How to Solve Geometry Problems Involving Parallel Lines and a Transversal

A very common task in geometry is to solve problems involving parallel lines and a transversal as shown. These problems are extremely easy to solve as long as you remember a few basic math facts.

This article shows the steps for solving the various types of questions that relate to this topic. Note that the article assumes that the reader has a basic knowledge of geometry concepts. It also presents the topic without proof, as is typically the case in middle and high school math classes.

Instructions

1. 

   First let's define parallel lines informally. Two lines are parallel if they will never meet no matter how far each is extended. They are both slanted at the same angle, but they need not be horizontal.

2. 

   We define a transversal as a line that "cuts through" a pair of parallel lines at an angle, as shown. Eight angles are formed. Typically, the transversal will be slanted such that four of these angles are less than 90° (think of them as "small"), and four of the angles are greater than 90° (think of them as "big"). Each small angle equals all of the other small angles in measure, and each big angle equals each of the other big angles. Make sure you see that if the two original lines were not parallel, this equality among the small and big angles would not be the case.

3. 

   There is one more math fact to review. We define a straight line (straight angle) as having 180°, which must be memorized. If we draw a line to meet that straight line, two angles will be formed. We say that such a pair of angles are supplementary--they add up to 180°. If one of the angles is x, the other one must be 180-x. Make sure that you understand that. In terms of the problems that we'll be looking at, just remember that "small" + "big" = 180. The "small" and "big" angles don't even have to be adjacent to each other.

4. 

   For this topic, you will usually be given problems in which the measure of only one of the eight angles are known. Just from knowing that one angle, you can easily determine the measure of any of the other angles that you may be asked about. You will also likely be asked to identify the name given to various pairs of angles.

5. 

   In this diagram, angles A and D are called vertical or opposite angles. They are equal in measure by the fact that "big equals big." Similarly, F and G are equal because "small equals small." If one of the small angles is given as 37°, then all of the other small angles, including the vertical or opposite one, are also 37°. If one of the big angles is given as 129°, then all of the other big angles are also 129°. For practice, try finding all of the pairs of vertical angles in the diagram. Also make sure that you fully understand that angles C and H couldn't possibly be equal in this diagram, since one is "small" and the other is "big."

6. 

   Important: You could be given a trick question which asks you under what circumstances angles C and H would be equal. Make sure you understand that the transversal would have to be perpendicular to the parallel lines. Under that circumstance, all eight angles formed would be 90°, and all would be equal to each other. For graduate exams such as the GRE or GMAT, you must be alert to the fact that the transversal actually could be like this, and the diagram is purposely not drawn to scale, but in other cases this should not be a concern.

7. 

   Look at angles A and B. They lie along a straight line, and so we know that they are supplementary. That means that they add up to 180°. If one is given as 30°, the other must be 180-30 or 150°. Notice how angles F and H are also supplementary. They don't lie along a horizontal line, but they still lie along a straight line, which means they add up to 180°. These pairs of supplementary angles are said to be adjacent, because they are next to each other. Try finding all of the other pairs of supplementary adjacent angles.

8. 

   Look at angles C and F. They are both small angles, and so we know they are equal. They are said to be alternate interior angles. They are on opposite sides of the transversal, and inside the two parallel lines. Angles D and E are also equal, since they are both big. They are also alternate interior angles. Those are the only pairs of alternate interior angles.

9. 

   Look at angles B and G. They are both small angles, and so we know they are equal. They are said to be alternate exterior angles. They are on opposite sides of the transversal, and outside the two parallel lines. Angles A and H are also equal, since they are both big. They are also alternate exterior angles. Those are the only pairs of alternate exterior angles.

10. 

    Look at angles A and E. They are both big angles, so we know they are equal, even though they aren't near each other. They are said to be corresponding angles because they are in corresponding positions. Both are above the parallel lines, and both are to the left of the transversal. Angles B and H are similarly corresponding to one another, and therefore are equal . Try to find the other pairs of corresponding angles, including the small angles that are corresponding.

11. 

    Look at angles A and G. One is small and one is big. That means that they are supplementary, since we know that small plus big equals 180°. However, they are not adjacent supplementary. Angles D and F are also non-adjacent supplementary. Don't get confused and say that they are alternate interior like the angles we looked at in Step 8.

12. 

    That's all you need to know to solve any typical problem involving parallel lines and a transversal. Make sure you remember all of the definitions in this article, and that the sum of supplementary angles is 180°. Keep studying! ☺

Resources

• Math with Larry (Free Online Math Help)