1. 8-4 Angles in Polygons Problem of the Day How many different rectangles are in the figure shown? 100

2. 8-4 Angles in Polygons Learn to find the measures of angles in polygons.

3. 8-4 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°.

4. 8-4 Angles in Polygons

5. 8-4 Angles in Polygons Find the measure of the unknown angle. Additional Example 1: Finding an Angle Measure of in a Triangle 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.

6. 8-4 Angles in Polygons Find the measure of the unknown angle. Check It Out: Example 1 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

7. 8-4 Angles in Polygons A diagonal is a line segment that connects two non-adjacent vertices of a polygon. Since the sum of the angle measures in each triangle is 180°, the sum of the angle measures in a four- sided figure is 2 · 180°, or 360°. Diagonal

8. 8-4 Angles in Polygons

9. 8-4 Angles in Polygons Find the unknown angle measure in the quadrilateral. Additional Example 2: Finding an Angle Measure of in a Quadrilateral 65° + 89° + 82° + x = 360° 236° + x = 360° –236° x = 124° The measure of the unknown angle is 124°. The sum of the measures of the angles is 360°. Combine like terms. Subtract 236° from both sides. 65° x 89° 82°

10. 8-4 Angles in Polygons Find the unknown angle measure in the quadrilateral. Check It Out: Example 2 67° + 92° + 89° + x = 360° 248° + x = 360° –248° x = 112° The measure of the unknown angle is 112°. The sum of the measures of the angles is 360°. Combine like terms. Subtract 248° from both sides. 67 ° 92 ° x 89 °

11. 8-4 Angles in Polygons 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.

12. 8-4 Angles in Polygons Divide each polygon into triangles to find the sum of its angle measures. Additional Example 3: Drawing Triangles to Find the Sum of Interior Angles There are 6 triangles. The sum of the angle measures of an octagon is 1,080°. 6 · 180° = 1080°

13. 8-4 Angles in Polygons Divide each polygon into triangles to find the sum of its angle measures. Check It Out: Example 3 There are 4 triangles. The sum of the angle measures of a hexagon is 720°. 4 · 180° = 720°

14. 8-4 Angles in Polygons Lesson Quiz 54° 37° 84° 720° 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 quadrilateral with angle measures of 144°, 84°, and 48°. 4. Divide a six-sided polygon into triangles to find the sum of its interior angles