A Guide to the GRE/Lines and Angles

Lines & Angles

Opposite angles are equal. Adjacent angles on a line add up to 180º.

Angles n and p are equal.
Angles q and o are equal.
Angles n and o add up to 180º.
Angles o and p add up to 180º.
Angles p and q add up to 180º.
Angles q and n add up to 180º.

A line intersecting two parallel lines forms the same angles with each.

Parallel lines are those which are a constant distance apart. A line intersecting them is known as a “transversal.”

Assume both sets of lines are parallel; therefore:

Angles f and g are equal.
Angles h and i add up to 180º.
Angles f and h are equal.
Angles g and i add up to 180º.

Practice

1. In this figure, both pairs of lines are parallel, and w is 10º greater than u. What are the measures of angles t, u, v and w?

Answers to Practice Questions

1. t, u, v, and w have measures of 95º, 95º, 85º, and 95º, respectively.

Because lines make the same angles through any parallel lines they intersect, angle w is identical to angle v as well as the angle directly opposite t. Because opposite angles are equal, angles t and w are equal.

All three of these angles are equal to either of the angles adjacent to angle u along the line (either the line running from t to u or from v to u); thus u + v = 180º. Since w = u + 10º, we can solve for u.

u + v = 180º Take the initial equation.

u + (u + 10º) = 180º Substitute for v in terms of u.

2u + 10º = 180º Add the variables.

2u = 170º Subtract 10º from both sides.

u = 85º Divide both sides by 2. u is 85º, making the other three angles 95º.