Download 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.
The sum of the measures of the three
angles of a triangle is 180.
B
A
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.
B
1
A
2
3
C
Because of the straight angle, 1  2  3  180 ,
.1  Aand 3  C (Parallel lines implies alt. int.
angles congruent). We can substitute A  2  C  180,
therefore, the mA  mB  mC  180
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.
Exterior AngleTheorem
(triangles only)
remote interior
angles
B
exterior angle
1
A
C
A  B  1
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
A
10
20
E
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
4x
3x
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
B
x
x
y
y
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