1. 5-5 Inequalities in Triangles
1/13/17
Objective: To use inequalities involving angles of triangles and sides of triangles.
COMPARISON PROPERTY OF INEQUALITY
If a = b + c and c > 0, then a > b
(If c were negative, b would be > a)

2. b + c > b + 0 2) Addition Property of Inequality (+b each side)
Do not write in notes
PROOF
Given: a = b + c, c > 0
Prove: a > b
STATEMENTS REASONS
c > ) Given
b + c > b ) Addition Property of
Inequality (+b each side)
b + c > b ) Simplify
a = b + c ) Given
a > b ) Substitution (a for b + c in
statement 3)

3. COROLLARY TO THE TRIANGLE EXTERIOR ANGLE THEOREM
Do not write the proof in notes
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
Informal Proof:
The Exterior Angle Theorem says that 1 = Since both 2 and 3 are > 0, the Comparison Property of Inequality shows that 1 is greater than both 2 and 3.

4. Ex: m 2 = m 1 by Isosceles Triangle Theorem. Explain why m 2 > m 3.
By the Corollary to the Exterior Angle Theorem,
m 1 > m So, m 2 > m 3 by Substitution. O 3 P 1 4 2
T Y
Explain why m OTY > m 3
We know 2 > 3, and OTY = , so OTY > 3

5. Two VERY important theorems. Do not write the proof, however.
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side Y
If XZ > XY, then m Y > m Z
X Z
THEOREM 5-11
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle B
If m A > m B, then BC > AC C A
GIVEN: m A > m B
PROVE: BC > AC by CONTRADICTION!!!
STEP 1: Assume BC > AC. That is, assume BC < AC or BC = AC.
STEP 2: If BC < AC, then m A < m B (Theorem 5-10). This contradicts the given fact that m A > m B. Therefore BC < AC must be false. If BC = AC, then m A = m B. (Isosceles Triangle Theorem). This also contradicts m A > m B. Therefore BC = AC must be false.
STEP 3: The assumption BC > AC is false so BC > AC.

6. Ex: In TUV, which side is the shortest? T
m T = 60 by Triangle Sum Theorem.
U is the smallest angle o 62o
Side TV is shortest by Theorem U V
List the sides of XYZ in order from shortest to longest Y
X o o Z .
80°
YZ < YX < XZ

7. Ex: Can a triangle have sides with the given lengths? Explain.
Another VERY important theorem.
THEOREM 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 Y
YZ + ZX > YX
ZX + XY > ZY X Z
Ex: Can a triangle have sides with the given lengths? Explain.
a) 3 ft, 7 ft, 8 ft b) 3 cm, 6 cm, 10 cm
3 + 7 > > 6
8 + 7 > > 3
3 + 8 > 7 YES > 10 NO
2 m, 7 m, 9 m
4 yd, 6 yd, 9 yd
2 + 7 = 9… NO: it must be greater
4 + 6 > 9… YES
Just check the two smallest sides. If their sum is > than the third side, it’s a Δ.

8. Longer than 2 cm and shorter than 18 cm
Ex: A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side.
x = the third side
x can be “at most”
x is “at least” 10 – 8
Longer than 2 cm and shorter than 18 cm
2 < x < 18 {do NOT use <}
1) A triangle has sides 3 in, 12 in. Describe the lengths possible for the third side.
9 < x < 15
(12 – 3) 
 (12 + 3)

9. Use ABC to describe the possible lengths of AC
1. The exterior angle is larger than either of the remote interior angles.
Angle 3 is opposite AB.
Angle 1 is opposite BC.
AB > BC so Angle 3 > Angle 1. A 1 25 o
B C D
Explain why m 4 > m 1
Explain why m 3 > m 1
Use ABC to describe the possible lengths of AC
_____________________________________________
Can a triangle have lengths of 2 mm, 3 mm, and 6 mm? Explain.
In XYZ, XY = 5, YZ = 8, and XZ = 7. Which angle is the largest?
In PQT, m P = 50 and m T = 70. Which side is shortest?
13 < AC < 37
NO: < 6
Angle X, since it’s opposite the largest side, YZ.
Angle Q = 60, so the side opposite Angle P is the shortest. That side is QT.

10. USING A VARIABLE pg 280 A B C Points A, B, and C are collinear.
BC is 6 less than twice AB, and AC = 30.
What is the length of BC?
Let x = AB AC = 30
“BC is 6 less than twice AB” so BC = 2x - 6
x + 2x – 6 = 30
3x – 6 = 30
3x = 36
x = 12, so AB = 12
BC = 2 • 12 – 6
BC = 18

11. Assignment:
Page 276 #4 – 21, 32, 34 – 36