Maths · 5 Geometry

5.5 Angles

Before you start

The types of angle: acute (less than $90 °$ ), right ( $90 °$ ), obtuse (between $90 °$ and $180 °$ ) and reflex (between $180 °$ and $360 °$ ).
A full turn is $360 °$ and a half turn is $180 °$ .
Solving linear equations, including ones with the unknown on both sides, such as $5 x - 20 = 3 x + 12$ .
The words isosceles, equilateral, quadrilateral and parallel, and what they mean.
Polygon names: pentagon (5 sides), hexagon (6), octagon (8), decagon (10).

Syllabus objectives covered (3)

Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°.
Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary).
Know and use angle properties of regular and irregular polygons.

Part 1 of 7

Naming an angle with three letters

An angle is the amount of turn between two lines that meet at a point. The point where they meet is the vertex of the angle, and the two lines are its arms.
If only two lines meet at a point $B$ , 'the angle at $B$ ' is clear. If three or more lines meet at $B$ , there are several angles at $B$ , and 'angle $B$ ' could mean any of them. That is why angles are named with three letters.
graph diagram — not drawn yet
A point B with three straight arms going out from it to points A, C and D, spread over about 150 degrees; angle ABC marked with one small arc and labelled 'angle ABC', and angle CBD marked with a second arc in a different colour and labelled 'angle CBD'.
Three lines meet at B, so 'angle B' is not clear. Angle ABC names exactly one of the angles there.
In angle $A B C$ the middle letter, $B$ , is the vertex. The first and last letters are points on the two arms. Read it as a path: start at $A$ , go in to the vertex $B$ , then go out along the other arm to $C$ .
Angle $A B C$ and angle $C B A$ are the same angle, because the vertex $B$ is still in the middle. Angle $B A C$ is a different angle: its vertex is $A$ .
You may see the same angle written as angle $A B C$ , $∠ A B C$ or $A \hat{B} C$ . All three mean the angle at $B$ between $B A$ and $B C$ .
Angle $A B C$ means the angle that is less than $180 °$ . If a question wants the angle on the other side, it says 'reflex angle $A B C$ '.

Part 2 of 7

Angles at a point, on a straight line, and vertically opposite

A full turn is $360 °$ . So angles that fit together all the way round one point, with no gaps and no overlaps, add up to $360 °$ . The reason to write is 'angles at a point add up to $360 °$ '.
$a + b + c + d = 360 °$
A straight line is half a turn. So angles that sit side by side at one point on a straight line, all on the same side of it, add up to $180 °$ . The reason to write is 'angles on a straight line add up to $180 °$ '.
The trap is using $180 °$ when the outer arms do not form one straight line. Use it only when the question tells you the line is straight, for example 'ABC is a straight line', or when the two arms are parts of one line that has been named.
When two straight lines cross, they make four angles. Each angle has one angle directly across the vertex from it. These pairs are called vertically opposite angles, and they are equal.
graph diagram — not drawn yet
Two straight lines crossing at a point O at about 60 degrees to each other, making four angles labelled a, b, c and d going anticlockwise round O; a and c shaded in one colour, b and d shaded in a second colour.
a and b are on a straight line, and so are b and c. So a and c are both what is left when b is taken from 180°.
Here is why. Call the four angles $a$ , $b$ , $c$ and $d$ going round. $a$ and $b$ lie on one straight line, so $a + b = 180 °$ . $b$ and $c$ lie on the other straight line, so $b + c = 180 °$ . Both sums equal $180 °$ and both contain $b$ , so $a$ must equal $c$ .
$a + b = b + c \Rightarrow a = c$
'Vertically' here has nothing to do with up and down. It comes from 'vertex': the two angles share a vertex and face opposite ways. They can lie side to side across the page.

