# GRE Math : Plane Geometry

## Example Questions

### Example Question #1 : Pentagons

In a five-sided polygon, one angle measures . What are the possible measurements of the other angles?

Explanation:

To find the sum of the interior angles of any polygon, use the formula , where n represents the number of sides of a polygon.

In this case:

The sum of the interior angles will be 540. Go through each answer choice and see which one adds up to 540 (including the original angle given in the problem).

The only one that does is 120, 115, 95, 105 and the original angle of 105.

### Example Question #1 : How To Find An Angle In A Pentagon

In a particular heptagon (a seven-sided polygon) the sum of four equal interior angles, each equal to  degrees, is equivalent to the sum of the remaining three interior angles.

Quantity A:

Quantity B:

Quantity B is greater

Quantity A is greater.

The relationship cannot be determined.

The two quantities are equal.

Quantity A is greater.

Explanation:

The sum of interior angles in a heptagon is  degrees. Note that to find the sum of interior angles of any polygon, it is given by the formula:

degrees, where  is the number of sides of the polygon.

Three interior angles (call them )  are unknown, but we are told that the sum of them is equal to the sum of four other equivalent angles (which we'll designate ):

Further more, all of these angles must sum up to  degrees:

We may not be able to find , or , indvidually, but the problem does not call for that, and we need only use their relation to , as stated in the first equation with them. Utilizing this in the second, we find:

### Example Question #3 : How To Find An Angle In A Pentagon

What is the value of  in the figure above?

Explanation:

Always begin working through problems like this by filling in all available information. We know that we can fill in two of the angles, giving us the following figure:

Now, we know that for any polygon, the total number of degrees in the figure can be calculated by the equation:

, where  is the number of sides.

Thus, for our figure, we have:

Based on this, we know:

Simplifying, we get:

Solving for , we get:

or

### Example Question #51 : Geometry

Quantity A: The measure of the largest angle in the figure above.

Quantity B:

Which of the following is true?

The two quantities are equal.

Quantity B is larger.

Quantity A is larger.

The relationship cannot be determined.

Quantity A is larger.

Explanation:

To begin, recall that the total degrees in any figure can be calculated by:

, where  represents the total number of sides. Thus, we know for our figure that:

Now, based on our figure, we can make the equation:

Simplifying, we get:

or

This means that  is . Quantity A is larger.

### Example Question #1 : Triangles

What is the perimeter of an isosceles triangle given that the sides 5 units long and half of the base measures to 4 units?

12
32
18
14
20
Explanation:

The base of the triangle is 4 + 4 = 8 so the total perimeter is 5 + 5 + 8 = 18.

### Example Question #1 : Triangles

An acute Isosceles triangle has two sides with length  and one side length . The length of side   ft. If the length of  half the length of side , what is the perimeter of the triangle?

foot

foot

inches

foot

inches

inches

Explanation:

This Isosceles triangle has two sides with a length of  foot and one side length that is half of the length of the two equivalent sides.

To find the missing side, double the value of side 's denominator:

. Thus, half of .

Therefore, this triangle has two sides with lengths of  and one side length of

To find the perimeter, apply the formula:

foot  inches

### Example Question #53 : Geometry

An acute Isosceles triangle has two sides with length  and one side length . The length of side  . If the length of  half the length of side , what is the perimeter of the triangle?

Explanation:

To solve this problem apply the formula: .

However, first calculate the length of the missing side by: .

Thus, the solution is:

### Example Question #54 : Geometry

Find the perimeter of the acute Isosceles triangle shown above.

Explanation:

To solve this problem apply the formula: .

However, first calculate the length of the missing side by:

### Example Question #55 : Geometry

An obtuse Isosceles triangle has two sides with length  and one side length . The length of side   ft. If the length of  half the length of side , what is the perimeter of the triangle?

ft

ft

ft

ft

ft

Explanation:

By definition, an Isosceles triangle must have two equivalent side lengths. Since we are told that  ft and that the sides with length  are half the length of side , find the length of  by:  and half of . Thus, both of sides with length  must equal  ft.

Now, apply the formula: .

Then, simplify the fraction/convert to mixed number fraction:

### Example Question #56 : Geometry

Find the perimeter of the acute Isosceles triangle shown above.

Explanation:

In order to solve this problem, first find the length of the missing sides. Then apply the formula:

Each of the missing sides equal:

Then apply the perimeter formula:

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