Types of Angles

Name an angle, sort angles into acute, right, obtuse, straight and reflex, estimate and measure them, and work with complementary and supplementary angles and the hands of a clock.

Before you start

A complete turn is $360^{\circ}$ , a half turn is $180^{\circ}$ and a quarter turn is $90^{\circ}$ . The corner of a sheet of paper is a quarter turn: a right angle.
An equation such as $5x = 90$ is solved by doing the same to both sides (Lesson 9.1): divide both sides by $5$ , so $x = 18$ .

What an Angle Is

A ray is part of a straight line that starts at a point and goes on without end in one direction.

An angle

An angle is formed by two rays, its arms, that start from the same point, its vertex.

Its size is the amount of turn from one arm to the other, about the vertex, measured in degrees (°). One complete turn is $360^{\circ}$ , so $1^{\circ}$ is $\frac{1}{360}$ of a complete turn.

The angle

AOB

: vertex

O

, arms

OA

and

OB

The length of the arms does not change the size of an angle. The two angles below are both $40^{\circ}$ : the arms of the second are longer, but the turn from one arm to the other is the same.

Short arms:

40^{\circ}

Long arms: also

40^{\circ}

Naming Angles

An angle is named by the points on it, with the angle sign $\angle$ .

Naming an angle

With three letters: a point on one arm, then the vertex, then a point on the other arm. The vertex is always the middle letter. $\angle AOB$ , which is the same angle as $\angle BOA$ , is read "angle $AOB$ ".

With one letter, the vertex alone, as $\angle O$ : only when there is just one angle at $O$ .

With a small letter written inside the angle in the figure, such as $a$ or $b$ . Then " $a = 35^{\circ}$ " means that the angle marked $a$ is $35^{\circ}$ .

An angle to be found is marked with a letter and the degree sign, such as $x^{\circ}$ or $2x^{\circ}$ . Here the letter is a number: $x^{\circ}$ means $x$ degrees, and $2x^{\circ}$ means $2x$ degrees. The answer is written " $x = 65$ ", and then "so the angle is $65^{\circ}$ ". When $x$ is found by solving an equation, the angle is worked out from it, as in Question 8.

Example 1: Naming one angle

Name the marked angle in four ways.

Solution

Three letters, the vertex in the middle: the vertex is $Q$ , and $P$ and $R$ are on the arms. So the angle is $\angle PQR$ , or $\angle RQP$ read from the other arm.

One letter: there is only one angle at $Q$ , so $\angle Q$ is clear.

The small letter in the angle: $a$ .

The angle is $\angle PQR$ , $\angle RQP$ , $\angle Q$ or $a$ .

Example 2: Several angles at one vertex

In the figure, $a = 35^{\circ}$ and $b = 50^{\circ}$ .

(a) Explain why " $\angle B$ " is not a good name for any angle in this figure.

(b) Name the angles $a$ and $b$ with three letters.

Solution

(a) Three rays start from $B$ , so there are three angles at $B$ : $a$ , $b$ , and the angle that $a$ and $b$ make together. " $\angle B$ " could mean any of them.

(b) Angle $a$ lies between the arms $BA$ and $BC$ , so $a = \angle ABC$ . Angle $b$ lies between the arms $BC$ and $BD$ , so $b = \angle CBD$ .

$\angle ABD = a + b = 35^{\circ} + 50^{\circ} = 85^{\circ}$

(a) There are three angles at $B$ . (b) $a = \angle ABC$ and $b = \angle CBD$ . (c) $\angle ABD = 85^{\circ}$ .

Acute, Right, Obtuse, Straight and Reflex Angles

Angles are sorted into types by their size, measured against a right angle ( $90^{\circ}$ ), a straight angle ( $180^{\circ}$ ) and a complete turn ( $360^{\circ}$ ).

Types of angles

An acute angle is greater than $0^{\circ}$ and less than $90^{\circ}$ .

A right angle is exactly $90^{\circ}$ . It is marked with a small square at the vertex.

An obtuse angle is greater than $90^{\circ}$ and less than $180^{\circ}$ .

