66-1-1 PropertiesProperties andand Attributes of of Polygons Polygons

Warm Up Lesson Presentation Lesson Quiz

HoltHolt McDougal Geometry Geometry 6-1 Properties and Attributes of Polygons Warm Up

1. A ? is a three-sided polygon. triangle 2. A ? is a four-sided polygon. quadrilateral

Evaluate each expression for n = 6. 3. (n – 4) 12 24 4. (n – 3) 90 270

Solve for a.

5. 12a + 4a + 9a = 100 4

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Objectives Classify polygons based on their sides and angles.

Find and use the measures of interior and exterior angles of polygons.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Vocabulary side of a polygon vertex of a polygon diagonal regular polygon concave convex

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

In Lesson 2-4, you learned the definition of a polygon. Now you will learn about the parts of a polygon and about ways to classify polygons.

Holt McDougal Geometry 6-1 Properties and Attributes 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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

Holt McDougal Geometry 6-1 Properties and Attributes of 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, hexagon

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 1B: Identifying Polygons

Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

polygon, heptagon

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 1C: Identifying Polygons

Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.

not a polygon

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 1b

Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.

polygon, nonagon

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 1c

Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.

not a polygon

Holt McDougal Geometry 6-1 Properties and Attributes of 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 2A: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, convex

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 2B: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, concave

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 2C: Classifying Polygons

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

regular, convex

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 2a

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

regular, convex

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 2b

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.

irregular, concave

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Remember! By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 3A: 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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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.

(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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 3C: 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. 35c + 18c + 32c + 32c + 18c = 540 Substitute. 135c = 540 Combine like terms. c = 4 Divide both sides by 135.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Example 3C Continued

mA = 35(4°) = 140° mB = mE = 18(4°) = 72°

mC = mD = 32(4°) = 128°

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 3a

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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 3b

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.

Holt McDougal Geometry 6-1 Properties and Attributes 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°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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 measure of one ext.  = has 20  ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

Check It Out! Example 4a Find the measure of each exterior angle of a regular dodecagon.

A dodecagon has 12 sides and 12 vertices.

sum of ext. s = 360°. Polygon  Sum Thm. A regular dodecagon measure of one ext. has 12  ext. s, so divide the sum by 12. The measure of each exterior angle of a regular dodecagon is 30°.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons

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.

Holt McDougal Geometry