Download Chapter 5 Lesson 5

Chapter 5 Lesson 5
Objective: To use inequalities
involving angles and sides of
triangles.
Comparison Property of Inequality
If a = b + c and c > 0, then a > b.
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
3
2
1
Example 1:
Applying the Corollary
In the diagram, m  2 = m  1 by the Isosceles
Triangle Theorem. Explain why m 2 > m  3.
By the corollary to the
Exterior Angle Theorem,
m  1 > m 3. So,
m  2 > m3 by substitution.
Theorem 5-10
If two sides of a triangle are not
congruent, then the larger angle lies
Y
opposite the longer side.
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
Example 2: Using Theorem 5-11
In ∆TUV, which side is shortest?
By the Triangle Angle-Sum
Theorem, m T = 60. The
smallest angle in ∆TUV is  U. U
It follows, by Theorem 5-11,
that the shortest side isTV .
T
V
Theorem 5-12
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
Example 3:
Using the Triangle Inequality Theorem
Can a triangle have sides with the given
lengths? Explain.
(a.) 3
3
8
3
ft., 7ft. 8ft.
+ 7 > 8
+ 7 > 3
+ 8 > 7
Yes; the sum of
any two length is
greater than the
third length.
(b.) 3 cm, 6 cm, 10 cm
3 + 6 > 10
No; the sum of 3 and
6 is not greater than
10.
Example 4:
Finding Possible Side Lengths
A triangle has sides of lengths 8 cm and 10 cm. Describe
the lengths possible for the third side.
Let x represent the length of the third side. By the
Triangle Inequality Theorem,
The third side must be longer than 2 cm and shorter than
18 cm.
Assignment
Page 276 – 277
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