6 1 Properties and Attributes of Polygons Warm

6 -1 Properties and Attributes of Polygons Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Geometry

Drill: Thurs, Feb 24 th 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. 12 a + 4 a + 9 a = 100 Obj: SWBAT find and use the measures of interior and exterior angles of polygons.

Objectives Classify polygons based on their sides and angles. 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

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.

Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.

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.

What do you notice about all of the convex polygons?

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.

SO WHAT IS A POLYGON? WHAT ARE THE PROPERTIES OF A POLYGON?

http: //www. brainpop. com/math/geometryandmeasure ment/polygons/preview. weml "Triangle" uses the Latin "angle" (angulus) rather than the Greek "gon" which means the same thing, so it's just the Latin equivalent of the Greek "trigon. " "Quadrilateral" is even more distinctive, since it not only comes from Latin but means "four sides" rather than "four angles"; and in fact we DO use the word "quadrangle" sometimes (and also "trilateral")

You can name a polygon by the number of its sides. The table shows the names of some common polygons.

Example 1 A: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon

Example 1 B: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, heptagon

Example 1 C: Identifying Polygons Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. not a polygon

Check It Out! Example 1 a Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon

Check It Out! Example 1 b Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. polygon, nonagon

Check It Out! Example 1 c Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides. not a polygon

Example 2 A: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, convex

Example 2 B: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

Example 2 C: Classifying Polygons Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex

Check It Out! Example 2 a Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. regular, convex

Check It Out! Example 2 b Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. irregular, concave

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.

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

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

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

Check It Out! Example 3 a Find the sum of the interior angle measures of a convex 15 -gon. (n – 2)180° Polygon Sum Thm. (15 – 2)180° A 15 -gon has 15 sides, so substitute 15 for n. 2340° Simplify.

Example 3 B: 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. (16 – 2)180° = 2520° Substitute 16 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are , so divide by 16.

Check It Out! Example 3 b Find the measure of each interior angle of a regular decagon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon Sum Thm. (10 – 2)180° = 1440° Substitute 10 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are , so divide by 10.

Drill: 1. Find the sum of the interior angle measures of a convex 18 -gon. 2. Find the measure of each interior angle of a regular octagon. • SWBAT Find and use the measures of interior and exterior angles of polygons.

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°.

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

Example 4 A: 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°. measure of one ext. = Polygon Sum Thm. A regular 20 -gon has 20 ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20 -gon is 18°.

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

Check It Out! Example 4 a Find the measure of each exterior angle of a regular dodecagon. A dodecagon has 12 sides and 12 vertices. sum of ext. s = 360°. measure of one ext. Polygon Sum Thm. A regular dodecagon has 12 ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°.

Check It Out! Example 4 b Find the value of r in polygon JKLM. 4 r° + 7 r° + 5 r° + 8 r° = 360° Polygon Ext. Sum Thm. 24 r = 360 Combine like terms. r = 15 Divide both sides by 24.

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.

Check It Out! Example 5 What if…? Suppose the shutter were formed by 8 blades instead of 10 blades. What would the measure of each exterior angle be? CBD is an exterior angle of a regular octagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°. A regular octagon has 8 ext. , so divide the sum by 8.

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