Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Warm Up Solve. 1. 72 + 18 + x = 180 2. 80 + 70 + x = 180 3. x + 42 + 90 = 180 4. 120 + x + 32 = 180 x = 90 x = 30 x = 48 x = 28

How many different rectangles are in the figure shown? Problem of the Day How many different rectangles are in the figure shown? 100

Learn to find the measures of angles in polygons.

Vocabulary diagonal

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°.

Additional Example 1: Finding an Angle Measure of in a Triangle 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°.

Check It Out: 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°.

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

Additional Example 2: Finding an Angle Measure of in a Quadrilateral 65° x 89° 82° Find the unknown angle measure in the quadrilateral. 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°.

Find the unknown angle measure in the quadrilateral. Check It Out: Example 2 92° Find the unknown angle measure in the quadrilateral. 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°.

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.

Additional Example 3: Drawing Triangles to Find the Sum of Interior Angles 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°.

Check It Out: Example 3 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°.

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

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°

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°

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°

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°