1. Do Now
Solve each equation.
x = 180
2. x = 180
x + 32 = 180
x = 90
x = 48
x = 28

2. Learn to find the measures of angles in polygons.

3. If you tear off the corners of a triangle and put them together, you will find that they form a straight angle.

4. The sum of the measures of the angles in a triangle is 180°
Triangle Sum Rule
The sum of the measures of the angles in a triangle is 180°

5. Example A
Write and solve an addition equation to find the missing angle measure.
55°
80° x
The sum of the measures of the angles is 180°.
80° + 55° + x = 180°
135° + x = 180°
Combine like terms.
–135°
–135°
Subtract 135° from both sides.
x = 45°
The measure of the unknown angle is 45°.

6. You Try
Write and solve an addition equation to find the missing angle measure.
30°
90° x
The sum of the measures of the angles is 180°.
90° + 30° + x = 180°
120° + x = 180°
Combine like terms.
–120°
–120°
Subtract 120° from both sides.
x = 60°
The measure of the unknown angle is 60°.

7. The sum of the angle measures in any four-sided figure can be found by dividing the figure into two triangles.
You can divide the figure by drawing a diagonal. A diagonal is a line segment that connects two non- adjacent vertices of a polygon.
Diagonal

8. Since the sum of the angle measures in each triangle is 180°, the sum of the angle measures in a four-sided figure is 360°

9. Quadrilateral Sum Rule
The sum of the measures of the angles in a quadrilateral is 360°

10. Example B
Write and solve an addition equation to find the missing angle measure.
65° x
89°
82°
The sum of the measures of the angles is 360°.
65° + 89° + 82° + x = 360°
236° + x = 360°
Combine like terms.
–236°
–236°
Subtract 236° from both sides.
x = 124°
The measure of the unknown angle is 124°.

11. You Try
Write and solve an addition equation to find the missing angle measure.
92°
89°
67° x
The sum of the measures of the angles is 360°.
67° + 92° + 89° + x = 360°
248° + x = 360°
Combine like terms.
–248°
–248°
Subtract 248° from both sides.
x = 112°
The measure of the unknown angle is 112°.

12. In a convex polygon, all diagonals can be drawn within the interior of the figure. By dividing any convex polygon into triangles, you can find the sum of its interior angle measures.

13. Example C
Divide each polygon into triangles to find the sum of its angle measures.
6 · 180° = 1080°
There are 6 triangles.
The sum of the angle measures
of an octagon is 1,080°.

14. You Try
Divide each polygon into triangles to find the sum of its angle measures.
4 · 180° = 720°
There are 4 triangles.
The sum of the angle measures
of a hexagon is 720°.

15. 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 quadrilateral with angle measures of 144°, 84°, and 48°.
84°
4. Divide a six-sided polygon into triangles to
find the sum of its interior angles
720°

16. Lesson Quiz for Student Response Systems
1. Identify the measure of the unknown angle for a triangle with angle measures of 35° and 53°.
A. 55°
B. 92°
C. 268°
D. 272°

17. Lesson Quiz for Student Response Systems
2. Identify the measure of the unknown angle for a right triangle with one angle measure of 62°.
A. 242°
B. 118°
C. 28°
D. 18°

18. Lesson Quiz for Student Response Systems
3. Divide a nine-sided polygon into triangles to identify the sum of its interior angles.
A. 1080°
B. 1260°
C. 1440°
D. 1620°