Geometry questions on the GRE are a logical puzzle. You will be given a tiny piece of information about a figure, and must deduce some piece of missing information. Geometry questions require you to follow a series of “if-then” statements, constantly solving the puzzle, piece by piece, in order to find the correct answer.

## Lines

The most basic element in geometry is the * point*.

Through any two points, there is exactly one * line*,

extending infinitely in both directions.

Lines can be named by two points or by a lower case letter.

A section of a line is called a * line segment* or a

*.*

**ray**## Angles

When two rays originate from the same point, they form an **angle**, which is represented by the symbol .

The point of intersection is the **vertex** and the two rays form the **sides** of the angle.

An angle can be named in a number of ways.

It can be called ∠BAC where the middle letter

is the vertex, ∠A where A is the vertex, or by

a number inside the angle.

Angles are measured in degrees, denoted bythe degree symbol °. For example, a 30 degree angle is written as 30°. A variable inside the angle, x°, stands for the measure of the angle.

Angles that have the same measure are congruent.

Congruent angles can be indicated with matching arcs. Congruent polygons can be indicated with the ≅ symbol.

## Types of Angles

Angles can be categorized by their angle measure.

acute

0° < x° < 90°

right

90°

obtuse

90° < x° < 180°

straight

180°

The box in the corner means that it is a right angle.

A straight line is a 180° angle.

Two angles are **complementary angles** or complements if the sum of their measures is 90°.

Two angles are **supplementary angles** or supplements if the sum of their measures is 180°.

The sum x + y = 90. **These two angles are complementary angles.**

The sum x + z = 180. Notice that x and z create a straight line. These two angles are supplementary angles.

## Example

In the figure, a beam of light is shown reflecting off a mirror. What is the value of x?

### Solution

Because these three angles form a straight line, they must add up to 180°.

x + 3x + x = 180

5x = 180

x = 36

## Example

The complement of an angle is one quarter of the supplement of the angle. What is the measure of the angle?

### Section

Let x be the measure of the angle.

Its complement c is such that x + c = 90, so x = 90 – c.

Its supplement s is such that x + s = 180, so x = 180 – s.

The complement is one quarter of the supplement, so c = s/4.

90 – c = 180 – s

Subtract 90 and add s to both sides.

s – c = 90

Substitute c = s/4.

s – s/4 = 90

3s/4 = 90

Multiply both sides by 4/3.

s = 120

**x + s = 180, so x = 180 – 120 = 60 so the measure of the angle is 60°.**

## Midpoint and Bisector

The **midpoint** is the center point of any line segment. In the figure below, point B is the midpoint of segment AC. The tick marks mean that the segments are the same length. They are *congruent*.

## Example

In segment EH, F is the midpoint of segment EH, and G is the midpoint of segment FH. If EH = 8, how long is segment EG?

### Solution

EH = 8, and the midpoint is F, so FH = 4.

FH = 4, and the midpoint is G, so GH = 2.

**EF + FG = EG, so EG = 4 + 2 = 6. **

Bisect means “to cut in half.” A bisector is a line that divides a line segment or an angle into two equal pieces.

## Example

Ray TR bisects ∠STQ. What is the measure of ∠RTP?

### Solution

∠STQ = ∠STR + ∠RTQ

∠STQ is a right angle, and since ∠STQ is bisected, ∠STR = ∠RTQ. So each angle is 45°.

The arcs show that ∠STR and ∠QTP are congruent.

So ∠STR = ∠QTP = ∠RTQ = 45°.

∠RTP = ∠RTQ + ∠QTP = 45° + 45° = 90°

So the measure of ∠RTP is 90° and is right angle.

Another method is to notice that ∠STR + ∠RTQ = ∠RTQ + ∠QTP.

Since ∠STR = ∠QTP, ∠STQ = ∠RTP.

∠STQ is a right angle, so ∠RTP is also a right angle.