Download Course 2 7-8

7-8 Angles in Polygons
Warm Up
Solve.
1. 72 + 18 + x = 180
2. 80 + 70 + x = 180
3. x + 42 + 90 = 180
4. 120 + x + 32 = 180
Course 2
7-8 Angles in Polygons
Learn to find the measures of angles in
polygons.
Course 2
7-8 Angles
Insert Lesson
Title Here
in Polygons
Vocabulary
diagonal
Course 2
7-8 Angles in Polygons
If you tear off the corners of a triangle and put
them together, you will find that they form a
straight angle. This suggests that the sum of the
measures of the angles in a triangle is 180°.
Course 2
7-8 Angles in Polygons
You can prove mathematically that the angle
measures in a triangle add up 180° by drawing a
diagram using the following steps.
a. Draw a triangle.
b. Extend the sides of
the triangle.
c. Draw a line through the vertex
opposite the base, so that the
line is parallel to the base.
Course 2
7-8 Angles in Polygons
1
4
2
3
5
1, 2, and 3 together form a straight angle.
Notice that
That is, the sum of their measures is 180°.
Notice also that the figure you have drawn consists of two
parallel lines cut by two transversals. So if you were to tear off
4 and 5 from the triangle, they would fit exactly
over 1 and 3. This shows that the sum of the measures
of the angles in the triangle are 180°.
Course 2
7-8 Angles in Polygons
Additional Example 1: Determining the Measure of
an Unknown Interior Angle
Find the measure of the
unknown angle.
55°
80°
x
80° + 55° + x = 180° The sum of the measures
of the angles is 180°.
135° + x = 180° Combine like terms.
–135°
–135° Subtract 135° from both sides.
x = 45°
The measure of the unknown angle is 45°.
Course 2
7-8 Angles in Polygons
Try This: Example 1
Find the measure of the
unknown angle.
30°
90°
x
90° + 30° + x = 180° The sum of the measures
of the angles is 180°.
120° + x = 180° Combine like terms.
–120°
–120° Subtract 120° from both sides.
x = 60°
The measure of the unknown angle is 60°.
Course 2
7-8 Angles in Polygons
The sum of the angle measures in other
polygons can be found by dividing the
polygon into triangles. A polygon can be
divided into triangles by drawing all of the
diagonals from one of its vertices.
Course 2
7-8 Angles in Polygons
A diagonal of a polygon is a segment that is
drawn from one vertex to another and is not one
of the sides of the polygon. You can divide a
polygon into triangles by using diagonals only if
all of the diagonals of that polygon are inside the
polygon. The sum of the angle measures in the
polygon is then found by combining the sums of
the angle measures in the triangles.
Course 2
7-8 Angles in Polygons
Number of
triangles
in pentagon
3
Course 2
Sum of angle
measures
in each
triangle
·
180°
Sum of angle
measures
in pentagon
=
540°
7-8 Angles in Polygons
Additional Example 2A: Drawing Triangles to Find
the Sum of Interior Angles
Divide each polygon into triangles to find the
sum of its angle measures.
A.
6 · 180° = 1080°
There are 6 triangles.
The sum of the angle measures
of an octagon is 1,080°.
Course 2
7-8 Angles in Polygons
Additional Example 2B: Drawing Triangles to Find
the Sum of Interior Angles
Divide each polygon into triangles to find the
sum of its angle measures.
B.
10 · 180° = 1,800°
There are 10 triangles.
The sum of the angle measures of a 12-sided
polygon is 1,800°.
Course 2
7-8 Angles in Polygons
Try This: Example 2A
Divide each polygon into triangles to find the
sum of its angle measures.
A.
4 · 180° = 720°
There are 4 triangles.
The sum of the angle measures
of a hexagon is 720°.
Course 2
7-8 Angles in Polygons
Try This: Example 2B
Divide each polygon into triangles to find the
sum of its angle measures.
B.
2 · 180° = 360°
There are 2 triangles.
The sum of the angle measures
of a square is 360°.
Course 2
7-8 Angles
Insert Lesson
in Polygons
Title Here
Lesson Quiz
Find the measure of the unknown angle for
each of the following.
1. a triangle with angle measures of 66° and 77°
37°
2. a right triangle with one angle measure of 36°
54°
3. an obtuse triangle with angle measures of 42°
and 32°
106°
4. Divide a seven-sided polygon into triangles to
find the sum of its interior angles
900°
Course 2
Assignment
• Page 384 – 385
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