle Bell Work: Simplify each expression. (Hint – simplify each radical first – then add or subtract)

1. 8 + 50 2. 20 + 80 3. 3 25 − 4

Indirect Proof In an indirect proof, you start by making an assumption that the desired conclusion is false. Then, you show that this assumption leads to a logical impossibility, so your original assumption must therefore be wrong by contradiction. Read the following algebraic example that illustrates this idea: Given: x = 2 Prove: 3x − 5 ≠ 10 Step 1: Assume the conclusion is false. The opposite of 3x − 5 ≠ 10 is

that 3x − 5 = 10 . Step 2: Find a contradiction. Solving this equation for x gives:

3x − 5 = 10 3x = 15 x = 5

Step 3: Point out the desired contradiction, and therefore the original conclusion must be true. x = 5 contradicts the given statement that x = 2 , therefore 3x − 5 ≠ 10 .

Read the next geometric example illustrating an indirect proof: Given: VABC is isosceles Prove: The base angles cannot be 92° . Step 1: Assume the conclusion is false. The opposite of “the base angles

cannot be 92° is that the base angles are each 92° .

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Step 2: Find a contradiction. If the base angles are each 92° , then they add up to 184° . This contradicts the Vangle sum theorem that says the angles of a triangle add to 180° .

Step 3: Point out the desired contradiction, and therefore the original conclusion must be true. The sum of the angles of a triangle cannot be 184° , therefore the base angles of an isosceles triangle cannot be 92° .

In each of the following examples, determine what assumption must be false to begin the indirect proof (you don’t have to complete the proof) Example 1: Example 2: Given: ∠A and ∠B are complementary Given: x is even, y is odd Prove: m∠A ≤ 90° Prove: x + y is odd ________________________ _____________________ Example 3: Example 4: Given: VABC is equilateral Given: VABC is a rt V Prove: The circumcenter and the Prove: Its sides cannot be incenter are the same point 3, 4, and 6 inches long.

________________________ _____________________ Now complete the indirect proof Example 5: Given: VABC Prove: VABC can have at most one right angle Step 1: Assume the conclusion is false. Step 2: Find a contradiction Step 3: Point out the contradiction, and desired conclusion

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Triangle Inequalities In VABC mark the longest sides and the largest angle. In VDEF mark the smallest side and the smallest angle. What do you notice?

Triangle Sides and Angles Theorem In a triangle, if one side is longer than another, then the longer side is opposite the larger angle, and the shorter side is opposite the smaller angle. Example 6: List the angles in order from smallest to largest. ______ ______ ______ Example 7: List the sides in order from shortest to longest. Example 8: In VABC , m∠A = 2x + 10 , m∠B = 3x − 20 and m∠C = x + 32 . List the sides of VABC in order from shortest to longest.

AB

C

DE

F

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Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side of the triangle.

Example 9: Determine whether or not it is possible to have a triangle with the given side lengths. Explain. a. 4, 9, 10 b. 8, 9, 19 c. 5, 7, 12 Example 10: A triangle has two sides with lengths 14 and 9. Give the possible values for the length of the third side.

C

BA

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6.5 Indirect Proof and Inequalities on One Triangle Homework Name_______________ Block______ List the angles from smallest to largest. 1. 2. 3. List the sides from shortest to longest 4. 5. 6. For 7 – 9, determine if it is possible to have a triangle with the given side lengths. Explain. 7. 15, 37, 53 8. 9, 16, 8 9. 17, 40, 23

10. Describe the possible values for x in the figure shown: 11. List the angles from smallest to largest. Show your work, and explain your reasoning.

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12. Write an indirect proof Given: VABC Prove: VABC has at most one obtuse angle 13. Write an indirect proof Given: VABC , m∠A = 100° Prove: ∠B is not a right angle

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Lesson 6 – Inequalities in Two Triangles Bell Work: 75+ 12 2. 30+ 120 3. 75− 5

The Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

The Converse of the Hinge Theorem The converse is also true. Write the converse: ______________________________________________________________ ______________________________________________________________ ______________________________________________________________

Example 1: Given that ST ≅ PR how does m∠PST compare to m∠SPR ? Example 2: Given that JK ≅ LK , how does JM compare to LM?

61°64°

K

M LJ

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Example 3: Complete the proof Given: AB ≅ BC , AD > DC Prove: m∠ABD > m∠CBD Example 4: Complete the proof Given: ∠XWY ≅ ∠XYW , WZ > YZ Prove: m∠WXZ > m∠YXZ

DA C

B

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Example 5: Two groups of bikers leave the same Circle K heading in opposite directions. Each group travels 2 miles, then changes direction and travels 1.2 miles. Group A starts due east, then turns 45° toward north. Group B starts due west and then turns 30° toward south. Which group is farther from the Circle K? Explain your reasoning. Example 6: In VDEF , DM is a median. Determine if each statement is always, sometimes, or never true.

a. if m∠2 > m∠1 , then ED > FD _____ b. If m∠E > m∠F , then ∠1 is obtuse _____ c. If ∠2 is acute, then m∠F > m∠E _____ d. If m∠E < m∠F , then m∠1 < m∠2 _____ e. m∠2 > m∠1 , then ED > FD _____ f. if m∠D = 90° , then FD > ED _____

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6.6 Inequalities in Two Triangles Homework Name ___________________ Block _____ Complete the statement with , , or .< > = Explain your reasoning 1. _____ AC DF 2. _____ m HGI m IGJ∠ ∠

3. 1 _____ 2m m∠ ∠ 4. _____ KL MN

Write and solve an inequality for the possible values of x. 5. 6.

7. Given: , TV UW TU VW≅ > 8. Given: 1 2,m m∠ > ∠ B is the midpt of AC.

Prove: m TVU m WUV∠ > ∠ Prove: AF CF>

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9. Two sailboats started at the same location. Sailboat A traveled 5 miles west, then turned 29° toward the north and continued for 8 miles. Sailboat B first went south for 8 miles, then turned 51° toward the east and continued for 5 miles. Which sailboat was farther from the starting point? Explain your reasoning.

10. You and your friend leave on different flights from the same airport. Your flight Flies 100 miles due west, then turns 20° toward north and flies 50 miles. Your friend’s flight Flies 100 miles due north, then turns 30° toward east and flies 50 miles. Determine which flight is farther from the airport. Explain your reasoning