Download Inequalities in Triangles

Inequalities in One Triangle
Geometry CP1 (Holt 5-5)
K. Santos
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
Example
List the angles from largest to smallest.
A
10
B
5
8
C
Largest angle: <C
<A
Smallest angle: <B
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
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
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
Examples:
Can you make a triangle with the following lengths?
4, 3, 6
4 + 3 > 6 add the two smallest numbers
7 > 6 triangle
2. 3, 7, 2
3+2>7
5 > 7 not a triangle
3. 5, 3, 2
3+2>5
5 >5 not a triangle
1.
Example:
The lengths of two sides of a triangle are given
as 5 and 8. Find the length of the third side.
8–5=3
5 + 8 = 13
The third side is between 3 and 13
3 < s < 13 where s is the missing side
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