Holt McDougal Geometry 6-1 Properties and Attributes of Polygons 6-1 Properties and Attributes of...

Holt McDougal Geometry

6-1 Properties and Attributes of Polygons6-1 Properties and Attributes of Polygons

Holt GeometryHolt McDougal 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



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


polygon, heptagon



Example 1C: Identifying Polygons


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


irregular, concave



Example 2C: Classifying Polygons


regular, convex





regular, convex





irregular, concave



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°




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.


(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.


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.




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.

Holt McDougal Geometry 6-1 Properties and Attributes of Polygons 6-1 Properties and Attributes of...

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