5.5 Use Inequalities in a Triangle

Theorem 5.10:

If one side of a triangle is longer than the other side, then the angle opposite the longest side is _______ than the angle opposite the shorter side.

larger A C

B

8 5

so ,BCAB .______ mm C A

5.5 Use Inequalities in a Triangle

Theorem 5.11:

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is _______ than the side opposite the smaller angle.

longer A C

B

50o 30o

C,A mm.________ so BC AB

5.5 Use Inequalities in a Triangle

Example 1 Write measurements in order from least to greatestWrite measurements in order from least to greatest

Solution

Write measurements of the triangle in Write measurements of the triangle in order from least to greatestorder from least to greatest.

57o

A

C

B

87o

36oa.

D

F

E

12

22 13b.

____B____ a. mCm Am____AC____ AB BC

5.5 Use Inequalities in a Triangle

Example 1 Write measurements in order from least to greatestWrite measurements in order from least to greatest

Solution

Write measurements of the triangle in Write measurements of the triangle in order from least to greatestorder from least to greatest.

57o

A

C

B

87o

36oa.

D

F

E

12

22 13b.

____D____ b. mEm Fm____EF____ DF DE

5.5 Use Inequalities in a TriangleCheckpoint. Write the measurements of the Checkpoint. Write the measurements of the triangle in order from least to greatest. triangle in order from least to greatest.

1.

34oA

C

B

99o

47o

Am Cm BmBC AB AC

5.5 Use Inequalities in a TriangleCheckpoint. Write the measurements of the Checkpoint. Write the measurements of the triangle in order from least to greatest. triangle in order from least to greatest.

2.

45oP R

Q

80o

55o

Pm Rm QmQR PQ PR

5.5 Use Inequalities in a Triangle

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.

AC______ ______AC

BCAB

___AC___

A

C

B

BC ABAB BC

5.5 Use Inequalities in a Triangle

Example 2 Find possible side lengthsFind possible side lengths

Solution

A triangle has one side of length 14 and another of length A triangle has one side of length 14 and another of length 10. Describe the possible lengths of the third side10. Describe the possible lengths of the third side.

Let x represent the length of the third side. Draw diagrams to help visualize the small and large values of x. Then use the Triangle Inequality Theorem to write and solve inequalities.Small values of x

x 10

14

____ x 10 14__x 4

Large values of xx

10 14

x ____10 14x__24 __or , x 24

The length of the third side must be _______________________________.greater than 4 and less than 24

5.5 Use Inequalities in a TriangleCheckpoint. Complete the following exercise Checkpoint. Complete the following exercise

3. A triangle has one side 23 meters and another of 17 meters. Describe the possible lengths of the third side. Small values of x

x 17

23

17x 23x 6

Large values of xx

17 23

2317 xx40 40or , x

The length of the third side must begreater than 6 meters or less than 40 meters

5.5 Use Inequalities in a Triangle

Theorem 5.13: Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles.

2____ m1m12

3

3____ m1m

5.5 Use Inequalities in a Triangle

Example 3 Relate exterior and interior angles Relate exterior and interior angles

Solution

So, by the Exterior Angle

Inequality Theorem,

_____ > 70o and ______>_____.

.C and B to1 relate that esinequaliti Write mmm

1

B

CA

o70

o60

.1 toanglesinterior t nonadjacen are C and B

1m 1m o60

5.5 Use Inequalities in a TriangleCheckpoint. Complete the following exercise Checkpoint. Complete the following exercise

.B andA

to1 relate that esinequaliti Write4.

mm

m

1B

C

A

o38

o112

.1 toanglesinterior t nonadjacen are B andA 1m o1121m o38

5.5 Use Inequalities in a Triangle

Pg. 299, 5.5 #1-23