A straight angle is exactly $180^{\circ}$ : its two arms make a straight line.

A reflex angle is greater than $180^{\circ}$ and less than $360^{\circ}$ .

Acute:

40^{\circ}

Right:

90^{\circ}

Obtuse:

130^{\circ}

Straight:

180^{\circ}

Reflex:

230^{\circ}

Example 3: Classifying angles by their size

State the type of each angle.

(a) $75^{\circ}$ (b) $90^{\circ}$ (c) $135^{\circ}$ (d) $180^{\circ}$ (e) $225^{\circ}$ (f) $4^{\circ}$

Solution

(a) $75^{\circ}$ is between $0^{\circ}$ and $90^{\circ}$ : acute.

(b) $90^{\circ}$ exactly: a right angle.

(d) $180^{\circ}$ exactly: a straight angle.

(e) $225^{\circ}$ is between $180^{\circ}$ and $360^{\circ}$ : reflex.

(f) $4^{\circ}$ is very small, but it is greater than $0^{\circ}$ and less than $90^{\circ}$ : acute.

(a) acute (b) right (c) obtuse (d) straight (e) reflex (f) acute

Reflex Angles

Two arms from one vertex make two angles: the angle between them on one side, and the rest of the complete turn on the other side. Unless the arms make a straight line, one of the two is less than $180^{\circ}$ and the other is a reflex angle.

When the arms do not make a straight line, the name $\angle AOB$ means the angle that is less than $180^{\circ}$ . The other is called reflex $\angle AOB$ , and in a figure its arc is drawn on the outside of the arms.

Reflex angles

An angle and the reflex angle on the other side of the same two arms together make a complete turn, so

$\text{reflex } \angle AOB = 360^{\circ} - \angle AOB$

Example 4: Finding a reflex angle

In the figure, $\angle AOB = 125^{\circ}$ . Find reflex $\angle AOB$ , marked $y^{\circ}$ .

Solution

The angle of $125^{\circ}$ and the reflex angle of $y^{\circ}$ have the same two arms, so together they make a complete turn.

$y = 360 - 125 = 235$

So reflex $\angle AOB = 235^{\circ}$ . Check: $125^{\circ} + 235^{\circ} = 360^{\circ}$ , and $235^{\circ}$ is between $180^{\circ}$ and $360^{\circ}$ , so it is reflex, as it must be.

$y = 235$ , so reflex $\angle AOB = 235^{\circ}$ .

Example 5: From the reflex angle back to the angle

In the figure, reflex $\angle PQR = 290^{\circ}$ . Find $\angle PQR$ , marked $x^{\circ}$ , and state its type.

Solution

The angle of $x^{\circ}$ and the reflex angle of $290^{\circ}$ have the same two arms, so together they make a complete turn.

$x = 360 - 290 = 70$

So $\angle PQR = 70^{\circ}$ . Check: $70^{\circ} + 290^{\circ} = 360^{\circ}$ . $70^{\circ}$ is less than $90^{\circ}$ , so $\angle PQR$ is acute.

$x = 70$ , so $\angle PQR = 70^{\circ}$ , an acute angle.

Estimating the Size of an Angle

Before measuring an angle, estimate it. Compare it with angles you know by sight. An estimate tells you the type of the angle at once, and it catches a wrong reading of a protractor later.

Angles to compare with

A right angle, $90^{\circ}$ : the corner of a page. Half of it, $45^{\circ}$ : a square of paper folded from corner to corner. A third of it, $30^{\circ}$ ; two thirds of it, $60^{\circ}$ .

A straight angle, $180^{\circ}$ , and three quarters of a turn, $270^{\circ}$ .

For a reflex angle, estimate the angle on the other side of the arms and take it from $360^{\circ}$ .

Example 6: Estimating and classifying from drawings

Estimate the size of each marked angle, then state its type.

(a)

(b)

(c)

(d)

Solution

(a) $p$ is less than a right angle, and a little less than half of one: about $35^{\circ}$ . Acute.

(b) $q$ is a little more than a right angle: about $100^{\circ}$ . Obtuse.

