Properties of Triangles

Unit 6

5.5 Inequalities in One Triangle

State Standards for Geometry6. Know and use the

Triangle Inequality Theorem.

13. Prove relationships between angles in a polygon.

Lesson Goals• Use triangle measurements

to decide which side is longest or which angle is largest.

• Apply the Triangle Inequality Theorem to determine if 3 lengths can form a triangle, and find possible lengths of a 3rd side given two.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

A

B

Cthen is the shortest side.ABIf is the smallest angle, C

then is the longest side. ACIf is the largest angle, B

theorem

Triangle Angle-Side Relationships Theorem The longest side of a triangle is opposite the largest angle The shortest side of a triangle is opposite the

smallest angle.

largest angle

longest side

shorte

st sid

e

smallest angle

http://www.mathopenref.com/common/appletframe.html?applet=trianglebigsmall&wid=600&ht=300

ACABBC

Write the measures for the sides of the triangle in order from least to greatest

A

B

C

111o

46o

23o

is opposite the smallest angle.BC

is opposite the middle-sized angle.AB

is opposite the largest angle.AC

example

m U m T m V

Write the measures for the angles of the triangle in order from least to greatest

is opposite the shortest side.V

is opposite the middle-length side.T

is opposite the longest side.U

You Try

T

U

10

V

7

11

theorem

Exterior Angle Inequality TheoremThe measure of an exterior angle of atriangle is greater than the measure of either of the two nonadjacent interior angles.

A

BC1

1m m A

theorem

Exterior Angle Inequality TheoremThe measure of an exterior angle of atriangle is greater than the measure of either of the two nonadjacent interior angles.

A

BC1

1m m A

1m m C

14

3

2

3 54 6

example

List all angles whose measures are less than 1.m

5

6

7

Write an equation or inequality to describe the relationship between the measures of all angles.

ao

doco

bo

180a b c

a b d

180c d d a

d b

example

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

A

BC

theorem

A C

CA B

AB BC AC

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

A

BC

theorem

AB AC BC

B C

CB A

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

A

BC

theorem

A B

CA B

AC BC AB

http://www.mathopenref.com/common/appletframe.html?applet=triangleinequality2&wid=600&ht=200

Can a triangle be constructed with sides of the following measures?

5, 7, 8

By the Triangle Inequality Theorem, the sum of the measures

of any two sides must by greater than the third side.

5 + 7 > 8 5 + 8 > 7 7 + 8 > 5

example

Yes

Can a triangle be constructed with sides of the following measures?

3, 6, 10

3 + 6 > 10

example

No

Can a triangle be constructed with sides ofthe following measures?

By the Triangle Inequality Theorem, the sum of the measures

of any two sides must by greater than the third side.

9 + 5 > 11

A triangle can be constructed.

You Try

9 in, 5 in , 11 in

Solve the inequality.

exampleAB AC BC

5 2 1 10x x x

3 4 10x x

4 4 10x

4 6x 3

2x

A triangle has one side of 8 cm and another of 17 cm. Describe the possible lengths of the third side.

By the Triangle Inequality Theorem, the sum of the measures

of any two sides must by greater than the third side.

x + 8 > 17 x + 17 > 8 8 + 17 > x

817

x

x > 9 x > -9 25 > x9 < x x < 25

example

x < 27

A triangle has one side of 11 in and another of 16 in. Describe the possible lengths of the third side.

By the Triangle Inequality Theorem, the sum of the measures

of any two sides must by greater than the third side.

x + 11 > 16 x + 16 > 11 11 + 16 > x

1116

x

x > 5 x > -5 27 > x5 < x

example

Today’s Assignment

p. 298: 1 – 6, 9, 12, 13, 16, 17, 20, 23 – 25, 27, 28

(worksheet in packet)