Holt Geometry

UNIT 6.1 POLYGONSUNIT 6.1 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

Classify polygons based on their sides and angles.

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

Objectives

side of a polygonvertex of a polygondiagonalregular polygonconcaveconvex

Vocabulary

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.

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

Remember!

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

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

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

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

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

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

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.

Example 2A: Classifying Polygons

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

irregular, convex

Example 2B: Classifying Polygons

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

irregular, concave

Example 2C: Classifying Polygons

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

regular, convex

Check It Out! Example 2a

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

regular, convex

Check It Out! Example 2b

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.

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

Remember!

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 trianglesis (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.

(n – 2)180°

(7 – 2)180°

900°

Polygon Sum Thm.

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

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.

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.

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.

mA + mB + mC + mD + mE = 540°Polygon Sum Thm.

35c + 18c + 32c + 32c + 18c = 540 Substitute.

135c = 540 Combine like terms.

c = 4 Divide both sides by 135.

Example 3C Continued

mA = 35(4°) = 140°

mB = mE = 18(4°) = 72°

mC = mD = 32(4°) = 128°

Check It Out! Example 3a

Find the sum of the interior angle measures of a convex 15-gon.

(n – 2)180°

(15 – 2)180°

2340°

Polygon Sum Thm.

A 15-gon has 15 sides, so substitute 15 for n.

Simplify.

Find the measure of each interior angle of a regular decagon.

Step 1 Find the sum of the interior angle measures.

Step 2 Find the measure of one interior angle.

Check It Out! Example 3b

(n – 2)180°

(10 – 2)180° = 1440°

Polygon Sum Thm.

Substitute 10 for n and simplify.

The int. s are , so divide by 10.

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°.

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

Remember!

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. =

Example 4B: Finding Interior Angle Measures and Sums in Polygons

Find the value of b in polygon FGHJKL.

15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360°

Polygon Ext. Sum Thm.

120b = 360 Combine like terms.

b = 3 Divide both sides by 120.

Find the measure of each exterior angle of a regular dodecagon.

Check It Out! Example 4a

A dodecagon has 12 sides and 12 vertices.

sum of ext. s = 360°.

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°.

Polygon Sum Thm.

measure of one ext.

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.

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.

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°

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