Parallel and perpendicular lines

In this section, we’ll cover the three main types of lines to know for the GRE:

• parallel lines
• perpendicular lines
• transversals

Parallel lines

Parallel lines have the same slope, and they never intersect.

Ways to know if lines are parallel

• Explicitly denoted using a math symbol: $A\parallel B$
• If the lines have the same slope, e.g.:
  • $y=4x-3$
  • $y=4x+7$
• If the lines are opposite sides of a parallelogram

Perpendicular lines

Perpendicular lines intersect at 90 degrees.

Ways to know if lines are perpendicular

• Explicitly denoted using a math symbol: $C\perp D$
• If the lines have reciprocal and opposite slopes, e.g.:
  • $y=2/3x+4$
  • $y=-3/2x-1$
• If the lines are adjacent sides of a rectangle or square
• If the angle between the lines is given to be $90^{\circ }$, sometimes shown as a little square box like in the figure above

Transversals

It’s likely that you’ll need to apply the concepts of parallel and perpendicular lines in relation to transversals. A transversal is a line that cuts through two or more other lines.

What happens when a transversal intersects a line?

When a transversal cuts through a line, it creates equal opposite angles.

In the figure above, $x$ is $75^{\circ }$ and $y$ is $105^{\circ }$.

What happens when a transversal intersects parallel lines?

If we add in a parallel line ($AB\parallel EF$), the transversal will create identical angles where it crosses the other lines. Note that the $AB\cap CD$ angles in the figure below are the same as the $EF\cap CD$ angles, just like they were copied and shifted along $CD$.

What happens when a transversal intersects perpendicular lines?

When a transversal cuts through perpendicular lines, it forms a right triangle.

Using the knowledge that the sum of the interior angles of a triangle is 180, and seeing that two of the angles are explicitly denoted in the figure above, we could find the value of the third angle $y$ to be $180-90-30=60$.

Example transversal question

Let’s try solving a GRE-level question involving perpendicular and parallel lines.

Given: $AB\parallel CD$ and $EF\perp CD$

Quantity A: $x$
Quantity B: $55$

Give it a try, and then we’ll walk through the explanation!

Let’s start by filling in the basic information: AB is parallel to CD, and EF is perpendicular to CD. This means that EF is creating right angles where it intersects AB and CD.

Additionally, we’re given that one angle on AB is $125^{\circ }$. The opposite angle will also be $125^{\circ }$, and since the sum of two angles forming a straight line is $180^{\circ }$, the adjacent angles must be $180^{\circ }-125^{\circ }=55^{\circ }$.

Since the diagonal line is a transversal crossing two parallel lines, we can copy over the measurements and shift them down the transversal.

This now gives us enough information to find the angle opposite $x$. It’s in the small triangle, so the interior angles will sum to $180^{\circ }$, making this angle $180^{\circ }-90^{\circ }-55^{\circ }=35^{\circ }$.

Now we just put it together: Quantity B (55) is greater than Quantity A (35).