7-5: Inequalities in a Triangle

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7-5: Inequalities in a Triangle Proof Geometry

Look at the following triangles. Which angle appears to be the smallest? Which side appears to be the shortest? Notice a pattern?

Opposite Angles theorem If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side In any ABC, if AB > AC, then C > B

Proof Given: ABC with AB > AC Prove: ACB > B 1. Let D be a point of , Point plotting Theorem such that AD = AC 2. 3  2 Isosc. Triangle Theorem 3. ACB > 2 Parts Theorem 4. ACB > 3 Substitution prop of ineq. 2 into 3 5. 3 > B Exterior  Ineq. theorem 6. ACB > B Transitive prop. of eq., 4 and 5

Example In Δ𝐴𝐵𝐶, rank the angles from smallest to largest.

Converse Opposite Angles Theorem If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. In any ABC, if C > B , then AB > AC

Proof (indirect) Given: ABC with C > B Prove: AB > AC Suppose: AB is not greater than AC (supposition) case 1: AB = AC Then C = B case 2: AB < AC Then C < B (conclusion resulting from supposition) But: It is given that C > B (the CONTRADICTION) So: AB > AC

Example In Δ𝐴𝐵𝐶, rank the sides from smallest to largest. m∠𝐴=59, 𝑚∠𝐵=60, 𝑚∠𝐶=61

[If you choose to do this, you may skip 2 proofs on the HW] Homework pg. 226 to 227: 1-8, 11, 13-17, Challenge problem: 18 (Hint: plot B' between D and C so that BD = B'D). [If you choose to do this, you may skip 2 proofs on the HW]