5.5 – use inequalities in a triangle. mn p measure each side of the triangle in centimeters and...

5.5 – Use Inequalities in a Triangle

M N

P

Measure each side of the triangle in centimeters and each angle in degrees. Write these measurements on your triangle.

2.9cm 5.4cm

6.5cm56º 26º

98º

Theorem:

If one side of the triangle is longer than the other side, then the angle opposite the longer side is larger than the angle opposite the shorter side

Theorem:

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

1. List the sides and angles in order from smallest to largest.

Sides Angles

C

A

AB

BC

AC B

Sides Angles

R

S

1. List the sides and angles in order from smallest to largest.

ST

RT

SR T

55°

1. List the sides and angles in order from smallest to largest.

Sides Angles

H

J

KJ

KH

JH K

33°

1. List the sides and angles in order from smallest to largest.

Sides Angles

E

D

DF

EF

DE F

Construct a triangle with sides of 2, 2, and 6 cm.

2cm

6cm

Doesn’t work!

Why?

6cm

2cm

Too short!

2cm

What if I tried 3, 3, and 6cm?

6cm3cm 3cm

Still doesn’t work!

Triangle Inequality Theorem:

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

a

b

ca + b > c

a + c > b

b + c > a

2. Can a triangle be formed with sides 1, 2, and 5?

No, 1 + 2 > 5

2. With 6, 7, and 11?

Yes, 6 + 7 > 117 + 11 > 66 + 11 > 7

2. With 8, 12, and 20?

No, 8 + 12 > 20

x + 9 > 21 x + 21 > 921 + 9 > xx > 12 30 > x

orx < 30

x > -12

12 < x < 30

3. What are the restrictions on x?

Trick!

Subtract the two sides to find the lowest and add to find the highest.

3. What are the possible values for the third side of a triangle with sides 9 and 12?

12 – 9 <

3 < x < 21

x < 12 + 9

5. What are the possible values for the third side of a triangle with sides 14 and 20?

20 – 14 <

6 < x < 34

x < 20 + 14

6. Solve the inequality AB + AC > BC for x.

3x + 1 + 4x – 2 > 5x + 7

7x – 1 > 5x + 7

2x – 1 > 7

2x > 8

x > 4