EXAMPLE 1 Find angle measures in a regular polygon

a. m AFB

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

AFB is a central angle, so m AFB = , or 72°.

360°

5

EXAMPLE 1 Find angle measures in a regular polygon

b. m AFG

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

FG is an apothem, which makes it an altitude of isosceles ∆AFB. So, FG bisects AFB and m AFG = m AFB = 36°.1

2

EXAMPLE 1 Find angle measures in a regular polygon

c. m GAF

In the diagram, ABCDE is a regular pentagon inscribed in F. Find each angle measure.

SOLUTION

The sum of the measures of right ∆GAF is 180°.So, 90° + 36° + m GAF = 180°, and m GAF = 54°.

GUIDED PRACTICE for Example 1

In the diagram, WXYZ is a square inscribed in P.

1. Identify the center, a radius, an apothem, and a central

angle of the polygon.

GUIDED PRACTICE for Example 1

SOLUTION

center – P

radius – PY or XP

an apothem – PQ

central angle – XPY

GUIDED PRACTICE for Example 1

2. Find m XPY, m XPQ, and m PXQ.

m XPY is a central angle so m XPY = 4

360

= XPQ = 45°

SOLUTION

m XPY = 90°

QP is an apothem, which make it an altitude of isosceles ∆ XPQ so QP bisects XPY and m XPQ = m 1

2

GUIDED PRACTICE for Example 1

90° + 45 + m PXQ = 180 and PXQ = 45°

The sum of measures of right ∆ PXQ is 180° so