Exercise 71 Page 756

Practice makes perfect

To calculate the measure of an interior angle of a regular n-gon, we can use the formula 180^(∘)(n-2)n, where n is the number of sides in the polygon. A pentagon is a polygon with 5 sides.

180^(∘)(n-2)/n

Substitute

n= 5

180^(∘)( 5-2)/5

▼

Evaluate

SubTerm

Subtract term

180^(∘)(3)/5

Multiply

540^(∘)/5

CalcQuot

Calculate quotient

108^(∘)

The interior angle of a regular pentagon is 108^(∘).

Next, we will draw an equilateral triangle with a central angle of a.

The sum of the central angles in any regular polygon equals 360^(∘). Since an equilateral triangle has 3 congruent central angles, we can determine the measure of a by dividing 360^(∘) by 3. 360^(∘)/3= 120^(∘) As we can see, a=120^(∘) and b=108^(∘). Therefore a is greater than b.

Notice that a and b are corresponding angles. If the two lines cut by the transversal were parallel, we could claim that a=b by the Corresponding Angles Theorem.

However, we do not have this information. Therefore, we have no way of saying which angle is greater or if they are the same.

With the given information, we can calculate ∠ a with the sine ratio and ∠ b with the cosine ratio.

Let's solve these equations for a and b.

sin a=8/12

▼

Solve for a

ReduceFrac

a/b=.a /4./.b /4.

sin a=2/3

sin^(-1)(LHS) = sin^(-1)(RHS)

a=sin^(-1)2/3

UseCalc

Use a calculator

a=41.8103148

RoundDec

Round to 2 decimal place(s)

a≈ 41.81^(∘)

Let's also solve for b in the second equation.

cos b=8/12

▼

Solve for b

ReduceFrac

a/b=.a /4./.b /4.

cos b=2/3

cos^(-1)(LHS) = cos^(-1)(RHS)

b=cos^(-1)2/3

UseCalc

Use a calculator

b=48.18968...

RoundDec

Round to 2 decimal place(s)

b≈ 48.19^(∘)

From our calculations we see that b is greater than a.

The equation tells us that to obtain a we have to add 3 to b. This must mean that a is greater than b.

a = b + 3 a is 3 greater than b

Like in Part A, we can use trigonometric ratios to write equations containing a and b. For the triangle on the left we need to use the tangent ratio. In the triangle on the right, we need to use the cosine ratio.