1. 7-5: Inequalities in a Triangle
Proof Geometry

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

3. 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

4. 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   Isosc. Triangle Theorem
3. ACB >  Parts Theorem
4. ACB >  Substitution prop of ineq. 2 into 3
5. 3 > B Exterior  Ineq. theorem
6. ACB > B Transitive prop. of eq., 4 and 5

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

6. 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

7. 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

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

9. [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]