(c) $r$ is nearly a straight angle; its arm is about $20^{\circ}$ short of a straight line: about $160^{\circ}$ . Obtuse.

(d) The arc of $s$ is on the outside, so $s$ is reflex. The angle on the other side of the arms is about $60^{\circ}$ , two thirds of a right angle, so $s$ is about $360^{\circ} - 60^{\circ} = 300^{\circ}$ . Reflex.

(a) about $35^{\circ}$ , acute (b) about $100^{\circ}$ , obtuse (c) about $160^{\circ}$ , obtuse (d) about $300^{\circ}$ , reflex. An estimate within about $10^{\circ}$ of these is a good one.

Measuring and Drawing Angles with a Protractor

A protractor measures angles in degrees. It has two scales, each from $0$ to $180$ . On most protractors, as on the one drawn below, the inner scale starts at $0$ on the right-hand end of the base line and the outer scale at $0$ on the left-hand end. Always check which scale reads $0$ on your arm.

To measure an angle

1. Place the centre of the protractor exactly on the vertex.

2. Turn the protractor so that its base line lies along one arm.

3. Choose the scale that reads $0$ on that arm.

4. Follow that scale round to the other arm, and read the number where the arm crosses it.

5. Check against the type of angle: an acute angle reads less than $90$ and an obtuse angle more than $90$ .

Example 7: Reading the right scale

A protractor is placed on $\angle AOB$ as shown. The arm $OB$ crosses $50$ on one scale and $130$ on the other. What is the size of $\angle AOB$ ?

Solution

The arm $OA$ lies along the base line to the right, where the inner (blue) scale reads $0$ . So the inner scale is the one to use.

Follow the inner scale from $0$ round to the arm $OB$ : it reads $130$ .

Check: $\angle AOB$ is wider than a right angle, so it is obtuse and must be more than $90^{\circ}$ . The reading $50$ , on the outer scale, cannot be right.

$\angle AOB = 130^{\circ}$ .

To draw an angle, for example $\angle ABC = 65^{\circ}$

1. Draw one arm, $BC$ , with a ruler.

2. Place the centre of the protractor on $B$ , with the base line along $BC$ .

3. On the scale that reads $0$ on $BC$ , find $65$ and mark a dot there. Call it $A$ .

4. Remove the protractor and join $B$ to $A$ with the ruler. Mark the angle $65^{\circ}$ .

A protractor reads only up to $180^{\circ}$ . To measure a reflex angle, measure the angle on the other side of its arms and take it from $360^{\circ}$ . To draw a reflex angle, work out the angle on the other side and draw that.

Example 8: Drawing a reflex angle

Explain how to draw reflex $\angle XYZ = 250^{\circ}$ with a protractor.

Solution

A reflex angle and the angle on the other side of the same arms add up to $360^{\circ}$ .

$\angle XYZ = 360^{\circ} - 250^{\circ} = 110^{\circ}$

Draw $\angle XYZ = 110^{\circ}$ in the usual way. The angle on the outside of its arms is reflex $\angle XYZ$ , which is $250^{\circ}$ . Mark it with an arc on the outside.

Draw $\angle XYZ = 110^{\circ}$ ; the reflex angle on the outside of its arms is $250^{\circ}$ .

Complementary and Supplementary Angles

Two pairs of angles have special names, from what the two angles add up to.

Complementary and supplementary angles

Two angles that add up to $90^{\circ}$ are complementary. Each is the complement of the other: the complement of an angle of $x^{\circ}$ is $(90 - x)^{\circ}$ .

Two angles that add up to $180^{\circ}$ are supplementary. Each is the supplement of the other: the supplement of an angle of $x^{\circ}$ is $(180 - x)^{\circ}$ .

The two angles need not be next to each other: $30^{\circ}$ and $60^{\circ}$ are complementary wherever they are drawn.

Example 9: Complements and supplements

(a) Find the complement of $38^{\circ}$ .

(b) Find the supplement of $38^{\circ}$ .

(d) Can two obtuse angles be supplementary? Explain.

