objective: after studying this section, you will be able to apply theorems about the interior...
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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
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