Points, Lines and Angles

Lessons

Senior 1 · Mathematics

Points, Lines and Angles

5 lessons

Lesson 3 of 5

Angles on a straight line and at a point

25 min

A junction on a straight road

Where a feeder road meets the straight Kigali–Muhanga highway, it makes two angles with the highway, one on each side. Together the two angles make up the straight line of the highway, a half turn, so if one angle is sharp the other must be wide. This lesson turns that idea into three rules, each with the reason it is true. You will use them to find unknown angles without a protractor.

Definition

Adjacent angles: two angles that share a vertex and one arm, and do not overlap.

Example: in the figure below, $\angle AOC$ and $\angle COB$ are adjacent: they share the vertex $O$ and the arm $O C$ .

$a + b = 58^\circ + 122^\circ = 180^\circ$ : together they make the straight angle $\angle AOB$ .

Important

Rule 1: angles on a straight line add up to $180^\circ$

a + b = 180^\circ

Why: the two adjacent angles together fill the straight angle $\angle AOB$ , which is a half turn, $180^\circ$ . The same is true for three or more adjacent angles that together fill one side of a straight line.

Definition

Supplementary angles: two angles whose sum is $180^\circ$ . Each is the supplement of the other.

Complementary angles: two angles whose sum is $90^\circ$ . Each is the complement of the other.

Example: the supplement of $38^\circ$ is $180^\circ - 38^\circ = 142^\circ$ ; its complement is $90^\circ - 38^\circ = 52^\circ$ .

Definition

Angles at a point: angles that share a vertex and together fill the whole turn around it, with no gaps and no overlaps.

$110^\circ + 80^\circ + 95^\circ + 75^\circ = 360^\circ$ : the angles fill one full turn.

Important

Rule 2: angles at a point add up to $360^\circ$

a + b + c + \cdots = 360^\circ

Why: going once round the point, from one arm through every angle back to the same arm, is one full turn.

Definition

Vertically opposite angles: when two straight lines cross, the four angles form two pairs of angles that face each other across the vertex. Each pair is called vertically opposite.

Example: in the figure below, $a$ and $c$ are vertically opposite, and so are $b$ and $d$ .

Two crossing lines: $a = c$ and $b = d$ . Here $a = c = 115^\circ$ and $b = d = 65^\circ$ .

Important

Rule 3: vertically opposite angles are equal

Why: $a$ and $b$ lie on a straight line, and so do $b$ and $c$ . So

a + b = 180^\circ \quad \text{and} \quad b + c = 180^\circ.

Both sums equal $180^\circ$ , so $a + b = b + c$ . Take $b$ from both sides: $a = c$ . In the same way, $b = d$ .

Give a reason for every angle

In geometry an answer is only complete when it says why. After each angle you find, write the rule you used in brackets:

Rule	What you write
Rule 1	(angles on a straight line)
Rule 2	(angles at a point)
Rule 3	(vertically opposite angles)

Worked example 1: three angles on a line

Three angles on a straight line.

Find $x$ , and the size of each angle.

Solution

x + 2x + 3x = 180^\circ \quad \text{(angles on a straight line)}

6x = 180^\circ, \quad \text{so} \quad x = 30^\circ.

The angles are $x = 30^\circ$ , $2x = 60^\circ$ and $3x = 90^\circ$ . Check: $30^\circ + 60^\circ + 90^\circ = 180^\circ$ . ✓

Worked example 2: angles at a point

Angles at a point: 150°, a right angle, x and x.

Find $x$ .

Solution The square marks a right angle, $90^\circ$ .

150^\circ + 90^\circ + x + x = 360^\circ \quad \text{(angles at a point)}

2x = 360^\circ - 240^\circ = 120^\circ, \quad \text{so} \quad x = 60^\circ.

Worked example 3: two crossing lines

Find p, q and r.