Part 3 of 7

Angles made by parallel lines

Parallel lines are always the same distance apart, so they never meet. On a diagram, matching arrowheads drawn on two lines show that they are parallel.
A line that crosses two or more other lines is called a transversal. Where a transversal crosses two lines it makes eight angles: four at each crossing.
The key idea is this. If the two lines are parallel, the transversal crosses both of them at the same slant. So the four angles at one crossing are an exact copy of the four angles at the other crossing.
Corresponding angles are in the same position at each crossing, for example both above the parallel line and to the right of the transversal. Because the second crossing is a copy of the first, corresponding angles are equal.
graph diagram — not drawn yet
Three side-by-side panels, each showing the same two horizontal parallel lines (each marked with one arrowhead) crossed by the same transversal sloping down to the right. Panel 1, titled 'corresponding: equal', shades the angle above the top line to the right of the transversal and the angle above the bottom line to the right of the transversal, with an F shape traced in colour. Panel 2, titled 'alternate: equal', shades the angle below the top line to the left of the transversal and the angle above the bottom line to the right of the transversal, with a Z shape traced. Panel 3, titled 'co-interior: add up to 180°', shades the angle below the top line to the right of the transversal and the angle above the bottom line to the right of the transversal, with a C shape traced.
Corresponding and alternate pairs are equal. A co-interior pair is one acute and one obtuse angle that add up to 180°.
Alternate angles are both between the parallel lines, on opposite sides of the transversal, one at each crossing. Alternate angles are equal. The reason: an alternate angle is vertically opposite one of a corresponding pair, so all three angles are equal.
Co-interior angles are both between the parallel lines, on the same side of the transversal. Co-interior angles add up to $180 °$ , which is what supplementary means. The reason: one angle of the pair lies on a straight line with the angle that corresponds to the other.
Letter shapes help you spot the pairs: corresponding angles sit in an F shape, alternate angles in a Z shape, co-interior angles in a C shape. The letter is only for spotting. In a reason, write the geometrical name, such as 'alternate angles are equal', because the syllabus asks for correct geometrical terminology.
These three facts are true only when the lines are parallel. If the lines are not marked with arrows or stated to be parallel, do not use any of them: corresponding angles on lines that are not parallel are not equal.
The facts also work the other way round. If a pair of alternate angles are equal, or a pair of corresponding angles are equal, or a pair of co-interior angles add up to $180 °$ , the lines are parallel. If co-interior angles add up to anything other than $180 °$ , the lines are not parallel.

Part 4 of 7

Triangles and quadrilaterals

The three angles inside any triangle add up to $180 °$ . The parallel-line facts show why this is always true.
Draw triangle $A B C$ with $B C$ along the bottom and $A$ at the top. Draw the line through $A$ that is parallel to $B C$ . At $A$ there are now three angles on that straight line: the triangle's own angle at $A$ in the middle, and one angle on each side of it.
graph diagram — not drawn yet
Triangle ABC with B bottom left, C bottom right and A at the top; a straight line drawn through A parallel to BC, with BC and the new line each marked with one arrowhead; angle ABC and the angle at A between the new line (to the left) and AB both marked with a single arc; angle ACB and the angle at A between the new line (to the right) and AC both marked with a double arc; angle BAC marked with a triple arc.
The three angles at A lie on a straight line, and they are copies of the three angles of the triangle.
The angle on the left of $A$ and angle $A B C$ are alternate angles, with $A B$ as the transversal, so they are equal. The angle on the right of $A$ and angle $A C B$ are alternate angles, with $A C$ as the transversal, so they are equal too.
So the three angles on the straight line at $A$ are equal to the three angles of the triangle. Angles on a straight line add up to $180 °$ , so the angles of the triangle do too.
$∠ A B C + ∠ B A C + ∠ A C B = 180 °$
An isosceles triangle has two equal sides, and the two angles opposite those sides are equal. An equilateral triangle has all three angles equal, so each one is $180 ° \div 3 = 60 °$ .
If one side of a triangle is extended, the angle between the extension and the next side is an exterior angle of the triangle. It equals the sum of the two interior angles at the other two vertices.
This follows from the two facts you already have. Call the interior angles $a$ , $b$ and $c$ , with $c$ next to the exterior angle. The exterior angle and the interior angle next to it lie on a straight line, so they add up to $180 °$ . All three interior angles also add up to $180 °$ . Take the shared interior angle away from both, and what is left on each side is equal. If you use this shortcut, you can always back it up with those two named facts.
$exterior angle = 180 ° - c = a + b$
Any quadrilateral can be cut into two triangles by one of its diagonals. The angles of the two triangles together make up the four angles of the quadrilateral exactly, so the angles of a quadrilateral add up to $2 \times 180 ° = 360 °$ .
$a + b + c + d = 360 °$
This is true of every quadrilateral, including one with a reflex angle, such as an arrowhead. For an arrowhead, use the diagonal that starts at the reflex angle, so that both triangles are inside the shape.

