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Core Connections Geometry, 2013

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Core Connections Geometry, 2013 View details

1. Section 8.1

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Exercise 44 Page 489

Practice makes perfect

a The sum of the interior angles of an n-gon can be written as the following expression.

180^(∘)(n-2)In this equation n is the number of sides (or vertices) in the polygon. From the exercise, we know that each interior angle in our polygon is 60^(∘). If this polygon has n interior angles, the sum of the interior angles should be 60^(∘) n. With this information, we can write an equation. 180^(∘)(n-2)=60^(∘) n Let's solve this equation for n.

180^(∘)(n-2)=60^(∘) n

▼

Solve for n

Distribute 180^(∘)

180^(∘) n-360^(∘) =60^(∘) n

LHS-60^(∘) n=RHS-60^(∘) n

120^(∘) n-360^(∘) =0

LHS+360^(∘)=RHS+360^(∘)

120^(∘) n=360^(∘)

.LHS /120^(∘).=.RHS /120^(∘).

n=3

The regular polygon has 3 sides, which means it is an equilateral triangle.

b Like in Part A, we equate 180^(∘)(n-2) with the product of the interior angle's measure and the number of sides, 156^(∘) n.

180^(∘)(n-2)=156^(∘) n Let's solve this equation for n.

180^(∘)(n-2)=156^(∘) n

▼

Solve for n

Distribute 180^(∘)

180^(∘) n-360^(∘) =156^(∘) n

LHS-156^(∘) n=RHS-156^(∘) n

24^(∘) n-360^(∘) =0

LHS+360^(∘)=RHS+360^(∘)

24^(∘) n =360^(∘)

.LHS /24^(∘).=.RHS /24^(∘).

n=15

The polygon has 15 sides.

c Like in Parts A and B, we have to equate 180^(∘)(n-2) with the product of the interior angle's measure and the number of sides, 90^(∘) n.

180^(∘)(n-2)=90^(∘) n Let's solve this equation for n.

180^(∘)(n-2)=90^(∘) n

▼

Solve for n

Distribute 180^(∘)

180^(∘) n-360^(∘) =90^(∘) n

LHS-90^(∘) n=RHS-90^(∘) n

90^(∘) n-360^(∘) =0

LHS+360^(∘)=RHS+360^(∘)

90^(∘) n =360^(∘)

.LHS /90^(∘).=.RHS /90^(∘).

n=4

The regular polygon has 4 sides, which means it is a square.

d Like in previous parts, we have to equate 180^(∘)(n-2) with the product of the interior angle's measure and the number of sides, 140^(∘) n.

180^(∘)(n-2)=140^(∘) n Let's solve this equation for n.

180^(∘)(n-2)=140^(∘) n

▼

Solve for n

Distribute 180^(∘)

180^(∘) n-360^(∘) =140^(∘) n

LHS-90^(∘) n=RHS-90^(∘) n

40^(∘) n-360^(∘) =0

LHS+360^(∘)=RHS+360^(∘)

40^(∘) n =360^(∘)

.LHS /40^(∘).=.RHS /40^(∘).

n=9

The regular polygon has 9 sides.

Subchapter links

8.1.1 Pinwheels and Polygons p.476-478

8.1.2 Interior Angles of Polygons p.481-482

8.1.3 Angles of Regular Polygons p.485-486

8.1.4 Regular Polygon Angle Connections p.488-490

8.1.5 Finding Areas of Regular Polygons p.494-496