7.1 Triangle application theorems

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. B A C

Proof 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 3 2 B Because of the straight angle, , . and (Parallel lines implies alt. int. angles congruent). We can substitute , therefore, the A C

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

Definition An exterior angle of a polygon is an angle that is adjacent to and supplementary to an interior angle of the polygon. Theorem (triangles only) The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

remote interior angles exterior angle 1 B A C

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) B D E 10 20 A C

Example 1 Find x, y, and z. 80 100 z 55 x y 60

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 3x 4x

Example 3 If 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. A 80 E x y x y B C

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. B 1 A C

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

7.1 Triangle application theorems