GRE Exam: A Simple Guide To Solving Geometry Problems - Lines and Angles

Geometry

About 20 percent of the questions on the GRE will deal with geometry. These questions will address your knowledge of geometric properties and formulas and your ability to use these formulas to construct algebraic relationships.

When answering geometry questions, you should generally follow these steps:
1. Draw the diagram and input all given information.
2. Infer all properties and relationships implied by the diagram.
3. Use these relationships to create algebraic relationships.
4. Solve for what the question is asking for.

Notice that the approach for geometry questions is in many ways similar to the approach for word problems. In both cases, your ultimate goal is to use the information given to construct algebraic relationships. The distinguishing element of geometry questions is that you will use geometric principles instead of words to create these algebraic relationships. This guide focuses on these principles and how they are tested on the exam.

Note that if a figure says “Not Drawn to Scale,” then the relationships between the sides and angles in the diagram do not match their visual appearance. However, the opposite is also true: if the figure does not say “Not Drawn to Scale,” then you can assume that the diagram is accurate. Nonetheless, you must be careful about making assumptions.

Lines and Angles
An angle is formed at the intersection of two rays or lines. The measurement of an angle can be between 0 degrees and 360 degrees. The angle below has a measure of 45 degrees.

One type of angle that occurs throughout geometry questions is a right angle. A right angle has a measure of 90 degrees and is denoted by the square at the intersection of the two lines or rays.

When two lines intersect to form a right angle, those lines are said to be perpendicular.

A straight line will always have an angle measure of 180 degrees. From this, you can infer Property 1: the angles on a line must add up to 180. Angles whose measures sum to 180 are termed supplementary angles.

Example: What is the value of (a + b) in the figure above?

SOLUTION: Since these four angles lie on line CD:


Intersecting Lines and Vertical Angles
The intersection of two lines will always create four angles.

These angles have two properties:
1. They will add up to 360.
2. The angles opposite each other will be equal. These angles are termed vertical angles.

Example: What is the value of x in the figure above?
SOLUTION: Since x and 2y lie on the same line, x + 2y = 180. The angle vertical to 2y = 80, so you can substitute 80 for 2y in the equation and solve: x + 80 = 180 → x = 100.

Note that the two preceding properties can apply to situations in which more than two lines intersect.

Here is the more general form of Property 1: the angles around a point must add up to 360.

Since these angles are all around a point, a + b + c + d + e + f = 360. In addition, a = f, b = e, c = d. Finally, a, b, and c are supplementary, and d, e, and f are supplementary.

Parallel Lines and Transversals
Parallel lines by definition will never intersect. On the GRE, you will be expected to understand the relationships among the angles of parallel lines cut by a transversal.

The best way to think about the relationship among the angles is in terms of the small angles and the large angles. All of the smaller angles will be acute (less than 90 degrees), and all of the larger angles will be obtuse (greater than 90 degrees). All the smaller angles will have the same measure, and all of the larger angles will have the same measure.

Thus in the preceding diagram, x < 90, and y > 90. Finally, the sum of any small angle and any large angle will always be 180.

Thus x + y = 180. Understanding and implementing these properties will equip you well on most questions that test parallel lines and transversals.

Example: In the diagram above, lines AB and CD are parallel. If the ratio of l to m is 3 to 2, what is l?
SOLUTION: When two parallel lines are cut by a transversal, the sum of the small and large angles is 180.

Thus l + m = 180. Since , you can let l = 3x and let m = 2x.

Substitute these terms into the first equation: