Holt Geometry 8-1 Find Angle Measures In Polygons 8-1 Find Angle Measures in Polygons Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation

Holt Geometry

8-1 Find Angle Measures In Polygons8-1 Find Angle Measures in Polygons

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

8-1 Find Angle Measures In 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


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

Objectives

Holt Geometry


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

Holt Geometry


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

Remember!

Holt Geometry


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 Geometry


Holt Geometry


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°.

Holt Geometry


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


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.

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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.

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Example 3C Continued

mA = 35(4°) = 140°

mB = mE = 18(4°) = 72°

mC = mD = 32(4°) = 128°

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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°.

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

Remember!

Holt Geometry


Holt Geometry


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


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.

Holt Geometry


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.

Holt Geometry


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.

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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

8-2 Properties of Parallelograms8-2 Properties of Parallelograms

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz

Holt Geometry

8-2 Properties of Parallelograms

Warm UpFind the value of each variable.

1. x 2. y 3. z2 184

Holt Geometry


Prove and apply properties of parallelograms.

Use properties of parallelograms to solve problems.

Objectives

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Any polygon with four sides is a quadrilateral. However, some quadrilaterals have special properties. These special quadrilaterals are given their own names.

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Holt Geometry


Holt Geometry


Example 2A: Using Properties of Parallelograms to Find Measures

WXYZ is a parallelogram. Find YZ.

Def. of segs.

Substitute the given values.

Subtract 6a from both sides and add 4 to both sides.

Divide both sides by 2.

YZ = XW

8a – 4 = 6a + 10

2a = 14

a = 7

YZ = 8a – 4 = 8(7) – 4 = 52

opp. s

Holt Geometry


Example 2B: Using Properties of Parallelograms to Find Measures

WXYZ is a parallelogram. Find mZ .

Divide by 27.

Add 9 to both sides.

Combine like terms.

Substitute the given values.

mZ + mW = 180°

(9b + 2) + (18b – 11) = 180

27b – 9 = 180

27b = 189

b = 7

mZ = (9b + 2)° = [9(7) + 2]° = 65°

cons. s supp.

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Check It Out! Example 2a

EFGH is a parallelogram.Find JG.

Substitute.

Simplify.

EJ = JG

3w = w + 8

2w = 8w = 4 Divide both sides by 2.

JG = w + 8 = 4 + 8 = 12

Def. of segs.

diags. bisect each other.

Holt Geometry


Check It Out! Example 2b

EFGH is a parallelogram.Find FH.

Substitute.

Simplify.

FJ = JH

4z – 9 = 2z

2z = 9

z = 4.5 Divide both sides by 2.

Def. of segs.

FH = (4z – 9) + (2z) = 4(4.5) – 9 + 2(4.5) = 18

diags. bisect each other.

Holt Geometry


Lesson Quiz: Part I

In PNWL, NW = 12, PM = 9, and mWLP = 144°. Find each measure.

1. PW 2. mPNW

18 144°

Holt Geometry


Lesson Quiz: Part II

QRST is a parallelogram. Find each measure.

2. TQ 3. mT

28 71°

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Holt Geometry 8-1 Find Angle Measures In Polygons 8-1 Find Angle Measures in Polygons Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation