Exercise 138 Page 525

180^(∘)(n-2)

Substitute

n= 12

180^(∘)( 12-2)

▼

Simplify

SubTerm

Subtract term

180^(∘)(10)

Multiply

Multiply

1800^(∘)

360^(∘)/n=35^(∘) Let's solve for n in this equation.

360^(∘)/n=35^(∘)

▼

Solve for n

MultEqn

LHS * n=RHS* n

360^(∘)=35^(∘) n

RearrangeEqn

Rearrange equation

35^(∘) n=360^(∘)

DivEqn

.LHS /35^(∘).=.RHS /35^(∘).

n=10.28571...

Since n is not an integer, this kind of regular polygon is not possible.

180^(∘)(n-2)=900^(∘)

DivEqn

.LHS /180^(∘).=.RHS /180^(∘).

n-2=5

AddEqn

LHS+2=RHS+2

n=7

The number of sides is 7.

Central angle=360^(∘)/n Let's call the measure of the interior angle θ. The measure of an interior angle plus its corresponding exterior angle is 180^(∘). With this information we can write and solve the following equation containing θ. θ+15^(∘) = 180^(∘) ⇔ θ =165^(∘) When we know the measure of the interior angle we can equate 180^(∘)(n-2) with the sum of the regular polygon's interior angles, 165^(∘) n, and solve for n.

180^(∘)(n-2)=165^(∘) n

▼

Solve for n

Distr

Distribute 180^(∘)

180^(∘) n- 360^(∘) = 165^(∘) n

AddEqn

LHS+360^(∘)=RHS+360^(∘)

180^(∘) n= 165^(∘) n+360^(∘)

SubEqn

LHS-165^(∘) n=RHS-165^(∘) n

15^(∘) n=360^(∘)

DivEqn

.LHS /15^(∘).=.RHS /15^(∘).

n=24^(∘)

When we know the number of sides of the regular polygon, we can find its central angle by substituting n= 24^(∘) in the formula we stated before. Central angle=360^(∘)/24^(∘) ⇔ Central angle=15^(∘) The central angle is 15^(∘).

180^(∘)(n-2)/n

Substitute

n= 10

180^(∘)( 10-2)/10

▼

Simplify

SubTerm

Subtract term

180^(∘)(8)/10

SimpQuot

Simplify quotient

18^(∘)(8)

Multiply

Multiply

144^(∘)

Let's call the measure of the exterior angle θ. The measure of an interior angle plus its corresponding exterior angle is 180^(∘). With this information we can write and solve the following equation containing θ. θ+144^(∘) = 180^(∘) ⇔ θ =36^(∘) The exterior angle is 36^(∘).