Measuring and Drawing Angles

An angle is formed by two rays sharing a vertex. Learn to identify angle types, use a protractor to measure angles, and draw angles of any given size.

What Is an Angle?

An angle is formed when two rays start at the same point. That shared point is the vertex, and the two rays are called the arms of the angle.

We label an angle by three letters — the middle letter is always the vertex. For example, $\angle ABC$ has vertex $B$ .

Angles are measured in degrees (°). A full turn is 360°.

Types of Angles

Type	Size	Quick picture
Acute	between 0° and 90°	less than a corner
Right	exactly 90°	a perfect corner
Obtuse	between 90° and 180°	wider than a corner
Straight	exactly 180°	a flat line
Reflex	between 180° and 360°	more than half a turn

Tip If you see a small square drawn at the vertex, it means the angle is exactly 90° — a right angle.

Using a Protractor to Measure an Angle

A protractor is a semi-circular tool with degree markings from 0° to 180°.

1
Place the protractor
Put the centre mark of the protractor exactly on the vertex of the angle.
2
Align the baseline
Rotate the protractor so that one arm of the angle lies along the 0° line.
3
Read the scale
Follow the arm of the angle to where it crosses the degree scale. Read the number — that is the angle size.
4
Choose the correct scale
Protractors have two scales (one reading left-to-right, one right-to-left). Always use the scale that starts at 0° on the arm you aligned.

Measuring angle ABC

Place the protractor centre on $B$ . Align the 0° mark with ray $BA$ . Read the number where ray $BC$ crosses the scale: say 55°. So $\angle ABC = 55°$ — an acute angle.

Drawing an Angle of a Given Size

1
Draw the first arm
Draw a ray starting at a point — this will be the vertex $A$ .
2
Place the protractor
Put the centre mark on $A$ and align the 0° line with the ray you just drew.
3
Mark the angle
Find the required degree mark on the scale and make a small dot there.
4
Draw the second arm
Remove the protractor and draw a ray from $A$ through the dot. Label the angle.

Drawing a 70° angle

Draw ray $AB$ . Place the protractor centre on $A$ , align 0° with $AB$ . Put a dot at 70°. Draw ray $AC$ through that dot. $\angle BAC = 70°$ — an acute angle.

Drawing a right angle (90°)

Follow the same steps but mark the dot at 90°. A right angle can also be drawn quickly with a set square (a triangular ruler with a 90° corner).

Full rotation

360° = \text{one full turn}

All angles around a point add up to 360°.

Straight angle

\alpha + \beta = 180°

Angles on a straight line always sum to 180°.

Real world: ramps

A wheelchair ramp rises at about 5° from the ground — a very small acute angle that keeps the slope gentle and safe.

Real world: clock hands

At 3:00 the clock hands form a 90° angle (right angle). At 6:00 they form a 180° angle (straight). At 2:00 the hour hand has moved $\frac{2}{12} \times 360° = 60°$ from 12 — an acute angle.

Real world: roof angles

A steep roof may have a pitch angle of 45° (acute), while a flatter roof might have only 20°. Engineers choose the angle based on rainfall and snow load.

Common Mistakes

Reading the wrong scale on the protractor (use the scale that starts from 0° on the aligned arm)
Not placing the centre mark exactly on the vertex
Forgetting that a right angle is 90°, not 100°
Drawing the second arm in the wrong direction when the angle is obtuse

Practice Problems

Using a protractor, you read 125° for angle PQR. What type of angle is it?

Show answer

Obtuse — it is between 90° and 180°.

Draw an angle of 45°. What type of angle is it? What is the size of its supplement?

Show answer

45° is an acute angle. Its supplement is 180° − 45° = 135° (an obtuse angle).