Solution

$p = 180^\circ - 65^\circ = 115^\circ$ (angles on a straight line)
$q = 65^\circ$ (vertically opposite angles)
$r = p = 115^\circ$ (vertically opposite angles)

Check: $65^\circ + 115^\circ + 65^\circ + 115^\circ = 360^\circ$ (angles at a point). ✓

Worked example 4: complementary angles

$\angle AOB = 90^\circ$ is split into $38^\circ$ and $y$ .

Find $y$ .

Solution $\angle AOB$ is a right angle, so $38^\circ$ and $y$ are complementary:

y = 90^\circ - 38^\circ = 52^\circ.

Warning

Three mistakes to avoid

Using Rule 1 when there is no straight line. Two angles add up to $180^\circ$ only if their outer arms form one straight line. Check the figure before you write "angles on a straight line".
Mixing up the pairs. The angles next to each other at a crossing add up to $180^\circ$ ; the angles opposite each other are equal.
Complement or supplement? Complementary angles make $90^\circ$ ; supplementary angles make $180^\circ$ . An obtuse angle has a supplement, but no complement.

Tip

Complementary angles make a Corner ( $90^\circ$ ); Supplementary angles make a Straight line ( $180^\circ$ ). In the alphabet, C comes before S, just as $90^\circ$ comes before $180^\circ$ .

Cognito: 5 Angle Rules to Know (8 min). Watch Rule 2, angles on a straight line (from 1:25), and Rule 4, angles around a point (from 2:47). The rules for triangles and quadrilaterals in the same video come back in Lesson 5.

✦

Practice

Give a reason for every angle you find. Questions 5 to 7 need an equation.

Find the supplement of (a) $47^\circ$ (b) $90^\circ$ (c) $163^\circ$ , and the complement of (d) $47^\circ$ (e) $5^\circ$ .
Two adjacent angles on a straight line are $74^\circ$ and $y$ . Find $y$ .
Four angles at a point are $95^\circ$ , $120^\circ$ , $85^\circ$ and $z$ . Find $z$ .
Two straight lines cross, and one of the four angles is $38^\circ$ . Find the other three.
Two adjacent angles on a straight line are $(2x + 15)^\circ$ and $(x + 30)^\circ$ . Find $x$ and both angles.
Two vertically opposite angles are $(3x + 10)^\circ$ and $(5x - 30)^\circ$ . Find $x$ and the size of the angles.
An angle is twice the size of its supplement. Find the angle.
Five equal angles fill the whole turn around a point. How big is each?

Show answersHide answers

(a) $133^\circ$ (b) $90^\circ$ (c) $17^\circ$ (d) $43^\circ$ (e) $85^\circ$
$y = 180^\circ - 74^\circ = 106^\circ$ (angles on a straight line)
$z = 360^\circ - 300^\circ = 60^\circ$ (angles at a point)
$142^\circ$ (angles on a straight line), $38^\circ$ (vertically opposite), $142^\circ$ (vertically opposite)
$3 x + 45 = 180$ , so $x = 45$ ; the angles are $105^\circ$ and $75^\circ$ .
$3 x + 10 = 5 x - 30$ , so $2 x = 40$ and $x = 20$ ; each angle is $70^\circ$ .
If the angle is $x$ , then $x = 2(180^\circ - x)$ , so $3x = 360^\circ$ and $x = 120^\circ$ . Check: its supplement is $60^\circ$ , and $120^\circ = 2 \times 60^\circ$ . ✓
$\frac{360^\circ}{5} = 72^\circ$ (angles at a point)

Sources and credits

10.2: Angles (adjacent, supplementary, complementary and vertical angles), from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: GCSE Maths - 5 Angle Rules to Know, Cognito, embedded from YouTube.

The proofs, the diagrams, the worked examples, the practice items and the quiz questions were written by Ganzaa.

1A junction on a straight road

2Give a reason for every angle

A junction on a straight road

Give a reason for every angle