What is the difference between complementary and supplementary angles?

Complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees. Angles that sit together along a straight line are always supplementary, and two angles forming a right angle are complementary.

Are vertical angles always equal?

Yes. When two straight lines cross, the two angles directly opposite each other (vertical angles) are always equal, regardless of the angle. The two pairs of adjacent angles are supplementary, summing to 180 degrees.

When are the angles on a transversal equal versus supplementary?

Only when the two lines are parallel. Corresponding angles and alternate interior angles are then equal, while co-interior (same-side interior) angles are supplementary. A quick check: two acute or two obtuse angles are equal; one acute and one obtuse are supplementary.

Can I assume lines are parallel or angles are right angles from a GRE figure?

No. GRE figures are not drawn to scale, so you may not assume lines are parallel, angles are right angles, or points are collinear unless the problem states it or you can prove it. Assuming parallelism is a common way to fall for a "cannot be determined" trap.

CramClub • Lines and Angles — GRE® General Test Guide & Practice

Supplementary sum to $180^\circ$ ; complementary sum to $90^\circ$

Vertical (opposite) angles are equal

Corresponding & alternate interior angles equal; co-interior sum to $180^\circ$

Why Angle Rules Win GRE Points

A large share of GRE geometry never asks you to compute an area or a length. Instead it hands you a tangle of lines and one or two given angles and expects you to chase the rest. These questions reward a small, fixed set of relationships that you must know cold — and, critically, the discipline to reason from the rules rather than from how the figure looks. GRE figures carry the note "not drawn to scale," so an angle that appears to be a right angle, or two lines that appear parallel, prove nothing unless the problem states it or you can derive it.

Angles at a Point and on a Line

Two angles are complementary when they sum to $90^\circ$ and supplementary when they sum to $180^\circ$ . Angles that lie along a single straight line are supplementary — the "straight angle" is $180^\circ$ . Angles that completely surround a point sum to a full turn, $360^\circ$ .

When two straight lines cross, they create four angles with two dependable properties. The two angles directly opposite each other — vertical angles — are always equal. Each pair of adjacent angles lies on a straight line, so it is supplementary. One consequence is powerful: a single given angle at an intersection determines all four. If one angle is $50^\circ$ , its vertical twin is $50^\circ$ and each neighbor is $180^\circ - 50^\circ = 130^\circ$ .

Relationship	Rule
Complementary angles	Sum to $90^\circ$
Supplementary angles	Sum to $180^\circ$
Angles on a straight line	Sum to $180^\circ$
Angles around a point	Sum to $360^\circ$
Vertical angles	Equal

Parallel Lines Cut by a Transversal

This single configuration is the richest source of GRE angle problems. When a transversal crosses two parallel lines, eight angles appear, but they take only two distinct values. The relationships worth naming:

Corresponding angles (matching positions at each intersection) are equal.
Alternate interior angles (opposite sides of the transversal, between the parallels) are equal.
Co-interior / same-side interior angles are supplementary — they sum to $180^\circ$ .

There is a fast mental shortcut that sidesteps the vocabulary entirely: on parallel lines, any two of the eight angles are either equal or supplementary, and you can tell which at a glance. If both angles look acute, or both look obtuse, they are equal; if one is acute and the other obtuse, they are supplementary. That one test resolves most transversal questions in seconds.

The trap the GRE plants here is worth memorizing: these equalities require the lines to be parallel. If a problem never states — and you cannot prove — that the two lines are parallel, then corresponding angles need not be equal, and the honest answer to a comparison may be "cannot be determined."

Angle-Chasing Strategy

Most multi-step angle problems dissolve into a chain of single moves. Label everything you know, then propagate:

Mark the given angle(s).
Fill in vertical angles (equal) and straight-line partners (supplement to $180^\circ$ ).
Across parallel lines, copy equal angles and fill supplements.
If a bent or "Z"-shaped transversal appears, draw an auxiliary line parallel to the two given parallels through the bend — the bend angle then splits into two alternate-interior pieces you already know.

