Holt Geometry

6-1 Properties and Attributes of Polygons

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

Holt Geometry

6-1 Properties and Attributes of Polygons

6-1 Properties and Attributes of Polygons

Holt Geometry

Holt 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 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 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 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 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 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 Geometry

6-1 Properties and Attributes of Polygons

A regular polygon is one that is both equilateral and equiangular.

If a polygon is not regular, it is called irregular.

Holt 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 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 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 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 Geometry

6-1 Properties and Attributes of Polygons

Holt 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°

(7 – 2)180°

900°

Polygon Sum Thm.

A heptagon has 7 sides, so substitute 7 for n.

Simplify.

Holt 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.

Step 2 Find the measure of one interior angle.

(n – 2)180°

(16 – 2)180° = 2520°

Polygon Sum Thm.

Substitute 16 for n and simplify.

The int. s are , so divide by 16.

Holt 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 Geometry

6-1 Properties and Attributes of Polygons

Holt 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°.

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°.

Polygon Sum Thm.

measure of one ext. =

Holt 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 Geometry

6-1 Properties and Attributes of Polygons

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.

Holt Geometry

6-1 Properties and Attributes of Polygons

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.

Lesson Quiz

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°

Holt Geometry

6-1 Properties and Attributes of Polygons

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.

Warm-Up(Pass back Papers!!!!!!!!!!!!!!!)

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°