Transcript 5-5 Inequalities in Triangles

Inequalities in Triangles
LESSON 5-5
Additional Examples
Quick Check
Explain why m4 > m5.
4 is an exterior angle of XYC. The Corollary to the Exterior Angle
Theorem states that the measure of an exterior angle of a triangle is
greater than the measure of each of its remote interior angles. The
remote interior angles are 2 and 3, so m4 > m2.
2 and 5 are corresponding angles formed by AB and XY cut by
transversal BC. Because AB || XY, you can conclude that 2
5
by the Corresponding Angles Postulate. Thus, m2 = m5.
Substituting m5 for m2 in the inequality m4 > m2 produces the
inequality m4 > m5.
HELP
GEOMETRY
Inequalities in Triangles
LESSON 5-5
Additional Examples
In RGY, RG = 14, GY = 12, and RY = 20. List the angles
from largest to smallest.
Theorem 5-10 states If two sides of a triangle are not congruent, then
the larger angle lies opposite the longer side. No two sides of RGY
are congruent, so the larger angle lies opposite the longer side.
Find the angle opposite each side.
The longest side is 20.
The opposite angle, G, is largest.
The shortest side is 12.
The opposite angle, R, is smallest.
From largest to smallest, the angles are
HELP
G,
Y,
R.
Quick Check
GEOMETRY
Inequalities in Triangles
LESSON 5-5
Additional Examples
In
ABC,
C is a right angle. Which is the longest side?
Theorem 5-11 states If two angles of a triangle are not congruent,
then the longer side lies opposite the larger angle.
Find the largest angle.
m A + m B + m C = 180
by the Triangle Angle-Sum Theorem.
m A + m B + 90 = 180 because the measure of a right angle is 90,
so m A + m B = 90.
m A < m C and m B < m
so C is the largest angle.
Because AB is opposite
C by the Comparison Property of Inequality,
C, the longest side in
ABC is AB .
Quick Check
HELP
GEOMETRY
Inequalities in Triangles
LESSON 5-5
Additional Examples
Can a triangle have sides with the given lengths? Explain.
a. 2 cm, 2 cm, 4 cm
b. 8 in., 15 in., 12 in.
According to the Triangle Inequality Theorem, the sum of the
lengths of any two sides of a triangle is greater than the length
of the third side.
a. 2 + 2 > 4
No.
The sum of 2 and 2 is not greater than 4, contradicting Theorem 5-12.
b. 8 + 15 > 12
8 + 12 > 15
15 + 12 > 8 Yes.
The sum of any two lengths is greater than the third length.
Quick Check
HELP
GEOMETRY
Inequalities in Triangles
LESSON 5-5
Additional Examples
In FGH, FG = 9 m and GH = 17 m. Describe the possible
lengths of FH.
The Triangle Inequality Theorem states the sum of the lengths of any
two sides of a triangle is greater than the length of the third side.
Solve three inequalities.
FH + 9 > 17
FH + 17 > 9
9 + 17 > FH
FH > 8
This can be written
“8 < FH.”
FH > –8
26 > FH
This can be written
“FH < 26.”
The length of FH must be longer than 8 m and shorter than 26 m.
Quick Check
HELP
GEOMETRY