**Angle Measures in Polygons :**

In this section, we are going to learn angle measures in polygons.

There are two types of aagle measures in polygons.

1. Interior Angles

2. Exterior Angles

**Interior Angle :**

An interior angle of a polygon is an angle inside the polygon at each of its vertices.

**Exterior Angle :**

An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side.

More clearly,

**Theorem : **

The sum of the measures of the interior angles of a convex n-gon is

(n - 2) ⋅ 180°

**Corollary to the above Theorem : **

The measure of each interior angle of a regular n-gon is

1/n ⋅ (n - 2) ⋅ 180°

or

[(n - 2) ⋅ 180°] / n

**Theorem : **

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is

360°

**Corollary to the above Theorem : **

The measure of each exterior angle of a regular n-gon is

1/n ⋅ 360°

or

360° / n

**Example :**

Find the value of x in the diagram shown below.

**Solution : **

The polygon shown in the diagram above has 6 sides. So it is hexagon.

The sum of the measures of the interior angles of any hexagon is

= (6 - 2) ⋅ 180°

= 4 ⋅ 180°

= 720°

We can add the measures of all interior angles of the above hexagon and the sum can be equated to 720°.

So, we have

136° + 136° + 88° + 142° + 105° + x° = 720°

Simplify.

607 + x = 720

Subtract 607 from each side.

x = 113

Hence, the measure of sixth interior angle of the hexagon is 113°.

**Example : **

The measure of each interior angle of a regular polygon is 140°. How many sides does the polygon have ?

**Solution : **

By Polygon Interior Angles Theorem, we have

[(n - 2) ⋅ 180°] / n = 140°

Multiply each side by n.

(n - 2) ⋅ 180 = 140n

Simplify.

180n - 360 = 140n

Subtract 140n from each side.

40n - 360 = 0

Add 360 to each side.

40n = 360

Divide each side by 40.

40n/40 = 360/40

n = 9

Hence, the polygon has 9 sides and it is a regular nonagon.

**Example 1 : **

Find the value of x in the regular polygon shown below.

**Solution : **

The polygon shown above is regular and it has 7 sides. So, it is a regular heptagon and the measure of each exterior angle is x°.

By the Polygon Exterior Angles Theorem, we have

x° = 1/7 ⋅ 360°

Simplify.

x ≈ 51.4

Hence, the measure of each exterior angle of a regular heptagon is about 51.4°.

**Example 2 : **

Find the value of x in the diagram shown below.

**Solution :**

The polygon shown in the diagram above has 5 sides. So it is pentagon.

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is

= 360°

We can add the measures of all exterior angles of the above pentagon and the sum can be equated to 360°.

So, we have

2x° + x° + 3x° + 4x° + 2x° = 360°

Simplify.

12x = 360

Divide each side by 12.

12x/12 = 360/12

x = 30

**Example :**

A home plate maker for a soft ball field is a pentagon. Three of the interior angles of the pentagon are right angles. The remaining two interior angles are congruent. What is the measure of each angle ?

**Solution : **

**Draw a Sketch : **

Sketch and label a diagram for the above plate maker. It is a non regular pentagon.

Let ∠A, ∠B and ∠D be the right angles.

Let ∠C and ∠E be the remaining two congruent angles.

So, we have

∠C ≅ ∠E

The sum of the measures of the interior angles of a pentagon is

= (5 - 2) ⋅ 180°

= 3 ⋅ 180°

= 540°

**Verbal Model :**

**Labels :**

Sum of measures of interior angles = 540°

Measure of each right angle = 90°

Measure of ∠C and ∠E = x°

**Reasoning : **

Write the equation.

540° = 3 ⋅ 90° + 2x°

Simplify.

540 = 270 + 2x

Subtract 270 from each side.

270 = 2x

Divide each side by 2.

270/2 = 2x/2

135 = x

Hence, the measure of each of the two congruent angles is 135°.

**Example : **

If you were designing the home plate marker for some new type of ball game, would it be possible to make a home plate marker that is a regular polygon with each interior angle having a measure of (a) 135° ? (b) 145° ?

**Solution : **

**Solution (a) :**

Let n be the number of sides of the regular polygon.

By Polygon Interior Angles Theorem, we have

[(n - 2) ⋅ 180°] / n = 135°

Multiply each side by n.

(n - 2) ⋅ 180 = 135n

180n - 360 = 135n

Subtract 135n from each side.

45n - 360 = 0

Add 360 to each side.

45n = 360

Divide each side by 45.

45n/45 = 360/45

n = 8

Yes, it would be possible. Because a polygon can have 8 sides.

**Solution (b) :**

Let n be the number of sides of the regular polygon.

By Polygon Interior Angles Theorem, we have

[(n - 2) ⋅ 180°] / n = 145°

Multiply each side by n.

(n - 2) ⋅ 180 = 145n

180n - 360 = 145n

Subtract 145n from each side.

35n - 360 = 0

Add 360 to each side.

35n = 360

Divide each side by 35.

35n/35 = 360/35

n ≈ 10.3

No, it would not be possible. Because, a polygon can not have 10.3 sides.

After having gone through the stuff given above, we hope that the students would have understood, "Angle Measures in Polygons".

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