Entry task The car at each vertex of a Ferris wheel holds 5 people. The sum of the interior angles of the Ferris wheel is 7740. What is the maximum number of people the Ferris wheel can hold? 225

6.1 Polygon Angle Sum Theorems Learning Target: I can classify polygons based on their sides and angles. Find and use the measures of interior and exterior angles of polygons.

Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

You can name a polygon by the number of its sides You can name a polygon by the number of its sides. The table shows the names of some common polygons. A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Remember!

Example 1A: Identifying Polygons Discuss with A/B partner whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a polygon polygon, hexagon polygon, heptagon not a polygon not a polygon

All the sides are congruent in an equilateral polygon All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular.

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

Check It Out! Example 2a Discuss with A/B partner whether the polygon is regular or irregular and whether it is concave or convex. regular, convex irregular, concave

With your table…. Work as a table and see if there is a way you can find the measure of a polygon with any number of sides. Complete the handout on 6.1

To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon.

The formula we use to find the sum of the interior angles of any polygon comes from the number of triangles in a figure

In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°.

Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon (7-sides). (n – 2)180° Polygon  Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify.

Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon  Sum Thm. Substitute 16 for n and simplify. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. Polygon  Sum Thm. (5 – 2)180° = 540° Polygon  Sum Thm. mA + mB + mC + mD + mE = 540° 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.

Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128°

Exterior Angles Activity Get graph paper or scratch paper. Using a straight edge draw a polygon, each person in your group must have different amount of sides. Discuss who will draw each shape. It does not have to be regular After the shape is drawn extend each consecutive side. Highlight the exterior angle Cut out each exterior angle Put the colored angles together What happened? What about the other polygons? What can you conclude about the exterior angle of polygons? 3 sides? 4 sides? 5 sides?

An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Remember!

Videos #1 #2 #3 #4 #5

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. Polygon  Sum Thm. A regular 20-gon has 20  ext. s, so divide the sum by 20. measure of one ext.  = The measure of each exterior angle of a regular 20-gon is 18°.

Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext.  Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 Combine like terms. b = 3 Divide both sides by 120.

Example 5: Art Application Ann is making paper stars for party decorations. What is the measure of 1? 1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular pentagon has 5  ext. , so divide the sum by 5.

Lesson Quiz 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. 2. Find the sum of the interior angle measures of a convex 11-gon. nonagon; irregular; concave 1620° 3. Find the measure of each interior angle of a regular 18-gon. 4. Find the measure of each exterior angle of a regular 15-gon. 160° 24°

Homework P. 356 #18-21 and 27, 29 –34, 37- 40