GED Math : Angles

Example Questions

Example Question #1 : Angles

What is the measure of each angle of a regular octagon?

Explanation:

The sum of the degree measures of the angles of a polygon with  sides is . Since an octagon has eight sides, substitute  to get:

Each angle of a regular polygon has equal measure, so divide this by 8 to get the measure of one angle:

, the degree measure of one angle.

Example Question #1 : Angles

Give the measure of each interior angle of a regular 72-sided polygon.

Explanation:

A regular polygon with  sides has interior angles of measure  each. Substitute 72 for .

Example Question #3 : Angles

Refer to the above diagram.

Which of these is a valid alternative name for  ?

Explanation:

When naming an angle after three points, the middle letter must be its vertex, or the point at which its sides meet - this is . The other two letters must refer to points on its two sides. Therefore,  includes  on one side, making one of its sides , and  on the other, making the other side .

An alternative name for this angle must be one of two things:

It can be named only after its vertex - that is,  - but only if there is no ambiguity as to which angle is being named. Since more than one angle in  the diagram has vertex  is  not a correct choice.

It can be named after three points. Again, the middle letter must be vertex , so we can throw out  and .

The only possible choice is .

Example Question #4 : Angles

Note: Figure NOT drawn to scale.

The above hexagon is regular. What is ?

Explanation:

Two of the angles of the quadrilateral formed are angles of a regular hexagon, so each measures

.

The four angles of the quadrilateral are . Their sum is , so we can set up an equation and solve for :

Example Question #5 : Angles

The above octagon is regular. What is ?

Explanation:

Three of the angles of the pentagon formed are angles of a regular octagon, so each measures

.

The five angles of the pentagon are . Their sum is , so we can set up an equation and solve for :

Example Question #6 : Angles

Note: Figure NOT drawn to scale.

Refer to the above figure.   is equilateral and Pentagon  is regular.

Evaluate .

Explanation:

First, we find .

.

is an angle of a regular pentagon and has measure .

, as an angle of an equilateral triangle, has measure

is equilateral, so ; Pentagon  is regular, so . Therefore, , and by the Isosceles Triangle Theorem, .

The degree measures of three angles of a triangle total , so:

Example Question #7 : Angles

Refer to the above figure, which shows Square   and regular Pentagon .

Evaluate .

Explanation:

.

is an angle of a regular pentagon and has measure .

is one of two acute angles of isosceles right triangle , so .

Example Question #8 : Angles

Refer to the above figure.   is equilateral, and Quadrilateral  is a square.

Evaluate

Explanation:

.

, as an angle of an equilateral triangle, has measure

, as an angle of a square, has measure .

Therefore,

.

Example Question #9 : Angles

Give the number of sides of a regular polygon whose interior angles have measure .

Explanation:

The easiest way to solve this is to look at the exterior angles, each of which have measure . Since each exterior angle of a regular polygon with  sides is , we solve for  in the following equation:

The polygon has 36 sides.

Example Question #10 : Angles

Three consecutive even angles add up to .  What must be the value of the second largest angle?

Explanation:

Let  be an even angle.  The next consecutive even values are .

Set up an equation such that all angles added equal to 180.

Divide by three on both sides.

The second largest angle is .

Substitute the value of  in to the expression.