Unit 2 MM1G3 b

Triangle Inequality, Side-Angle Inequality and Exterior Angle

Inequality

http://www.mathopenref.com/triangleinequality.html

http://www.mathopenref.com/trianglesideangle.html

Exterior Angle Inequality

Triangle Inequality

Examples

Triangle Inequality The sum of the measures of any two sides of a triangle

must be greater than the measure of the third side.

R

ST

RS + ST > TR ST + TR > RS RS + TR > ST

Example 1: Can the measurements 3, 8, and 12

form a triangle?

Solution:

Begin with the measures of the two smaller sides, 3 and 8. To form a triangle,

their sum must be greater than the measure of the third side, which is 12.

However, 3+8<12. Therefore, these measurements cannot form a triangle.

3 8

12

Example 2: Can the measurements 1, 1, and 2

form a triangle?

Solution: Begin with the measures of the two

smaller sides, 1 and 1. To form a triangle,their sum must be greater than the measure of

the third side, which is 2.

However, 1+1=2. Therefore, these measurements cannot form a triangle.

11

2

Example 3: Can the measurements 3, 4, and 5

form a triangle?

Solution:Begin with the measures of the two

smaller sides, 3 and 4. To form a triangle, their sum must be greater than the measure of

the third side, which is 5.

3+4>5

These measurements satisfy the triangle inequality. Therefore, a triangle can be formed.

35

4

It is not necessary to verify the inequality for each pair of sides. As long as the sum of the measures of the two smaller sides is greater than the measure of the third side, then the sum of the measures of any other pair must be greater than the measure of the remaining side.

Example:

Since 3+4>5,

then 3+5>4 and 4+5>3

5 3

4

If we know the lengths of two sides of a triangle, we can determine the possible lengths for the third side.

Example 4: A triangle has two sides with lengths 5 cm and 8 cm. What are the possible lengths for the third side?

8 cm

5 cm

8 cm

5 cm

? ?

To find the possible lengths of the third side, find the sum and the difference of the two given lengths.

8 cm - 5 cm = 3 cm 8 cm + 5cm = 13cm

The length of the third side must be between these two values, but cannot include them.

3 cm < third side < 13 cm

Let’s test this result.

The triangle has two sides that measure 5 cm and 8 cm. And:

3 cm < third side < 13 cm

• Can the third side measure 4 cm?

The triangle would have sides 5 cm, 8 cm, and 4 cm. Adding the two smaller sides,

5 cm + 4 cm > 8 cm

Remember: The sum of any two sides must be greater than the third side.

Try These:

D

B

A

C

C

Side Angle Inequality

Side-Angle Inequality

Side-Angle Inequality

Examples

In a triangle, the largest angle is opposite the longest side and vice versa.

Likewise, the smallest angle is opposite the shortest side and vice versa.

CB

A

85º

60º

35º

side.shortest theis AB Therefore,

angle.smallest theis C Also,

side.longest theis AC Therefore,

angle.largest theis , In

BABC

Example 1: In the triangle below, list the sides in order from longest to shortest.

Solution:

25º50º

105º

ZY

X

XY ,XZ ,YZ

isshortest longest to from sides theoforder theTherefore,

side.shortest theis XY thenangle,smallest theis Z Since

side.longest theis YZ thenangle,largest theis X Since

Example 2: In the triangle below, list the angles in order from largest to smallest.

Solution:

6 cm

5 cm

10 cm

R Q

P

P Q, R,

issmallest largest to from angles theoforder theTherefore,

angle.smallest theis P thenside,shortest theis RQ Since

angle.largest theis R thenside,longest theis PQ Since

Example 3:

Solution:

angle.largest

heIdentify t inches. 90 is triangle theofperimeter The

.13 and ,162 ,37 , In xEFxDFxDEDEF

F

ED

x + 132x - 16

37 - x

angle.largest theis D thenside,longest theis EF Since

in. 411328 in. 4016)28(2 in. 92837

28

90342

901316237

EFDFDE

x

x

xxx

Summary

In a triangle, the largest angle is opposite the longest side and vice versa.

Likewise, the smallest angle is opposite the shortest side and vice versa.

Side-Angle Inequality

Side-Angle Inequality

Try These:

B

Try These:

B

Try These:

C

Try These: