Transcript 5-5 PPT Inequalities in Triangles

Section 5-5 Inequalities in Triangles
SPI 22C: apply the Triangle Inequality Property to determine which
sets of side lengths determine a triangle
SPI 32E: solve problems involving congruent angles given angle
measures expressed algebraically
Objectives:
• use inequalities involving angles and sides of triangles
Comparison Property of Inequality
If a = b + c and c > 0, then a > b
6 = 2 + 4, with c = 4, then 6 > 2
Using Property to Prove Corollary
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.
Write a paragraph proof given the following information.
Given: 1 is an exterior angle
Prove: m1 > m2 and m1 > m3
3
1
2
Proof: By the Exterior Angle Theorem , m1 = m2 + m3.
Since the m2 > 0 and the m3 > 0, you can apply the
Comparison Property of Inequality and conclude that
m1 > m2 and m1 > m3.
Applying the Corollary
In ∆ PQR, m<Q = 45º, and m<R = 72º. Find the measure of
an exterior angle at P.
It is always helpful to draw a
diagram and label it with the given
information.
Then, using the theorem set the
exterior angle ( x ) equal to the sum
of the two non-adjacent interior
angles
(45º and 72º.)
So, an exterior angle at P
measures 117º.
x = 45 + 72
x = 117
Triangle Properties
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 (Converse of Thm 5-10)
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.
Real-World Connection
A landscaper is designing a triangular deck. She wants
to place benches in the two larger corners. Which
corners have the largest angles?
Angles B and C have the
larger angles, since they are
opposite the two longer
sides.
Properties of Triangles
Theorem 5-12 Triangle Inequality
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
Apply Properties of Triangles
Can a triangle have sides with the given lengths? Explain.
a. 2 m, 7 m, and 9 m
NO
2 + 7 is not greater than 9
b. 4 yd, 6 yd, and 9 yd
YES
4+6>9
6+9>4
4+9>6