Solution

(a) $90^{\circ} - 38^{\circ} = 52^{\circ}$ . Check: $38^{\circ} + 52^{\circ} = 90^{\circ}$ .

(b) $180^{\circ} - 38^{\circ} = 142^{\circ}$ . Check: $38^{\circ} + 142^{\circ} = 180^{\circ}$ .

(d) Each obtuse angle is more than $90^{\circ}$ , so two of them add up to more than $90^{\circ} + 90^{\circ} = 180^{\circ}$ . They cannot be supplementary.

(a) $52^{\circ}$ (b) $142^{\circ}$ (c) supplementary (d) No: two obtuse angles add up to more than $180^{\circ}$ .

Example 10: Two angles that make a right angle

In the figure, $\angle AOC$ is a right angle and $\angle AOB = 28^{\circ}$ . Find $x$ .

Solution

$\angle AOB$ and $\angle BOC$ together make the right angle $AOC$ , so they are complementary.

$x = 90 - 28 = 62$

So $\angle BOC = 62^{\circ}$ . Check: $28^{\circ} + 62^{\circ} = 90^{\circ}$ .

$x = 62$ , so $\angle BOC = 62^{\circ}$ .

Coming next

Two angles side by side whose outer arms make a straight line are supplementary. Finding such angles, with the reason written out, is the work of Lesson 11.2.

Angles Turned by the Hands of a Clock

The hands of a clock turn about its centre. The angle between the hands, and the angle each hand turns through in a given time, can be worked out from three facts.

The hands of a clock

The $12$ numbers divide a complete turn into $12$ equal parts: $360^{\circ} \div 12 = 30^{\circ}$ between two numbers next to each other.

The minute hand turns $360^{\circ}$ in $60$ minutes: $360^{\circ} \div 60 = 6^{\circ}$ every minute.

The hour hand turns $30^{\circ}$ in $60$ minutes, from one number to the next: $30^{\circ} \div 60 = 0.5^{\circ}$ every minute.

Example 11: The angle a hand turns through

Through what angle does the minute hand of a clock turn between 3.10 p.m. and 3.35 p.m.? What type of angle is it?

Solution

From 3.10 p.m. to 3.35 p.m. is $35 - 10 = 25$ minutes.

The minute hand turns $6^{\circ}$ each minute: $25 \times 6^{\circ} = 150^{\circ}$ .

$150^{\circ}$ is between $90^{\circ}$ and $180^{\circ}$ : obtuse.

The minute hand turns through $150^{\circ}$ , an obtuse angle.

Example 12: The angle between the hands at 4 o'clock

Find the angle between the hands of a clock at 4 o'clock, marked $x^{\circ}$ , and the reflex angle between them.

Solution

At 4 o'clock the minute hand points to $12$ and the hour hand points to $4$ .

From $12$ round to $4$ there are $4$ spaces of $30^{\circ}$ : $x = 4 \times 30 = 120$ . So the angle between the hands is $120^{\circ}$ , an obtuse angle.

The reflex angle between the hands is $360^{\circ} - 120^{\circ} = 240^{\circ}$ . Check: $120^{\circ} + 240^{\circ} = 360^{\circ}$ .

$x = 120$ : the angle between the hands is $120^{\circ}$ , an obtuse angle; the reflex angle between them is $240^{\circ}$ .

Example 13: At half past two

Find the angle between the hands of a clock at 2.30, marked $x^{\circ}$ .

Solution

At 2.30 the minute hand points to $6$ . The hour hand is not on $2$ : in the $30$ minutes since 2 o'clock it has turned $30 \times 0.5^{\circ} = 15^{\circ}$ , half way from $2$ towards $3$ .

Measure each hand from $12$ , turning clockwise. Minute hand: $6 \times 30^{\circ} = 180^{\circ}$ . Hour hand: $2 \times 30^{\circ} + 15^{\circ} = 75^{\circ}$ .

$x = 180 - 75 = 105$

So the angle between the hands is $105^{\circ}$ . Check: the hour hand at $75^{\circ}$ turned on by $105^{\circ}$ reaches $180^{\circ}$ , the minute hand.

