7.1 TRIANGLE APPLICATION

THEOREMS

Objective:After studying this section, you will be able to apply theorems about the interior angles,

the exterior angles, and the midlines of triangles.

Theorem

The sum of the measures of the three angles of a triangle is 180.

A

B

C

Proof

A

B

C

According to the Parallel Postulate, there exists exactly one line parallel to line AC

passing through point B, so we can draw the following figure.

1 2 3

Because of the straight angle, , . and (Parallel lines implies alt. int. angles congruent). We can substitute , therefore, the

180321 A1 C3

1802 CA180 CmBmAm

What is an exterior angle?An exterior angle is an angle that is

formed by extending one of the sides of a polygon.

Angle 1 is an exterior angle in the following polygons.

1

1

1

DefinitionAn exterior angle of a polygon is an angle that is adjacent to

and supplementary to an interior angle of the polygon.

The measure of an exterior angle of a triangle is equal to

the sum of the measures of the remote interior angles.

Theorem (triangles only)

A

B

C

1exterior angle

remote interior angles

1 BA

Exterior AngleTheorem (triangles only)

Theorem

A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-

half the length of the third side. (Midline Theorem)

A

B

C

D E10

20

Example 1

Find x, y, and z.

80

55

x

z

y 60

100

Example 2

The measures of the three angles of a triangle are in the ratio 3:4:5. Find the

measure of the largest angle.

5x

4x

3x

Example 3If one of the angles of a triangle is 80

degrees. Find the measure of the angle formed by the bisectors of the other two

angles.

80

xyx

y CB

E

A

Example 4

Angle 1 = 150 degrees, and the measure of angle B is twice that of angle A. Find the

measure of each angle of the triangle.

A

B

C

1

Summary

Explain how you can find the measure of an exterior angle.

Homework

Worksheet 7.1