In this section, you’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, for example:
  • $y=4x-3$
  • $y=4x+7$
• If the lines are opposite sides of a parallelogram
• The lines have similar arrows drawn on them, like in the diagram above

Perpendicular lines

Perpendicular lines intersect at $90^{\circ }$.

Ways to know if lines are perpendicular

• Explicitly denoted using a math symbol: $C\perp D$
• If the lines have reciprocal and opposite slopes, for example:
  • $y=2/3x+4$
  • $y=-3/2x-1$
• 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 small square box like in the figure above

Transversals

You’ll often apply parallel and perpendicular line facts using 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 you add in a parallel line ($AB\parallel EF$), the transversal creates identical angles where it crosses each parallel line. Notice that the $AB\cap CD$ angles in the figure below match the $EF\cap CD$ angles - they’re the same angles “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.

In the figure above, the triangle has one right angle ($90^{\circ }$) and another angle labeled $30^{\circ }$. Since the interior angles of a triangle sum to $180^{\circ }$, the third angle is

$180-90-30=60$

So $y=60^{\circ }$.

Example transversal question

Let’s solve 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.

Start with the given relationships:

• $AB$ is parallel to $CD$.
• $EF$ is perpendicular to $CD$.

Because $EF\perp CD$, line $EF$ forms right angles where it intersects $CD$. Since $AB\parallel CD$, line $EF$ is also perpendicular to $AB$, so it forms right angles where it intersects $AB$ as well.

Next, use the given $125^{\circ }$ angle on $AB$:

• The opposite angle is also $125^{\circ }$.
• Adjacent angles on a straight line sum to $180^{\circ }$, so each adjacent angle is $180^{\circ }-125^{\circ }=55^{\circ }$.

Now use the fact that the diagonal line is a transversal crossing two parallel lines. That means the corresponding angles at the two intersections are equal, so you can copy the $55^{\circ }$ angle down to the intersection with the other parallel line.

At this point, look at the small triangle containing $x$. Two of its angles are known:

• one right angle ($90^{\circ }$)
• one angle of $55^{\circ }$

So the third angle (the one opposite $x$) is

$180^{\circ }-90^{\circ }-55^{\circ }=35^{\circ }$.

So Quantity A is $35$, and Quantity B is $55$. Quantity B is greater.

Common themes

• Never assume two lines are parallel or perpendicular to each other based on appearance alone. The problem should either state this information directly, or there should be enough information to prove it.
• Find as many angles as possible before looking for the specific angle(s) to solve a problem. The solution is usually much clearer once all of the knowable information is filled in. This is especially relevant for questions that involve transversals.