Angle relationships

Complementary and supplementary

Complementary angles add up to $90^\circ$. To find the complement of an angle, subtract it from $90^\circ$ — the complement of $35^\circ$ is $90^\circ - 35^\circ = 55^\circ$.

Supplementary angles add up to $180^\circ$. The supplement of $60^\circ$ is $180^\circ - 60^\circ = 120^\circ$.

Angles at a point

Three facts cover most “find the pronumeral” questions:

• Angles on a straight line add to $180^\circ$.
• Angles at a point (a full turn) add to $360^\circ$.
• Vertically opposite angles — the angles facing each other when two straight lines cross — are equal.

Adjacent angles share a common vertex and a common arm. To find a missing angle, add up the angles you already know, then subtract from $180^\circ$ or $360^\circ$. For example, if angles of $150^\circ$, $130^\circ$ and $x^\circ$ meet at a point:

$$x = 360 - (150 + 130) = 80$$

Parallel lines and transversals

A transversal is a line that cuts across two or more other lines. When those lines are parallel, three angle pairs appear:

• Alternate angles are equal — inside the parallel lines, on opposite sides of the transversal (a Z shape).
• Corresponding angles are equal — in the same position at each crossing (an F shape).
• Co-interior angles add to $180^\circ$ — inside, on the same side (a C shape).

These rules also work in reverse: if a pair of alternate angles are equal, the lines must be parallel. That is how to answer “are these two lines parallel?” questions.

If an angle sits between two parallel lines and touches neither, draw an extra parallel line through its vertex. The angle splits into two parts, and each part is an alternate angle with one of the marked angles.

Always give reasons

Most angle questions say “give reasons”, and the reason earns marks. Write the rule in full, naming the relationship:

• vertically opposite angles are equal
• angles on a straight line add to $180^\circ$
• alternate angles on parallel lines are equal

If a rule depends on the lines being parallel, your reason must say “on parallel lines”.