**Angle Measures in Polygons Worksheet :**

Worksheet given in this section will be much useful for the students who would like to practice problems on angles measures in polygons.

Before look at the worksheet, if you would like to know the stuff related to angle measures in polygons,

**Problem 1 :**

Find the value of x in the diagram shown below.

**Problem 2 :**

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

**Problem 3 :**

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

**Problem 4 :**

Find the value of x in the diagram shown below.

**Problem 5 :**

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 ?

**Problem 6 :**

If we 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° ?

**Problem 1 :**

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°.

**Problem 2 :**

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.

**Problem 3 :**

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°.

**Problem 4 :**

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

**Problem 5 :**

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°.

**Problem 6 :**

If we 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 how to solve problems on angle measures in polygons.

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