### All HiSET: Math Resources

## Example Questions

### Example Question #1 : Angle Measure, Central Angles, And Inscribed Angles

A quadrilateral is shown, and the angle measures of 3 interior angles are given. Find *x*, the missing angle measure.

**Possible Answers:**

**Correct answer:**

The sum of the measures of the interior angles of a quadrilateral is 360 degrees. The sum of the measures of the interior angles of any polygon can be determined using the following formula:

, where is the number of sides.

For example, with a quadrilateral, which has 4 sides, you obtain the following calculation:

Solving for requires setting up an algebraic equation, adding all 4 angles to equal 360 degrees:

Solving for is straightforward: subtract the values of the 3 known angles from both sides:

### Example Question #2 : Angle Measure, Central Angles, And Inscribed Angles

is the center of the above circle. Calculate .

**Possible Answers:**

**Correct answer:**

is the central angle that intercepts , so

.

Therefore, we need to find to obtain our answer.

If the sides of an angle with vertex outside the circle are both tangent to the circle, the angle formed is half the difference of the measures of the arcs. Therefore,

Letting , since the total arc measure of a circle is 360 degrees,

We are also given that

Making substitutions, and solving for :

Multiply both sides by 2:

Subtract 360 from both sides:

Divide both sides by :

,

the measure of and, consequently, that of .

### Example Question #3 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular seven-sided polygon, or *heptagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Consider the figure below, which adds some radii of the heptagon (and circle):

, as a radius of a regular polygon, bisects . The measure of this angle can be calculated using the formula

,

where :

Consequently,

,

the correct response.

### Example Question #4 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular seven-sided polygon, or *heptagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Examine the diagram below, which divides into three congruent angles, one of which is :

The measure of a central angle of a regular -sided polygon which intercepts one side of the polygon is ; setting , the measure of is

. has measure three times this; that is,

### Example Question #5 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular seven-sided polygon, or *heptagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Examine the diagram below, which divides into two congruent angles, one of which is :

The measure of a central angle of a regular -sided polygon which intercepts one side of the polygon is ; setting , the measure of is

. has measure twice this; that is,

### Example Question #6 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular ten-sided polygon, or *decagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Examine the diagram below, which divides into two congruent angles, one of which is :

The measure of a central angle of a regular -sided polygon which intercepts one side of the polygon is ; setting , the measure of is

. has measure twice this; that is,

.

### Example Question #1 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular ten-sided polygon, or *decagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Consider the triangle . Since and are radii, they are congruent, and by the Isosceles Triangle Theorem, .

Now, examine the figure below, which divides into three congruent angles, one of which is :

. has measure three times this; that is,

.

The measures of the interior angles of a triangle total , so

Substituting 108 for and for :

### Example Question #2 : Angle Measure, Central Angles, And Inscribed Angles

The above figure shows a regular ten-sided polygon, or *decagon*, inscribed inside a circle. is the common center of the figures.

Give the measure of .

**Possible Answers:**

**Correct answer:**

Through symmetry, it can be seen that Quadrilateral is a trapezoid, such that . By the Same-Side Interior Angle Theorem, and are supplementary - that is,

.

The measure of can be calculated using the formula

,

where :

Substituting:

### Example Question #9 : Angle Measure, Central Angles, And Inscribed Angles

If two angles are supplementary and one angle measures , what is the measurement of the second angle?

**Possible Answers:**

**Correct answer:**

Step 1: Define supplementary angles. Supplementary angles are two angles whose sum is .

Step 2: Find the other angle by subtracting the given angle from the maximum sum of the two angles.

So,

The missing angle (or second angle) is

### Example Question #10 : Angle Measure, Central Angles, And Inscribed Angles

and are complementary angles.

and are supplementary angles.

Evaluate .

**Possible Answers:**

**Correct answer:**

and are supplementary angles, so, by definition,

, so substitute and solve for :

and are complementary angles, so, by definition,

Substitute and solve for :

- that is, the angles have the same measure. Therefore,

.