For example, suppose parallel lines are crossed by a path that bends at point $P$ ; the angle above the bend (measured to the upper parallel) is $35^\circ$ and the angle below the bend (to the lower parallel) is $50^\circ$ . Drawing a line through $P$ parallel to both splits the bend into $35^\circ$ and $50^\circ$ by alternate interior angles, so the full bend angle is $35^\circ + 50^\circ = 85^\circ$ .

Turning Diagrams into Equations

The hardest GRE angle items give you algebraic expressions instead of numbers. The method never changes: name the relationship, write the equation it forces, solve, then evaluate.

(3x + 10)^\circ &= (5x - 30)^\circ \quad\text{(corresponding angles, so equal)} \\ 40 &= 2x \\ x &= 20 \end{aligned}$$ Then the angle is $3(20) + 10 = 70^\circ$. Notice the last step: the problem usually wants the **angle measure**, not the value of $x$. Stopping at $x = 20$ is the single most common way to lose an otherwise-solved problem. ## The Habits That Matter Read what is actually stated, not what is drawn. Never assume a right angle, parallel lines, or that a point lies on a line unless told. Fill the figure with every angle you can deduce before hunting for the target. And always re-read the question to confirm whether it wants an angle, a sum of angles, or a variable — the GRE writes distractors for each of those stopping points.

An angle measures $37^\circ$ . What are its complement and its supplement?

Two lines intersect. One of the four angles is $(2x + 15)^\circ$ and the angle vertically opposite it is $(3x - 25)^\circ$ . What is the common measure of these two angles?

Lines $\ell$ and $m$ are parallel and are crossed by a transversal. One interior angle on line $\ell$ measures $68^\circ$ . What is the measure of the co-interior (same-side interior) angle on line $m$ ?

A transversal crosses parallel lines. The angle above the upper line is $(3x + 10)^\circ$ and the corresponding angle above the lower line is $(5x - 30)^\circ$ . Find the measure of the angle.

Parallel lines $\ell$ and $m$ are crossed by a path that bends at point $P$ between them. The angle from the bend up to $\ell$ is $42^\circ$ and the angle from the bend down to $m$ is $63^\circ$ . What is the measure of the bend angle at $P$ ?

Three rays from a single point on a straight line divide the $180^\circ$ straight angle into three angles whose measures are in the ratio $2 : 3 : 4$ . What is the measure of the largest of the three angles?

Assuming two lines are parallel (or an angle is a right angle) because the figure looks that way.

GRE figures are "not drawn to scale." Corresponding/alternate-angle equalities hold only when the problem states the lines are parallel or you can prove it. Otherwise, a comparison may genuinely be "cannot be determined."

Solving for the variable and stopping — reporting $x = 20$ when the question asked for the angle measure.

Re-read the final sentence. After finding $x$ , substitute back to get the angle, sum, or length the question actually requests.

Confusing complementary ( $90^\circ$ ) with supplementary ( $180^\circ$ ), so a supplement is computed as $90^\circ$ minus the angle.

Fix the two totals in memory: comple‑ment ↔ ninety (both start with the corner idea of a right angle), supple‑ment ↔ straight line ↔ $180^\circ$ .

Treating co-interior (same-side interior) angles as equal instead of supplementary.

Only corresponding and alternate interior angles are equal. Same-side interior angles sum to $180^\circ$ . Use the acute/obtuse glance test: one acute + one obtuse ⇒ supplementary.

Lines $\ell$ and $m$ are parallel and are cut by a transversal. One interior angle measures $68^\circ$ . What is the measure of the co-interior angle on the other line?

Two angles are supplementary. The larger angle is $30^\circ$ more than twice the smaller. What is the measure of the smaller angle?

Two lines intersect. One angle is $(2x + 15)^\circ$ and the angle vertically opposite it is $(3x - 25)^\circ$ . What is the measure of that angle?

Two distinct lines intersect at a point, forming four angles. Angles $x$ and $y$ are two of these four angles.\n\nQuantity A: $x + y$ \n\nQuantity B: $180$

Parallel lines are crossed by a path that bends at point $P$ between them. The angle from the bend to the upper line is $35^\circ$ and the angle from the bend to the lower line is $45^\circ$ . What is the measure of the bend angle at $P$ ?

Three rays from a point on a straight line divide the straight angle into three angles in the ratio $2 : 3 : 4$ . What is the measure of the largest angle?