1. 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

2. Learn to find the measures of angles in polygons. Course 2 7-8 Angles in Polygons

3. Vocabulary diagonal Insert Lesson Title Here Course 2 7-8 Angles in Polygons

4. 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°.

5. 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.

6. Course 2 7-8 Angles in Polygons Notice that 1,2, 3 and together form a straight angle. That is, the sum of their measures is 180°. 1 2 3 4 5 Notice also that the figure you have drawn consists of two parallel lines cut by two transversals. So if you were to tear off 4and 5 from the triangle, they would fit exactly over 1 3. and This shows that the sum of the measures of the angles in the triangle are 180°.

7. Find the measure of the unknown angle. Additional Example 1: Determining the Measure of an Unknown Interior Angle Course 2 7-8 Angles in Polygons 55° 80° x 80° + 55° + x = 180° 135° + x = 180° –135° x = 45° The measure of the unknown angle is 45°. The sum of the measures of the angles is 180°. Combine like terms. Subtract 135° from both sides.

8. Find the measure of the unknown angle. Try This: Example 1 Course 2 7-8 Angles in Polygons 90° + 30° + x = 180° 120° + x = 180° –120° x = 60° The measure of the unknown angle is 60°. The sum of the measures of the angles is 180°. Combine like terms. Subtract 120° from both sides. 90° 30° x

9. 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.

10. 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.

11. Course 2 7-8 Angles in Polygons Number of triangles in pentagon Sum of angle measures in each triangle Sum of angle measures in pentagon 3 · 180°=540°

12. Divide each polygon into triangles to find the sum of its angle measures. Additional Example 2A: Drawing Triangles to Find the Sum of Interior Angles Course 2 7-8 Angles in Polygons A. There are 6 triangles. The sum of the angle measures of an octagon is 1,080°. 6 · 180° = 1080°

13. Divide each polygon into triangles to find the sum of its angle measures. Additional Example 2B: Drawing Triangles to Find the Sum of Interior Angles Course 2 7-8 Angles in Polygons 10 · 180° = 1,800°There are 10 triangles. The sum of the angle measures of a 12-sided polygon is 1,800°. B.

14. Divide each polygon into triangles to find the sum of its angle measures. Try This: Example 2A Course 2 7-8 Angles in Polygons A. There are 4 triangles. The sum of the angle measures of a hexagon is 720°. 4 · 180° = 720°

15. Divide each polygon into triangles to find the sum of its angle measures. Try This: Example 2B Course 2 7-8 Angles in Polygons B. There are 2 triangles. The sum of the angle measures of a square is 360°. 2 · 180° = 360°

16. Lesson Quiz 54° 37° Insert Lesson Title Here 106° 900° Course 2 7-8 Angles in Polygons Find the measure of the unknown angle for each of the following. 1. a triangle with angle measures of 66° and 77° 2. a right triangle with one angle measure of 36° 3. an obtuse triangle with angle measures of 42° and 32° 4. Divide a seven-sided polygon into triangles to find the sum of its interior angles

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