6.4 Triangle Inequalities

Angle and Side Inequalities Sketch a good size triangle in your notebook

(about a third of the page). Using a ruler find the approximate length of each

side (in inches or centimeters). How is the largest side related to the largest

angle?

How is the smallest angle related to the smallest side?

Name the angles in ascending order.

m A

m C21

19

18

A

B

C

m B

Name the longest side. AB

AB

58

79A

B

C

43

BCName the shortest side.

o

o

o

TRIANGLE INEQUALITY THEOREM The sum of the lengths of any two

sides of a triangle is greater than the length of the third side.

Is it possible for a triangle to have the following lengths? 3, 6, 8 10, 10, 0.5

3 + 6 = 9 > 83 + 8 = 11 > 66 + 8 = 14 > 3 YES

0.5 + 10 = 10.5 > 100.5 + 10 = 10.5 > 1010 + 10 = 20 > 0.5 YES

Get on your “Thinking Caps”

Can you think of three lengths that cannot make a triangle?

More Triangle Inequality Practice

The lengths of two sides of a

triangle are 3 and 5. The length of

the third side must be greater than

and less than .5 - 32 5 + 38

2

1

1312

5Y

W

Z

15

9X

NOTE: We DO NOT KNOW that !!ZW XYThere are 3 triangles involved!

Put the following angles in ascending order.

1, 2, , ,X Y XZY

Section 6.4 #17

2

1

1312

5Y

W

Z

15

9X

NOTE: We DO NOT KNOW that !!ZW XY

Label XWZ and XZW

3

4

2

1

1312

5Y

W

Z

15

9X

There are 3 triangles involved!

3

4

2

1

1312

5Y

W

Z

15

9X

Using we get:WYZ

YW

Z

15

9

1

12

2

2 > Y > 13

4

2

1

1312

5Y

W

Z

15

9X

Using we get:WXZ

2

X W

Z

13

5

4

12

3

3 > X > 4

And... 2 > X

(Ext. Angle)

2 > X

3

4

2

1

1312

5Y

W

Z

15

9X

Using we get:XYZ

X > XZY > Y 13

14Y

Z

15

X

Now we can make our list:

2 X > XZY > Y > 1>m m m m m

PRACTICE MAKES PERFECT!

Page 221

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Extra Credit Opportunity (SAT style!)

What is the smallest integer, x, for which x, x + 5, and 2x – 15 can be the lengths of the sides of a triangle?

Hint: Use the Triangle Inequality Theorem