Exercise 88 Page 635

a The sum of the exterior angles of any polygon, one at each vertex, is 360^(∘). Our 6 -gon is regular, so the interior angles are all congruent. This means that their supplements, the exterior angles, are also congruent. Therefore, the measure of one exterior angle is 360^(∘) divided by 6.

360^(∘)/6=60^(∘) Therefore, x = 60^(∘).

b Recall that the measure of each interior angle of a regular n -gon is ( n-2)180^(∘) n. To find the measure of an interior angle of a 8 -gon, we have to substitute 8 for n in this expression.

( 8-2)180^(∘)/8= 135^(∘) Therefore, x = 135^(∘).

c Let's consider the given diagram.

The diagram shows a regular pentagon with a right triangle. This right triangle has the pentagon's apothem as one of its legs and the pentagon's radius as the hypotenuse. First, let's reflect this right triangle though the apothem.

A rotation by 360^(∘) ÷ 5 = 72^(∘) around the pentagon's center does not change the shape of the pentagon. For this reason, we can create similar two triangles for each of the other sides of the pentagon.

We have 10 angles measuring x degrees around the center of the pentagon. Since the sum of angles around a point equals 360^(∘), we can find the value of x by dividing 360^(∘) by 10. x= 360^(∘)/10 = 36^(∘)

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Exercise 88 Page 635

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