Part 5 of 7

Setting out a chain of reasons

Many angle questions need more than one step. You cannot reach the angle you want directly, so you find a stepping-stone angle first and use that.
Start from what you know. Mark each given angle, then look for one fact that links a known angle to an unknown one: next to it on a line, across a vertex, in the same triangle, or in a parallel-line pair.
Write each step on its own line: the angle, named with three letters, its value, and the reason in brackets. For example: angle $D E F = 180 ° - 72 ° = 108 °$ (angles on a straight line add up to $180 °$ ).
Make each reason a full fact, not one word. 'Straight line' is only a label. 'Angles on a straight line add up to $180 °$ ' is the property that justifies the number.
When the angles are given as expressions in $x$ , use the fact to write an equation, and solve it. If $(2 x + 40) °$ and $(x + 20) °$ lie on a straight line:
$(2 x + 40) + (x + 20) = 180 \Rightarrow 3 x + 60 = 180 \Rightarrow x = 40$
$x$ is not the angle. Substitute back to get the angles: $2 (40) + 40 = 120 °$ and $40 + 20 = 60 °$ . Then check them against the fact: $120 ° + 60 ° = 180 °$ .
Do not measure with a protractor. The marking rules give no marks for a scale drawing where a calculation is required, so every angle has to come from a fact.

Part 6 of 7

The interior angles of any polygon

A polygon is a closed shape with straight sides. A polygon with $n$ sides has $n$ vertices and $n$ interior angles, one inside each vertex.
Choose one vertex and draw every diagonal from it. In a polygon with no reflex angle, this cuts the shape into triangles. From one vertex you can draw a diagonal to every other vertex except its two neighbours, and that gives $n - 2$ triangles.
graph diagram — not drawn yet
An irregular convex hexagon with vertices labelled A to F, with the three diagonals AC, AD and AE drawn from vertex A, dividing it into four triangles numbered 1, 2, 3 and 4.
6 sides give 6 − 2 = 4 triangles, so the interior angles add up to 4 × 180° = 720°.
The angles of these triangles fit together exactly to make the interior angles of the polygon. Each triangle gives $180 °$ , so:
$sum of interior angles = (n - 2) \times 180 °$
Check it against what you know. A triangle ( $n = 3$ ) gives $1 \times 180 ° = 180 °$ . A quadrilateral ( $n = 4$ ) gives $2 \times 180 ° = 360 °$ . A pentagon gives $540 °$ , a hexagon $720 °$ , an octagon $1080 °$ .
The formula holds for every polygon, regular or irregular. For a polygon with a reflex angle the triangles have to be drawn more carefully, but the total is the same. If you forget the formula, rebuild it from the triangles.
A regular polygon has all its sides equal and all its angles equal. So each interior angle is the sum shared equally between the $n$ vertices. For a regular hexagon: $720 ° \div 6 = 120 °$ .
$each interior angle of a regular polygon = \frac{(n - 2) \times 180 °}{n}$
An irregular polygon does not have all its angles equal, so you cannot divide. Use the sum as an equation instead: the angles you know plus the unknown ones equal $(n - 2) \times 180 °$ .

