HiSET: Math : Angle measure, central angles, and inscribed angles

Example Questions

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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.

Explanation:

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

is the center of the above circle. Calculate .

Explanation:

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 .

Explanation:

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 #2 : 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 .

Explanation:

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 #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 .

Explanation:

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 #4 : 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 .

Explanation:

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 #5 : 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 .

Explanation:

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 :

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,

.

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

Substituting 108 for  and  for :

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 .

Explanation:

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 #7 : Angle Measure, Central Angles, And Inscribed Angles

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

Explanation:

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 .

Explanation:

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,

.

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