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Indirect Proof and Inequalities 5-5 in One Triangle Learning Targets I will identify the first step in an indirect proof. I will apply inequalities in one triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Vocabulary indirect proof Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Helpful Hint When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Identify the assumption for the following indirect proof: A triangle cannot have two right angles. FIRST: Identify the conjecture to be proven. Given: A triangle’s interior angles add up to 180°. Prove: A triangle cannot have two right angles. ASSUME: Assume the opposite of the conclusion. An angle has two right angles. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Continued FIND THE CONTRADICTION Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that a triangle can have two right angles is false. Therefore a triangle cannot have two right angles. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The shortest side is smallest angle is F. The longest side is , so the , so the largest angle is G. The angles from smallest to largest are F, H and G. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48° The smallest angle is R, so the shortest side is . The largest angle is Q, so the longest side is The sides from shortest to longest are Holt McDougal Geometry . Indirect Proof and Inequalities 5-5 in One Triangle A triangle is formed by three segments, but not every set of three segments can form a triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle A certain relationship must exist among the lengths of three segments in order for them to form a triangle. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 3A: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 3B: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6 Yes—the sum of each pair of lengths is greater than the third length. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Check It Out! Example 3a Tell whether a triangle can have sides with the given lengths. Explain. 8, 13, 21 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Example 4: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side. Let x represent the length of the third side. Then apply the Triangle Inequality Theorem. x + 8 > 13 x>5 8 + 13 > x 21 > x Combine the inequalities. So 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Homework: Pg 348, #16 – 31* *For the indirect proofs, write only the assumption. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. C, B, A 2. Write the sides in order from shortest to longest. Holt McDougal Geometry Indirect Proof and Inequalities 5-5 in One Triangle Lesson Quiz: Part II 3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side. 5 cm < x < 29 cm 4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain. No; 2.7 + 3.5 is not greater than 9.8. 5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain. Yes; the sum of any two lengths is greater than the third length. Holt McDougal Geometry