GRE Geometry - Lines and Angles

When two lines intersect, then the angles that are formed are special. A geometry question on the GRE exam that is meant to measure your knowledge of lines and angles can assume one of many forms. For starters, you should be familiar with the terms transversal, supplementary, and complimentary angles.

If two lines intersect, then the angles that are opposite of each other are equal, and the angles that are next to each other are supplemental, meaning that they add up to 180 degrees. Also, in the case that two lines intersect at 90 degrees, then all four angles are equal and are 90 degrees, and the two lines are said to be perpendicular to each other. Below, when lines L1 and L2 intersect, angle A is equal to angle C and angle B is equal to angle D. Also, angle A+B=180 degrees because they form a straight line, and likewise, B+C=180, C+D=180, and D+A=180. For Lines L3 and L4, the two lines intersect at 90 degrees, so all four angles formed by their intersection are 90 degrees and are hence equal:

GRE supplemental and complimentary angles

If a line intersects two other lines, then it is called a transversal, and there are a total of 8 angles formed. Four of the angles are called interior angles, and four of the angles are called exterior angles. These angles have special relationships. It's best to explain this visually. In the 3 lines below, line t is the transversal of lines p and r, and the interior angles are 3, 4, 5 and 6, while the exterior angles are 1, 2, 7 and 8. And IF lines p and r are parallel, then there are several angle relationships, namely:

  • angle 2 is equal to angle 3
  • angle 1 is equal to angle 4
  • angle 5 is equal to angle 8
  • angle 6 is equal to angle 7
  • angle 2 is equal to angle 6
  • angle 4 is equal to angle 8
  • angle 3 is equal to angle 7
  • angle 1 is equal to angle 5
  GRE geometry complimentary angles

In the above three lines, since angles 6 and 8 form a straight line, then they add to 180 degrees. Using substitution, since angle 4 is equal to angle 8, then angles 6 and 4 also add to 180 degrees, and hence are complimentary. Likewise, similar substitution relationships can be derived from the graph.