1. Inequalities in Triangles
Skill 28

2. Objective
HSG-C.3/10: Students are responsible for using properties of bisectors, medians and altitudes. Also, using these properties to understand problems.

3. Theorem 32: Bigger Angle – Longer Side Theorem
If two sides of a triangle are not congruent,
then the larger angle lies opposite the longest side.
Example B
Biggest angle is ∠B 17 10 C A 19

4. Theorem 33: Converse of Bigger Angle – Longer Side Thm.
If two angles of a triangle are not congruent,
then the longer side lies opposite the largest angle.
Example B
80ᵒ
Longest side is 𝑨𝑪
60ᵒ
40ᵒ C A

5. Theorem 34: Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length than the length of the third side.
Example B
𝟖+𝟏𝟎>𝟏𝟑
𝟖+𝟏𝟑>𝟏𝟎 8 10
𝟏𝟎+𝟏𝟑>𝟖 C A 13

6. Theorem 35: The Hinge Theorem
If the two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.
Example A E
85ᵒ
88ᵒ B C D F
𝑫𝑭>𝑩𝑪

7. Theorem 36: Converse of the Hinge Theorem
If the two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.
Example A E B C D F 25 20
𝒎∠𝑨>𝒎∠𝑬

8. Example 1; Using Bigger Angle – Longer Side
A town park is triangular. A landscape architect wants to place a bench at the corner with the largest angle. Which two streets form the corner with the largest angle? Explain.
Hollingsworth Rd.
175 yd.
MLK Blvd.
105 yd.
Valley Rd.
120 yd.
Hollingsworth Road is the longest side
Bench should be the opposite corner
Corner of Valley Road and MLK Boulavard

9. Example 2; Using Converse
List the sides of ∆𝑇𝑈𝑉 in order from shortest to longest T U V
58ᵒ
62ᵒ
𝒎∠𝑻=𝟏𝟖𝟎−𝟓𝟖−𝟔𝟐
𝒎∠𝑻=𝟔𝟎
𝟓𝟖ᵒ<𝟔𝟎ᵒ<𝟔𝟐ᵒ
𝑻𝑽 < 𝑼𝑽 < 𝑻𝑼

10. Example 3; Using Triangle Inequality Theorem
Can a triangle have sides with the given lengths?
a) 3 𝑓𝑡., 7 𝑓𝑡., 8 𝑓𝑡. b) 5 𝑚, 10 𝑚, 15 𝑚
𝟑+𝟕=𝟏𝟎>𝟖
𝟓+𝟏𝟎=𝟏𝟓≯𝟏𝟓
𝟑+𝟖=𝟏𝟏>𝟕
No the sum of 5 and 10 is not greater than the third side.
𝟕+𝟖=𝟏𝟓>𝟑
Yes, the sum of the length of any two sides is greater than the third side.

11. Example 4; Finding Possible Side Lengths
Suppose two sides of a triangular sandbox are 5 and 8 feet long. What is the range of possible lengths for the third side? Explain.
𝟓+𝒙>𝟖
𝟖+𝒙>𝟓
𝟓+𝟖>𝒙
𝒙>𝟑
𝒙>−𝟑
𝟏𝟑>𝒙
𝟑<𝒙<𝟏𝟑
So the third side of the sandbox is between 3 feet and 13 feet.

12. Example 5; Using Converse of Hinge Theroem
What is the range of possible x values? Explain. R S T U 15 10
5𝑥−20 °
60ᵒ
𝟔𝟎>𝟓𝒙−𝟐𝟎
𝟓𝒙−𝟐𝟎>𝟎
𝟓𝒙>𝟐𝟎
𝟖𝟎>𝟓𝒙
𝟏𝟔>𝒙
𝒙>𝟒
𝟒<𝒙<𝟏𝟔

13. Example 6; Prove Relationships in Triangles
Statement Reason
1) 𝐵𝐴=𝐷𝐸 and 𝐵𝐸>𝐷𝐴
1) Given
2) 𝐴𝐸=𝐴𝐸
2) Reflexive Prop.
3) 𝑚∠𝐵𝐴𝐸>𝑚∠𝐷𝐸𝐴
3) Con. of Hinge Thm.
4) 𝑚∠𝐷𝐸𝐴=𝑚∠𝐷𝐸𝐵+𝑚∠𝐵𝐸𝐴
4) Angle Addition Prop.
5) 𝑚∠𝐷𝐸𝐴>𝑚∠𝐵𝐸𝐴
5) Comparison Prop.
6) 𝑚∠𝐵𝐴𝐸>𝑚∠𝐵𝐸𝐴
6) Transitive Prop. A B D E

14. #28: Inequalities in Triangles
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