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Section 5-5 INDIRECT PROOF AND INEQUALITIES IN ONE TRIANGLE Indirect Proof – you begin by assuming that the conclusion in false. Then, you show your assumption is false. 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. Example 1 Write an indirect proof that a triangle cannot have two right angles. Step 1 Identify the conjecture to be proven. Given: A triangle’s interior angles add up to 180°. Prove: A triangle cannot have two right angles. Step 2 Assume the opposite of the conclusion. An angle has two right angles. Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° A triangle cannot have an angle with a measure of zero degrees, therefore this conjecture is false. Therefore, a triangle cannot have two right angles. In other words, the longest side is opposite of the largest angle. The shortest side is opposite of the smallest angle. And the medium side is opposite of the medium angle. The converse is also true! The largest angle is opposite the largest side, etc. Example 2: 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. Example 3: Write the sides in order from shortest to longest. First…find the missing angle… 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 . Write all of this down… In other words, the two shortest sides of a triangle must add up to be LARGER than the third side. As shown below…. (you don’t have to write this down) 4+4=8, which is larger than 7, so this can be a triangle. 3-3=6, which is NOT larger than 7, so this CANNOT be a triangle. Example 4: 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. Example 5: 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. Example 6: 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 x + 13 > 8 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. Now…watch this video for a little extra help… Section 5-5 Video Assignment #57 DUE TOMORROW! Pg. 335 #2-14 even