$x = 105$ , so the angle between the hands is $105^{\circ}$ .

Common Mistakes

Putting the vertex at the end of the name. The angle at $O$ between $OA$ and $OB$ is $\angle AOB$ , not $\angle ABO$ or $\angle OAB$ .
Writing $\angle B$ when several angles meet at $B$ . Use three letters.
Thinking that longer arms make a bigger angle. The size of an angle is the turn between its arms, not their length.
Reading the wrong scale of the protractor. Use the scale that reads $0$ on the first arm, then check: an acute angle reads less than $90$ , an obtuse angle more than $90$ .
Taking $\angle AOB$ to mean the reflex angle. $\angle AOB$ is never the reflex angle; that one is written "reflex $\angle AOB$ ".
Mixing up the two words. Complementary angles make a corner, $90^{\circ}$ ; supplementary angles make a straight line, $180^{\circ}$ .
Leaving the hour hand on the number. At 2.30 the hour hand is half way between $2$ and $3$ , not on $2$ .

Practice

Try each question, then open its solution to check your working. The figures are drawn to scale.

Question 1

State the type of each angle.

(a) $47^{\circ}$ (b) $90^{\circ}$ (c) $91^{\circ}$ (d) $180^{\circ}$ (e) $199^{\circ}$ (f) $345^{\circ}$

ShowHide solution

(a) $47^{\circ}$ is between $0^{\circ}$ and $90^{\circ}$ : acute.

(b) Exactly $90^{\circ}$ : right.

(d) Exactly $180^{\circ}$ : straight.

(e) $199^{\circ}$ is between $180^{\circ}$ and $360^{\circ}$ : reflex.

(f) $345^{\circ}$ is between $180^{\circ}$ and $360^{\circ}$ : reflex.

(a) acute (b) right (c) obtuse (d) straight (e) reflex (f) reflex

Question 2

True or false? Give a reason for each answer.

(a) An angle of $89^{\circ}$ is obtuse.

(b) A right angle is exactly $90^{\circ}$ .

(d) Every angle less than $180^{\circ}$ is acute.

(e) An acute angle and an obtuse angle can be supplementary.

(f) Two acute angles can be supplementary.

ShowHide solution

(a) False: $89^{\circ}$ is less than $90^{\circ}$ , so it is acute.

(b) True: that is what a right angle is.

(d) False: $90^{\circ}$ (right) and $120^{\circ}$ (obtuse) are both less than $180^{\circ}$ , and neither is acute.

(e) True: for example, $60^{\circ} + 120^{\circ} = 180^{\circ}$ .

(f) False: two angles each less than $90^{\circ}$ add up to less than $90^{\circ} + 90^{\circ} = 180^{\circ}$ .

(a) False (b) True (c) True (d) False (e) True (f) False

Question 3

In the figure, $p = 30^{\circ}$ , $q = 45^{\circ}$ and $r = 60^{\circ}$ .

(a) Name the angles $p$ , $q$ and $r$ with three letters.

(b) Name the angle that is equal to $p + q$ , and find its size.

ShowHide solution

(a) $p$ lies between $OA$ and $OB$ , $q$ between $OB$ and $OC$ , $r$ between $OC$ and $OD$ . So $p = \angle AOB$ , $q = \angle BOC$ and $r = \angle COD$ .

(b) $p$ and $q$ together lie between the arms $OA$ and $OC$ : $\angle AOC = 30^{\circ} + 45^{\circ} = 75^{\circ}$ .

$\angle AOB = 30^{\circ}$ , acute; $\angle AOC = 30^{\circ} + 45^{\circ} = 75^{\circ}$ , acute; $\angle AOD = 30^{\circ} + 45^{\circ} + 60^{\circ} = 135^{\circ}$ , obtuse;

$\angle BOC = 45^{\circ}$ , acute; $\angle BOD = 45^{\circ} + 60^{\circ} = 105^{\circ}$ , obtuse; $\angle COD = 60^{\circ}$ , acute.

