Find the sum of the measures of the interior angles of a convex octagon.
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Find the sum of the measures of the interior angles of a convex octagon. ( n – 2) 180 ° =. ( 8 – 2) 180 °. = 6 180°. ANSWER. The sum of the measures of the interior angles of an octagon is 1080° . Find the sum of angle measures in a polygon. EXAMPLE 1. SOLUTION.

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Example 1

Find the sum of the measures of the interior angles of a convex octagon.

(n – 2) 180° =

(8 – 2) 180°

= 6 180°

ANSWER

The sum of the measures of the interior angles of an octagon is 1080°.

Find the sum of angle measures in a polygon

EXAMPLE 1

SOLUTION

An octagon has 8 sides. Use the Polygon Interior Angles Theorem.

Substitute 8 for n.

Subtract.

= 1080°

Multiply.


Example 1

(n–2) 180° =

900°

n–2 =

5

7

n =

ANSWER

The polygon has 7 sides. It is a heptagon.

Find the number of sides of a polygon

EXAMPLE 2

The sum of the measures of the interior angles of a convex polygon is 900°. Classify the polygon by the number of sides.

SOLUTION

Use the Polygon Interior Angles Theorem to write an equation involving the number of sidesn. Then solve the equation to find the number of sides.

Polygon Interior Angles Theorem

Divide each side by 180°.

Add 2 to each side.


Example 1

360°

x° + 2x° + 89° + 67° =

360

3x + 156 =

68

x =

The correct answer is B.

ANSWER

Find the variable based on the exterior angles

EXAMPLE 3

SOLUTION

Use the Polygon Exterior Angles Theorem to write and

solve an equation.

Polygon Exterior Angles Theorem

Combine like terms.

Solve for x.


Example 1

The trampoline shown is shaped like a regular dodecagon. Find (a) the measure of each interior angle and (b)the measure of each exterior angle.

a.

Use the Polygon Interior Angles Theorem to find the sum of the measures of the interior angles.

(n –2) 180° =

(12 – 2) 180° =

Find angle measures in regular polygons

EXAMPLE 4

TRAMPOLINE

SOLUTION

1800°

Since a dodecagon has 12 congruent interior angles:

1800°

Divide

= 150 °

12


Example 1

The trampoline shown is shaped like a regular dodecagon. Find (a) the measure of each interior angle and (b)the measure of each exterior angle.

b.

By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is 360°.

Find angle measures in regular polygons

EXAMPLE 4

TRAMPOLINE

SOLUTION

A dodecagon has 12 exterior angles

360°

Divide

= 30 °

12


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