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1. 7-8 Angles in Polygons Course 2 Warm Up Problem of the Day Lesson Presentation

2. Warm Up Solve. 1.72 + 18 + x = 180 2. 80 + 70 + x = 180 3.x + 42 + 90 = 180 4. 120 + x + 32 = 180

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

4. Insert Lesson Title Here Vocabulary diagonal

5. 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°.

6. 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.

7. 1, 2, 3 and Notice that together form a straight angle. 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 from the triangle, they would fit exactly 5 4 and 1 over and 3. This shows that the sum of the measures of the angles in the triangle are 180°. 1 3 2 5 4

8. Additional Example 1: Determining the Measure of an Unknown Interior Angle 55° Find the measure of the unknown angle. 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°.

9. Try This: Example 1 30° Find the measure of the unknown angle. 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°.

10. 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.

11. Adiagonalof 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.

12. Number of triangles in pentagon Sum of angle measures in each triangle Sum of angle measures in pentagon 3 180° = 540° ·

13. 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°.

14. 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°.

15. 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°.

16. 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°.

17. Insert Lesson 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°