Warm Up(Add to Notes) 1.Write a conditional from the sentence “An isosceles triangle has two congruent sides.” 2. Write the contrapositive of the conditional “If it is Tuesday, then John has a piano lesson.” 3. Show that the conjecture “If x > 6, then 2x > 14” is false by finding a counterexample. If a ∆ is isosc., then it has 2 sides. If John does not have a piano lesson, then it is not Tuesday. x = 7
5-5 Indirect Proof and Inequalities in One Triangle Holt Geometry
In an indirect proof, you assume 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.
Check It Out! 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.
Check It Out! 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° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°.
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.
The shortest side is , so the smallest angle is F. The longest side is , so the largest angle is G. Write the angles in order from smallest to largest. The angles from smallest to largest are F, H and G.
The smallest angle is D, so the shortest side is . The largest angle is F, so the longest side is . The sides from shortest to longest are Check It Out! Example 2b Write the sides in order from shortest to longest. mE = 180° – (90° + 22°) = 68°
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.
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 + 13 > 8 8 + 13 > x x > 5 x > –5 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.
Example 5: Travel Application The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland? Let x be the distance from San Francisco to Oakland. x + 46 > 51 x + 51 > 46 46 + 51 > x Δ Inequal. Thm. x > 5 x > –5 97 > x Subtr. Prop. of Inequal. 5 < x < 97 Combine the inequalities. The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.
Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. 2. Write the sides in order from shortest to longest. C, B, A
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. 4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain. 5 cm < x < 29 cm 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.