Points, Lines and Angles

Lessons

Senior 1 · Mathematics

Points, Lines and Angles

5 lessons

Lesson 2 of 5

Angles and their types

25 min

Measuring an amount of turn

In one hour, the minute hand of the classroom clock turns all the way round and returns to 12. In 15 minutes it makes a quarter of that turn. An angle measures an amount of turn like this, and mathematicians divide one full turn into 360 equal parts called degrees. So in 15 minutes the minute hand turns through $\frac{1}{4} \times 360^\circ = 90^\circ$ .

Definition

Angle: the figure formed by two rays that start at the same point. The common starting point is the vertex, and the two rays are the arms of the angle. The size of the angle is the amount of turn from one arm to the other.

Notation: the angle with vertex $B$ and arms $B A$ and $B C$ is written $\angle ABC$ , $A\hat{B}C$ , or simply $\hat{B}$ when only one angle has its vertex at $B$ . An angle can also be named by a small letter such as $x$ , or a Greek letter such as $\theta$ ("theta").

$\angle ABC = 50^\circ$ . The vertex $B$ is the middle letter.

Definition

Degree: the unit of angle, one 360th part of a full turn, so that

1 \text{ full turn} = 360^\circ, \qquad \tfrac{1}{2} \text{ turn} = 180^\circ, \qquad \tfrac{1}{4} \text{ turn} = 90^\circ.

Example: the minute hand turns $360^\circ$ in 60 minutes, so it turns $6^\circ$ every minute.

Measuring with a protractor

A protractor has two scales, one running each way. To measure $\angle PQR$ :

Estimate first. Is the angle smaller or bigger than a right angle, the corner of a sheet of paper?
Put the centre mark of the protractor exactly on the vertex $Q$ .
Turn the protractor so that its base line lies along one arm, here $Q R$ .
Follow the scale that reads 0 on that arm, round to where the other arm crosses it. Read the number, and check it against your estimate.

The inner scale starts at 0 on the arm $Q R$ , so $\angle PQR = 65^\circ$ . The same arm crosses 115 on the outer scale: that is $180^\circ - 65^\circ$ , the wrong reading.

To draw an angle of, say, $65^\circ$ : draw one arm, put the centre on its endpoint and the base line along it, mark a dot at 65 on the scale that starts from 0 on your arm, and join the dot to the vertex.

Six kinds of angle

Angles are named by their size, compared with a right angle ( $90^\circ$ ) and a straight angle ( $180^\circ$ ).

Definition

Acute angle: an angle between $0^\circ$ and $90^\circ$ , that is $0^\circ \lt x \lt 90^\circ$ .

Right angle: an angle of exactly $90^\circ$ , a quarter turn. It is marked with a small square, not an arc.

Obtuse angle: an angle between $90^\circ$ and $180^\circ$ , that is $90^\circ \lt x \lt 180^\circ$ .

Definition

Straight angle: an angle of exactly $180^\circ$ , a half turn. Its two arms form a straight line.

Reflex angle: an angle between $180^\circ$ and $360^\circ$ , that is $180^\circ \lt x \lt 360^\circ$ .

Full turn (revolution): an angle of exactly $360^\circ$ ; the turning arm comes back to where it started.

Each angle is marked from the horizontal arm, turning anticlockwise. The reflex angle is the one on the long side of the arms.

Type	Size	Where you see one
acute	$0^\circ \lt x \lt 90^\circ$	the tip of a slice of cake
right	$x = 90^\circ$	the corner of a page or a door frame
obtuse	$90^\circ \lt x \lt 180^\circ$	a laptop screen opened wide
straight	$x = 180^\circ$	the hands of a clock at 6:00
reflex	$180^\circ \lt x \lt 360^\circ$	the outside of the corner of a room
full turn	$x = 360^\circ$	a wheel turning once

Important

Every pair of arms makes two angles, and they add up to $360^\circ$ . If one of them is $x$ , the other is $360^\circ - x$ .

Why: turning from one arm to the other the short way, and then carrying on the long way back to the first arm, is exactly one full turn. So an angle of $40^\circ$ comes with a reflex angle of $360^\circ - 40^\circ = 320^\circ$ . Unless a question says "reflex", "the angle between two arms" means the smaller one.

Worked example 1: classifying angles

Name the type of each angle: (a) $37^\circ$ (b) $90^\circ$ (c) $162^\circ$ (d) $180^\circ$ (e) $205^\circ$ (f) $89.5^\circ$ .

Solution Compare each size with $90^\circ$ , $180^\circ$ and $360^\circ$ .