(a) $p = \angle AOB$ , $q = \angle BOC$ , $r = \angle COD$ (b) $\angle AOC = 75^{\circ}$ (c) $\angle AOB = 30^{\circ}$ , $\angle BOC = 45^{\circ}$ , $\angle COD = 60^{\circ}$ , $\angle AOC = 75^{\circ}$ (all acute); $\angle BOD = 105^{\circ}$ , $\angle AOD = 135^{\circ}$ (both obtuse)

Question 4

Estimate the size of each marked angle, then state its type.

(a)

(b)

(c)

(d)

(e)

ShowHide solution

(a) $a$ is a little less than a right angle: about $80^{\circ}$ . Acute.

(b) $b$ is small, about two thirds of $30^{\circ}$ : about $20^{\circ}$ . Acute.

(c) $c$ is between a right angle and a straight angle, nearer to the straight angle; its arm is about $35^{\circ}$ short of a straight line: about $145^{\circ}$ . Obtuse.

(d) The arc of $d$ is on the outside, so $d$ is reflex. Its second arm has turned about $30^{\circ}$ past a straight line: about $180^{\circ} + 30^{\circ} = 210^{\circ}$ . Reflex.

(e) The arc of $e$ is on the outside, so $e$ is reflex. The angle on the other side of the arms is about $30^{\circ}$ : $e$ is about $360^{\circ} - 30^{\circ} = 330^{\circ}$ . Reflex.

(a) about $80^{\circ}$ , acute (b) about $20^{\circ}$ , acute (c) about $145^{\circ}$ , obtuse (d) about $210^{\circ}$ , reflex (e) about $330^{\circ}$ , reflex. Within about $10^{\circ}$ is a good estimate.

Question 5

(a) $\angle AOB = 48^{\circ}$ . Find reflex $\angle AOB$ .

(b) Reflex $\angle PQR = 205^{\circ}$ . Find $\angle PQR$ and state its type.

ShowHide solution

(a) Reflex $\angle AOB = 360^{\circ} - 48^{\circ} = 312^{\circ}$ .

(b) $\angle PQR = 360^{\circ} - 205^{\circ} = 155^{\circ}$ , which is between $90^{\circ}$ and $180^{\circ}$ : obtuse.

$\begin{aligned} x + 3x &= 360 \\ 4x &= 360 \\ x &= 90 \end{aligned}$

Check: $\angle XYZ = 90^{\circ}$ and reflex $\angle XYZ = 3 \times 90^{\circ} = 270^{\circ}$ ; $90^{\circ} + 270^{\circ} = 360^{\circ}$ .

(a) $312^{\circ}$ (b) $155^{\circ}$ , obtuse (c) $x = 90$ , so $\angle XYZ = 90^{\circ}$ , a right angle

Question 6

(a) A protractor is placed on $\angle POQ$ as shown, with the arm $OP$ along the base line to the left. Ravi reads $\angle POQ = 115^{\circ}$ . Is he right? Explain.

(b) Explain how to draw reflex $\angle DEF = 300^{\circ}$ with a protractor.

ShowHide solution

(a) The arm $OP$ lies on the left-hand end of the base line, where the outer scale reads $0$ . Follow the outer scale round to $OQ$ : it reads $65$ .

Check: $\angle POQ$ is narrower than a right angle, so it is acute and must be less than $90^{\circ}$ . Ravi read $115$ on the inner scale, which starts at $0$ on the right, not on $OP$ .

(b) $\angle DEF = 360^{\circ} - 300^{\circ} = 60^{\circ}$ . Draw $\angle DEF = 60^{\circ}$ : draw $ED$ , put the centre of the protractor on $E$ with the base line along $ED$ , mark $F$ at $60$ on the scale that reads $0$ on $ED$ , and join $E$ to $F$ . The angle on the outside of the arms is reflex $\angle DEF = 300^{\circ}$ ; mark it with an arc on the outside.

(a) No: $\angle POQ = 65^{\circ}$ ; Ravi read the wrong scale. (b) Draw $\angle DEF = 60^{\circ}$ ; the reflex angle on the outside of its arms is $300^{\circ}$ .

Question 7

(a) Find the complement of $17^{\circ}$ .

