Parallel and perpendicular lines

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.