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Learn how to classify polygons based on their sides and angles, find the measures of interior and exterior angles, and solve problems involving polygons.
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Warm Up 1.A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Evaluate each expression for n = 6. 3. (n – 4) 12 4. (n – 3) 90 Solve for a. 5. 12a + 4a + 9a = 100 triangle quadrilateral 24 270 4
Learning Targets I will classify polygons based on their sides and angles. I will find and use the measures of interior and exterior angles of polygons.
Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex
Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.
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. The table shows the names of some common polygons.
Example 1A: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, heptagon polygon, hexagon not a polygon
Check It Out! Example 1a Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon
A regular polygonis a polygon 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. The easy way to remember: If it caves in, it is concave. If it does not cave in, it is convex.
Example 2A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex regular, convex irregular, concave
Check It Out! Example 2a Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex
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.
Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.
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 the following convex polygons. 1. Heptagon 2. Decagon 3. Pentagon 1. 900° 2. 1,440° 3. 540 °
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. (16 – 2)180° = 2520° Step 2 Find the measure of one interior angle.
Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. c= 4 mA = 35(4°)= 140° mB = mE = 18(4°)= 72° mC = mD = 32(4°)= 128°
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. The measure of each exterior angle of a regular 20-gon is 18°.
Check It Out! Example 4b Find the value of r in polygon JKLM. 4r° + 7r° + 5r° + 8r°= 360° Polygon Ext. Sum Thm. 24r= 360 Combine like terms. r= 15 Divide both sides by 24.
