Chapter 5 Lesson 5 (Lesson 5-5): Inequalities in Triangles
Objective: To use inequalities involving angles of triangles and To use inequalities
involving sides of triangles
Comparison Property of Inequality
If a=b + c and c>0, then a>b
Proof…
Given a=b + c, c>0
Prove a>b
1.
2.
3.
4.
5.
Statements
c>0
b+ c> b + 0
b+c >b
a=b+c
a>b
Reasons
Given
Addition Property of Inequality
Simplify
Given
Substitute a for b +c in Statement 3
The Comparison Property of Inequality will assist you in proving the following…
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of each of
its remote interior angles.
m1 > m2 and m1 > m3
Theorem 5-10
If two sides of a triangle are not congruent, then the larger angle lies opposite the
longer side.
If XZ > XY, then mY > mZ
Theorem 5-11
If two angles of a triangle are not congruent, then the longer side lies opposite the
larger angle.
If mA > mB, then BC > AC
Theorem 5-12-Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
XY + YZ >XZ
YZ +ZX > YX
ZX + XY > ZY
Practice…
I have already given you the proof to the Comparison Property of Inequality . Now
try proving the theorems above (Theorem 5-10 –Theorem 5-12).
Theorem 5-10 Proof
Theorem 5-11 Proof
Theorem 5-12 Proof
1. Can a triangle have sides with the side lengths 2m, 7m and 9m? __________________
2. Can a triangle have sides with the side lengths 4 yd, 6y and 9yd? __________________
3. Can a triangle have sides with the side lengths 11cm, 12cm, and 15
cm?________________
4. Can a triangle have sides with the side lengths 2ft, 5ft and 12ft?
__________________
5. List the angles of the below triangle in order from smallest to largest.
6. List the angles of the triangle below in order from smallest to largest.
△ABC, where AB = 8, BC = 5 and CA = 7
7. A triangle has the following side lengths. Describe the possible side lengths
for the third side of the triangle.
5 in and 16 in
8. A triangle has the following side lengths. Describe the possible side lengths
for the third side of the triangle.
18m and 23 m
List the sides in order from smallest to largest.
9. ____________________________
10. _____________________________
Scroll Down for the Answers
1. No
2. No
3. Yes
4. No
5. D, C,E
6. A, B, C
7. 11 < s < 21
8. 5 < s < 41
9. ED, DF, EF
10. AB, BC, AC