$0^\circ \lt 37^\circ \lt 90^\circ$ : acute.
exactly $90^\circ$ : right.
$90^\circ \lt 162^\circ \lt 180^\circ$ : obtuse.
exactly $180^\circ$ : straight.
$180^\circ \lt 205^\circ \lt 360^\circ$ : reflex.
$89.5^\circ \lt 90^\circ$ : acute, even though it looks like a right angle. Only an angle of exactly $90^\circ$ is a right angle.

Worked example 2: the hands of a clock

At 4:00 the hands are 4 hour-spaces apart: $4 \times 30^\circ = 120^\circ$ .

Find the angle between the hands of a clock at (a) 4:00 (b) 3:00 and (c) 6:00, and name each type. At 4:00, what is the reflex angle between the hands?

Solution The 12 hour marks divide the full turn into 12 equal parts, so the angle between neighbouring marks is

\frac{360^\circ}{12} = 30^\circ.

At 4:00 the hands point to 12 and 4, which are 4 spaces apart: $4 \times 30^\circ = 120^\circ$ , an obtuse angle.
At 3:00: $3 \times 30^\circ = 90^\circ$ , a right angle.
At 6:00: $6 \times 30^\circ = 180^\circ$ , a straight angle.

The reflex angle at 4:00 is $360^\circ - 120^\circ = 240^\circ$ .

Worked example 3: finding a mistake

Mugisha measures an angle with a protractor and writes $145^\circ$ , but the angle is clearly smaller than the corner of his exercise book. What has gone wrong, and what is the correct size?

Solution An angle smaller than a right angle is acute, so $145^\circ$ (obtuse) cannot be right. Mugisha read the scale that starts from 0 on the other side. The two scales always add up to $180^\circ$ , so the correct reading is

180^\circ - 145^\circ = 35^\circ.

Warning

Two common mistakes

Reading the wrong scale. Always estimate first. If your estimate says acute and the protractor says $145^\circ$ , use the other scale.
Putting the vertex in the wrong place in the name. The middle letter is always the vertex: $\angle ABC$ has its vertex at $B$ , while $\angle BAC$ has its vertex at $A$ . They are different angles.

Tip

Carry a right angle with you: the corner of any page is $90^\circ$ . Hold it against an angle. If the angle fits inside the corner, it is acute; if it is wider than the corner but not yet a straight line, it is obtuse.

Math Antics: Angle Basics (8 min). Watch how an angle is measured as an amount of turn, why a full turn is 360°, and how right, acute, obtuse and straight angles compare.

✦

Practice

Estimate before you calculate. The last two questions are harder.

Name the type of each angle: (a) $72^\circ$ (b) $118^\circ$ (c) $300^\circ$ (d) $6^\circ$ (e) $180^\circ$ .
For $\angle PQR$ , name (a) the vertex, (b) the two arms.
Find the reflex angle that goes with an angle of (a) $40^\circ$ (b) $155^\circ$ (c) $90^\circ$ .
Through how many degrees does the minute hand of a clock turn in (a) 15 minutes, (b) 20 minutes, (c) 45 minutes?
Find the smaller angle between the hands of a clock at (a) 2:00 (b) 5:00 (c) 10:00.
Keza's protractor shows 50 on one scale and 130 on the other where the second arm crosses. The angle is wider than the corner of her book. What is its size?
Find the smaller angle between the hands of a clock at 2:30. (The hour hand moves too!)

Show answersHide answers

(a) acute (b) obtuse (c) reflex (d) acute (e) straight
(a) $Q$ (b) $Q P$ and $Q R$
(a) $320^\circ$ (b) $205^\circ$ (c) $270^\circ$
The hand turns $6^\circ$ per minute: (a) $90^\circ$ (b) $120^\circ$ (c) $270^\circ$
(a) $2 \times 30^\circ = 60^\circ$ (b) $5 \times 30^\circ = 150^\circ$ (c) 10 and 12 are 2 spaces apart: $60^\circ$
Wider than a right angle means obtuse, so $130^\circ$ .
The minute hand points to 6, which is $180^\circ$ from 12. The hour hand is halfway between 2 and 3, which is $2.5 \times 30^\circ = 75^\circ$ from 12. The angle between them is $180^\circ - 75^\circ = 105^\circ$ .

Sources and credits

10.2: 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: Math Antics: Angle Basics, mathantics, embedded from YouTube.

The protractor, clock and angle-type diagrams, the worked examples, the practice items and the quiz questions were written by Ganzaa.

1Measuring an amount of turn

2Measuring with a protractor

3Six kinds of angle

Measuring an amount of turn

Measuring with a protractor

Six kinds of angle