Angle Relationships: Complementary, Supplementary and Vertical Angles

Two angles that add to a fixed total

Relationship	They add to	Picture
complementary	$90°$	a square corner
supplementary	$180°$	a straight line

$$\text{complementary}: \; a + b = 90° \qquad \text{supplementary}: \; a + b = 180°$$

If two angles are complementary and one is $35°$:

$$35° + b = 90° \quad\Rightarrow\quad b = 55°$$

C for corner and $90$, S for straight and $180$. The alphabet gives the same order: complementary before supplementary, $90$ before $180$.

Adjacent angles on a straight line

Adjacent angles sit side by side, sharing a vertex and one arm. When two adjacent angles form a straight line, they are supplementary:

$$a + b = 180°$$

That is what “a straight angle is $180°$” means in practice, and it is the fact almost every angle problem eventually leans on.

Vertical angles are equal

When two straight lines cross, four angles appear. The pairs opposite each other are vertical angles, and they are always equal.

Label them $a$, $b$, $c$, $d$ going round:

Pair	Relationship
$a$ and $c$	vertical — equal
$b$ and $d$	vertical — equal
$a$ and $b$	adjacent — supplementary
$b$ and $c$	adjacent — supplementary

Why vertical angles have to be equal

This one has a short proof, and it is worth following rather than accepting.

$a$ and $b$ sit together on a straight line:

$$a + b = 180°$$

$b$ and $c$ also sit together on a straight line:

$$b + c = 180°$$

Two quantities that both equal $180°$ must equal each other:

$$a + b = b + c$$

Subtract $b$ from both sides:

$$a = c$$

Nothing about the particular angles was used, so it holds for every pair of crossing lines. That is what makes it a theorem rather than something to check each time — and it means one measured angle at a crossing gives you all four.

Finding a missing angle

Each relationship turns the picture into an equation, and from there it is ordinary solving.

Two supplementary angles, one $130°$:

$$130 + x = 180 \quad\Rightarrow\quad x = 50°$$

An angle and its complement, where the second is twice the first:

$$x + 2x = 90 \quad\Rightarrow\quad 3x = 90 \quad\Rightarrow\quad x = 30°$$

So the angles are $30°$ and $60°$.

Check the answer against the relationship. $30 + 60 = 90$, so the pair really is complementary. That check catches most slips in one line.

Angles round a point

A full turn is $360°$, so angles meeting at a point add to that:

$$a + b + c + d = 360°$$

At a crossing of two lines this agrees with everything above: two pairs of supplementary angles give $180 + 180 = 360$.

Worked examples

Common mistakes

Practice problems