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Warm-Up:
EOC Review
What is the negation of
x ≤ 10?
A) x ≤ 10
B) –x ≤ 10
C) –x > 10
D) x > 10
What is the inverse of
p  q?
A) q  p
B) ~q  ~p
C) p  q
D) ~p  ~q
Inequalities in
Triangles
5.5
Today’s Goals
By the end of class today, YOU should be able to…
1. Use inequalities involving angles of
triangles.
2. To use inequalities involving sides of
triangles.
Comparison Property of
Inequality
If a = b + c and c > 0, then a > b.
Proof of the Comparison
Property of Inequality
Given: a = b + c, and c > 0
Prove: a > b
Statement 1: c > 0
Statement 2: b + c > b + 0
Given
Addition Property of Inequality
Statement 3: b + c > b
Simplify
Statement 4: a = b + c
Given
Statement 5: a > b
Substitute a for b + c in
Statement 3
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.
Ex.1: Using Theorem 5-11

In TUV, which side is shortest?
Ex.1: Solution
By the Triangle Angle-Sum Theorem, m<T = 60.
The smallest angle in TUV is U. It follows, by
Theorem 5-11, that the shortest side is TV.
You Try…

Which side is shortest?

Which side is longest?
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
Ex.2: Using the Triangle
Inequality Theorem
Can a triangle have sides with the given
lengths?
3ft, 7ft, 8ft
Ex.2: Solution
3+7>8
8+7>3
3+8>7
Yes, the sum of any two lengths is greater than
the third length
You Try…
Can a triangle have sides with the given
lengths?
3ft, 6 ft, 10 ft
Ex.3: Using the Triangle
Inequality Theorem
A triangle has sides of lengths 8 cm and
10 cm. Describe the lengths possible for
the third side.
Ex.3: Solution
Let x represent the length of the third side.
x + 8 > 10
x>2
x + 10 > 8
x > -2
8 + 10 > x
x < 18
The third side must be longer than 2 cm
and shorter than 18 cm.
You Try…
A triangle has sides of lengths 7 in and 4
in. Describe the lengths possible for the
third side.
Homework
 Page
276 #s 1, 6, 7, 14, 17, 20, 24, 25
 Page 277 # 34
 The
assignment can also be found at:
• http://www.pearsonsuccessnet.com/snpap
p/iText/products/0-13-037878-X/Ch05/0505/PH_Geom_ch05-05_Ex.pdf