Part 7 of 7

Exterior angles and regular polygons

At a vertex of a polygon, extend one side beyond the vertex. The angle between the extension and the next side is the exterior angle at that vertex.
The interior angle and the exterior angle at a vertex lie on a straight line, so they add up to $180 °$ .
$interior angle + exterior angle = 180 °$
Imagine walking round a polygon with no reflex angle, along each side in turn. At each vertex you turn through the exterior angle. When you are back at the start, you face the way you began, so your turns add up to exactly one full turn.
graph diagram — not drawn yet
An irregular convex pentagon with each side extended past its vertex in the same direction round the shape (anticlockwise), the five exterior angles marked with arcs and labelled e1 to e5, and a small curved arrow at each vertex showing the turn onto the next side.
Going once round the shape, the five turns make one full turn: e1 + e2 + e3 + e4 + e5 = 360°.
So the exterior angles of a polygon with no reflex angle add up to $360 °$ , whatever the number of sides, regular or not.
$sum of exterior angles = 360 °$
In a regular polygon all the exterior angles are equal, so each one is $360 °$ shared between $n$ vertices. For a regular 12-sided polygon: exterior angle $= 360 ° \div 12 = 30 °$ , so interior angle $= 180 ° - 30 ° = 150 °$ .
$each exterior angle = \frac{360 °}{n} n = \frac{360 °}{each exterior angle}$
To find the number of sides from an interior angle, go through the exterior angle: take the interior angle from $180 °$ , then divide $360 °$ by the result. For an interior angle of $140 °$ : $180 ° - 140 ° = 40 °$ and $360 ° \div 40 ° = 9$ sides.
If $360 °$ divided by the exterior angle is not a whole number, no regular polygon has that angle, because a polygon cannot have part of a side. An interior angle of $130 °$ gives an exterior angle of $50 °$ , and $360 \div 50 = 7.2$ , so there is no regular polygon with interior angle $130 °$ .

Worked examples

Example 1. The straight lines $A B$ and $C D$ cross at $O$ . Angle $A O C = (3 x + 12) °$ and angle $B O D = (5 x - 20) °$ . Find the value of $x$ , then find angle $C O B$ . Give a reason for each step.

Show the working

Angle $A O C$ is between the arms $O A$ and $O C$ . Angle $B O D$ is between $O B$ and $O D$ . $O B$ continues the line from $A$ through $O$ , and $O D$ continues the line from $C$ through $O$ , so the two angles face each other across $O$ : they are vertically opposite.
Vertically opposite angles are equal, so set the two expressions equal to each other.
$5 x - 20 = 3 x + 12$
Subtract $3 x$ from both sides to collect the $x$ terms, then add $20$ to both sides.
$2 x = 32 \Rightarrow x = 16$
$x$ is not an angle yet. Substitute to find angle $A O C$ , and check with angle $B O D$ .
$∠ A O C = 3 (16) + 12 = 60 °, ∠ B O D = 5 (16) - 20 = 60 °$
Angle $C O B$ is next to angle $A O C$ , and together they fill one side of the straight line $A B$ at $O$ . Angles on a straight line add up to $180 °$ .
$∠ C O B = 180 ° - 60 ° = 120 °$

Answer: $x = 16$ , and angle $C O B = 120 °$ (vertically opposite angles are equal; angles on a straight line add up to $180 °$ ).

Example 2. In triangle $P Q R$ , $P Q = P R$ and angle $Q P R = 48 °$ . The side $Q R$ is extended beyond $R$ to a point $S$ , so that $Q$ , $R$ and $S$ lie on a straight line. Find angle $P R S$ . Give a reason for each step.

Show the working

$P Q = P R$ , so the triangle is isosceles. The equal angles are the ones opposite the equal sides: angle $P R Q$ is opposite $P Q$ , and angle $P Q R$ is opposite $P R$ . So angle $P Q R$ = angle $P R Q$ .
Angles in a triangle add up to $180 °$ . After the $48 °$ at $P$ , the two equal angles share what is left.
$∠ P Q R = ∠ P R Q = \frac{180 ° - 48 °}{2} = 66 °$
$Q$ , $R$ and $S$ are on a straight line, so angles $P R Q$ and $P R S$ add up to $180 °$ (angles on a straight line).
$∠ P R S = 180 ° - 66 ° = 114 °$
Check with the exterior angle. Angle $P R S$ is the exterior angle of the triangle at $R$ , so it equals the sum of the interior angles at $P$ and $Q$ : $48 ° + 66 ° = 114 °$ . The two methods agree.