(b) Find the supplement of $17^{\circ}$ .

(d) Explain why an angle of $110^{\circ}$ has no complement.

(e) Which angle is its own complement?

ShowHide solution

(a) $90^{\circ} - 17^{\circ} = 73^{\circ}$ .

(b) $180^{\circ} - 17^{\circ} = 163^{\circ}$ .

(d) A complement makes the sum $90^{\circ}$ , but $110^{\circ}$ is already more than $90^{\circ}$ . It would need an angle of $90^{\circ} - 110^{\circ} = -20^{\circ}$ , and no angle is negative.

(e) An angle that is its own complement is half of $90^{\circ}$ : $90^{\circ} \div 2 = 45^{\circ}$ . Check: $45^{\circ} + 45^{\circ} = 90^{\circ}$ .

(a) $73^{\circ}$ (b) $163^{\circ}$ (c) $90^{\circ}$ (d) $110^{\circ}$ is more than $90^{\circ}$ on its own. (e) $45^{\circ}$

Question 8

In the figure, $\angle PQS$ is a right angle. Find $x$ , and then the sizes of $\angle PQR$ and $\angle RQS$ .

ShowHide solution

$\angle PQR$ and $\angle RQS$ together make the right angle $PQS$ , so they are complementary:

$\begin{aligned} 2x + 3x &= 90 \\ 5x &= 90 \\ x &= 18 \end{aligned}$

$\angle PQR = 2 \times 18^{\circ} = 36^{\circ}$ and $\angle RQS = 3 \times 18^{\circ} = 54^{\circ}$ .

Check: $36^{\circ} + 54^{\circ} = 90^{\circ}$ .

$x = 18$ ; $\angle PQR = 36^{\circ}$ and $\angle RQS = 54^{\circ}$ .

Question 9

(a) Two angles are supplementary, and one of them is $40^{\circ}$ more than the other. Find the two angles.

(b) Two angles are complementary, and one of them is $5$ times the other. Find the two angles.

ShowHide solution

(a) Let the smaller angle be $x^{\circ}$ . Then the larger is $(x + 40)^{\circ}$ , and the two add up to $180^{\circ}$ :

$\begin{aligned} x + (x + 40) &= 180 \\ 2x + 40 &= 180 \\ 2x &= 140 \\ x &= 70 \end{aligned}$

The angles are $70^{\circ}$ and $70^{\circ} + 40^{\circ} = 110^{\circ}$ . Check: $70^{\circ} + 110^{\circ} = 180^{\circ}$ , and $110^{\circ}$ is $40^{\circ}$ more than $70^{\circ}$ .

(b) Let the smaller angle be $y^{\circ}$ . Then the larger is $5y^{\circ}$ , and the two add up to $90^{\circ}$ :

$\begin{aligned} y + 5y &= 90 \\ 6y &= 90 \\ y &= 15 \end{aligned}$

The angles are $15^{\circ}$ and $5 \times 15^{\circ} = 75^{\circ}$ . Check: $15^{\circ} + 75^{\circ} = 90^{\circ}$ .

(a) $70^{\circ}$ and $110^{\circ}$ (b) $15^{\circ}$ and $75^{\circ}$

Question 10

Wei Jie opens his laptop so that the angle between the screen and the keyboard is $115^{\circ}$ .

(a) What type of angle is $115^{\circ}$ ?

(b) His laptop's screen can fold all the way back. He pushes it back until it lies flat on the table, in line with the keyboard. What is the angle between the screen and the keyboard now, and what type of angle is it?

ShowHide solution

(a) $115^{\circ}$ is between $90^{\circ}$ and $180^{\circ}$ : obtuse.

(b) The screen and the keyboard now make one straight line: the angle is $180^{\circ}$ , a straight angle.

(a) obtuse (b) $180^{\circ}$ , a straight angle (c) $65^{\circ}$

Question 11

Siti starts her homework at 7.15 p.m. and stops at 7.50 p.m.

(a) Through what angle does the minute hand of her clock turn in that time? What type of angle is it?

(b) Through what angle does the hour hand turn in the same time?

