5.6 Comparing Measures of a Triangle Up to now, all proofs have used the Laws of Syllogism and Detachment to obtain conclusions directly. LAW OF SYLLOGISM If p → q and q → r are true conditional statements, then p → r is true. LAW OF DETACHMENT If p → q is a true statement and p is true, then q is true. 5.6 Inequalities in Two Triangles

5.6 Comparing Measures of a Triangle In this lesson, you will study indirect proofs. An indirect proof is a proof that a statement is true by first assuming that its opposite is true. If this assumption leads to an impossibility, then you have proved that the original statement is true. GUIDELINES FOR WRITING AN INDIRECT PROOF 1) Identify the statement that you want to prove is true. 2) Begin by assuming the statement is false; assume its opposite is true. 3) Obtain statements that logically follow from your assumption. 4) If you obtain a contradiction, then the original statement must be true. 5.6 Inequalities in Two Triangles

5.6 EF > BC Indirect Proof and Inequalities in Two Triangles The proof of Theorem 5.14 HINGE THEOREM Theorem 5.14 Hinge Theorem IF 2 sides of 1 Δ are  to 2 sides of another Δ, and the included  of the first is larger than () the included  of the second, THEN the third side of the first is longer than the third side of the second. EF > BC 5.6 Inequalities in Two Triangles

5.6 m D > m  A Indirect Proof and Inequalities in Two Triangles CONVERSE OF HINGE THEOREM Theorem 5.14 Converse of the Hinge Theorem IF 2 sides of 1 Δ are  to 2 sides of another Δ, and the third side of the first is longer than the third side of the second, THEN the included  of the first is larger than () the included  of the second. m D > m  A 5.6 Inequalities in Two Triangles

5.6 Writing Measurements in Order from Least to Greatest Write the measurements of the triangles in order from least to greatest. 100˚ H J G 45˚ 35˚ SOLUTION mG < m H < mJ JH < JG < GH 5.6 Inequalities in Two Triangles

5.6 Using the Triangle Inequality The result of the previous example is summarized as Theorem 5.13. THEOREM Theorem 5.13 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. C A B AB + BC > AC AC + BC > AB AB + AC > BC 5.6 Inequalities in Two Triangles

5.6 Finding Possible Side Lengths A triangle has one side of 10 centimeters and another of 14 centimeters. Describe the possible lengths of the third side. SOLUTION Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities. x + 10 > 14 10 + 14 > x x > 4 24 > x So, the length of the third side must be greater than 4 centimeters and less than 24 centimeters. 5.6 Inequalities in Two Triangles