Learning Targets I will identify the first step in an indirect proof. I will apply inequalities in one triangle.

Vocabulary indirect proof

So far you have written proofs using direct reasoning 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.

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Helpful Hint

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.

Check It Out! Example 1 Continued FIND THE CONTRADICTION Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 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 positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The shortest side is , so the smallest angle is F. The longest side is , so the largest angle is G. The angles from smallest to largest are F, H and G.

Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 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

A triangle is formed by three segments, but not every set of three segments can form a triangle.

A certain relationship must exist among the lengths of three segments in order for them to form a 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.

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.

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.

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 8 + 13 > x 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.

Homework: Pg 348, #16 – 31* *For the indirect proofs, write only the assumption.

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.