ShowHide solution

(a) From 7.15 p.m. to 7.50 p.m. is $50 - 15 = 35$ minutes. The minute hand turns $35 \times 6^{\circ} = 210^{\circ}$ , which is between $180^{\circ}$ and $360^{\circ}$ : reflex.

(b) The hour hand turns $0.5^{\circ}$ a minute: $35 \times 0.5^{\circ} = 17.5^{\circ}$ .

(a) $210^{\circ}$ , a reflex angle (b) $17.5^{\circ}$

Question 12

Find the angle between the hands of a clock (the one that is not reflex), and state its type, at

(a) 5 o'clock (b) 9 o'clock (c) 3.30.

ShowHide solution

(a) At 5 o'clock the hands point to $12$ and $5$ : $5$ spaces of $30^{\circ}$ , so $5 \times 30^{\circ} = 150^{\circ}$ . Obtuse.

(b) At 9 o'clock the hands point to $9$ and $12$ . Going from $9$ round to $12$ is $3$ spaces: $3 \times 30^{\circ} = 90^{\circ}$ . A right angle.

(c) The minute hand points to $6$ : $6 \times 30^{\circ} = 180^{\circ}$ from $12$ . The hour hand has turned $30 \times 0.5^{\circ} = 15^{\circ}$ past $3$ : $3 \times 30^{\circ} + 15^{\circ} = 105^{\circ}$ from $12$ . The angle between them is $180^{\circ} - 105^{\circ} = 75^{\circ}$ . Acute.

(a) $150^{\circ}$ , obtuse (b) $90^{\circ}$ , a right angle (c) $75^{\circ}$ , acute

Key Points

An angle is made by two rays (its arms) from one point (its vertex); its size is the turn from one arm to the other, in degrees. A complete turn is $360^{\circ}$ .
Name an angle with three letters, the vertex in the middle ( $\angle AOB$ ), or with the vertex alone ( $\angle O$ ) only when there is just one angle there.
Acute: between $0^{\circ}$ and $90^{\circ}$ . Right: $90^{\circ}$ . Obtuse: between $90^{\circ}$ and $180^{\circ}$ . Straight: $180^{\circ}$ . Reflex: between $180^{\circ}$ and $360^{\circ}$ .
$\angle AOB$ alone means the angle that is not reflex; reflex $\angle AOB = 360^{\circ} - \angle AOB$ .
Estimate an angle against $90^{\circ}$ , $45^{\circ}$ and $180^{\circ}$ before you measure it.
To measure with a protractor, put its centre on the vertex and its base line along one arm, and read the scale that reads $0$ on that arm.
Complementary angles add up to $90^{\circ}$ ; supplementary angles add up to $180^{\circ}$ .
On a clock, neighbouring numbers are $30^{\circ}$ apart; the minute hand turns $6^{\circ}$ a minute and the hour hand $0.5^{\circ}$ a minute.

Before you start

What an Angle Is

An angle

Naming Angles

Naming an angle

Example 1: Naming one angle

Example 2: Several angles at one vertex

Acute, Right, Obtuse, Straight and Reflex Angles

Types of angles

Example 3: Classifying angles by their size

Reflex Angles

Reflex angles

Example 4: Finding a reflex angle

Example 5: From the reflex angle back to the angle

Estimating the Size of an Angle

Angles to compare with

Example 6: Estimating and classifying from drawings

Measuring and Drawing Angles with a Protractor

To measure an angle

Example 7: Reading the right scale

To draw an angle, for example ∠ABC=65∘\angle ABC = 65^{\circ}∠ABC=65∘

Example 8: Drawing a reflex angle

Complementary and Supplementary Angles

Complementary and supplementary angles

Example 9: Complements and supplements

Example 10: Two angles that make a right angle

Coming next

Angles Turned by the Hands of a Clock

The hands of a clock

Example 11: The angle a hand turns through

Example 12: The angle between the hands at 4 o'clock

Example 13: At half past two

Common Mistakes

Practice

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Key Points

To draw an angle, for example $\angle ABC = 65^{\circ}$