1. Inequalities in One Triangle
Geometry CP1 (Holt 5-5) K. Santos

2. Theorem 5-5-1
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Y X Z (In a triangle, larger angle opposite longer side) If XZ > XY then m<Y > m<Z

3. Example
List the angles from largest to smallest. A 10 B 5 8 C Largest angle: <C <A Smallest angle: <B

4. Theorem 5-5-2
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. A B C (In a triangle, longer side is opposite larger angle) If m<A > m<B then BC > AC

5. Example
List the sides from largest to smallest. A 50° 60° B C Find the missing angle first: 180 – (50 +60) m < C = 70° Largest side: 𝐴𝐵 𝐴𝐶 Smallest side: 𝐵𝐶

6. Triangle Inequality Theorem (5-5-3)
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. X Y Z XY + YZ > XZ YZ + ZX > YX ZX + XY > ZY add two smallest sides together first

7. Examples: Can you make a triangle with the following lengths? 4, 3, 6
4 + 3 > 6 add the two smallest numbers
7 > 6 triangle
3, 7, 2
3 + 2 > 7
5 > 7 not a triangle
5, 3, 2
3 + 2 > 5
5 >5 not a triangle

8. Example:
The lengths of two sides of a triangle are given as 5 and 8. Find the length of the third side. 8 – 5 = = 13 The third side is between 3 and 13 3 < s < 13 where s is the missing side

9. Example:
Use the lengths: a, b, c, d, and e and rank the sides from largest to smallest. d 59° Hint: find missing angles a e c 61° 59° 60° b In the left triangle: e > b > a In the right triangle: c > d > e But notice side e is in both inequality statements. So, by using substitution property you can write one big inequality statement. c > d > e > b > a