Answer: Angle $P R S = 114 °$ (base angles of an isosceles triangle are equal; angles in a triangle add up to $180 °$ ; angles on a straight line add up to $180 °$ ).

Example 3. $A B$ and $C D$ are parallel straight lines, with $A B$ above $C D$ and with $A$ and $C$ on the left. A straight line crosses $A B$ at $G$ and $C D$ at $H$ , and $H$ is below and to the right of $G$ . Angle $B G H = (2 x + 10) °$ and angle $G H D = (3 x - 5) °$ . (a) Find the value of $x$ . (b) Find angle $A G H$ , giving a reason.

Show the working

Locate the two angles. Angle $B G H$ is at $G$ , between $G B$ (to the right) and $G H$ (down the transversal). Angle $G H D$ is at $H$ , between $H G$ (up the transversal) and $H D$ (to the right). Both are between the parallel lines, and both are on the right of the transversal, where $B$ and $D$ are. That makes them co-interior angles.
graph diagram — not drawn yet
Two horizontal parallel lines, the top one labelled A on the left and B on the right, the bottom one labelled C on the left and D on the right, each marked with one arrowhead; a transversal sloping down to the right crossing AB at G and CD at H; angle BGH marked with an arc and labelled (2x + 10)°, angle GHD marked with an arc and labelled (3x − 5)°, and angle AGH marked with a dashed arc and a question mark.
BGH and GHD are on the same side of the transversal and between the parallel lines, so they are co-interior. AGH and GHD are on opposite sides, so they are alternate.
(a) Co-interior angles add up to $180 °$ , so add the expressions and set the total equal to $180$ .
$(2 x + 10) + (3 x - 5) = 180$
Collect the terms and solve.
$5 x + 5 = 180 \Rightarrow 5 x = 175 \Rightarrow x = 35$
Find both angles and check the fact: angle $B G H = 2 (35) + 10 = 80 °$ and angle $G H D = 3 (35) - 5 = 100 °$ , and $80 ° + 100 ° = 180 °$ .
(b) Angle $A G H$ is at $G$ , between $G A$ (to the left) and $G H$ . It is between the parallel lines, on the left of the transversal. Angle $G H D$ is between them on the right. So angles $A G H$ and $G H D$ are alternate angles, and alternate angles are equal.
$∠ A G H = ∠ G H D = 100 °$
Check another way: angles $A G H$ and $B G H$ lie on the straight line $A B$ , and $100 ° + 80 ° = 180 °$ .

Answer: (a) $x = 35$ (b) angle $A G H = 100 °$ (alternate angles are equal).

Example 4. Each interior angle of a regular polygon is $156 °$ . (a) Find the number of sides of the polygon. (b) Find the sum of its interior angles.

Show the working

(a) Go through the exterior angle. The interior and exterior angle at a vertex lie on a straight line.
$exterior angle = 180 ° - 156 ° = 24 °$
The exterior angles of a regular polygon are all equal and add up to $360 °$ . Divide $360 °$ by one of them to count how many there are, which is the number of sides.
$n = 360 ° \div 24 ° = 15$
15 is a whole number, so such a polygon exists.
(b) Use the angle sum with $n = 15$ .
$(15 - 2) \times 180 ° = 13 \times 180 ° = 2340 °$
Check: 15 equal angles of $156 °$ give $15 \times 156 ° = 2340 °$ . The two answers agree.

Answer: (a) 15 sides (b) $2340 °$

Where people go wrong

Mistake: Reading angle $A B C$ as the angle at $A$ , because $A$ comes first.

Instead: The vertex is always the middle letter: angle $A B C$ is at $B$ , between $B A$ and $B C$ .

The outer letters only name the arms. Where several lines meet, the angle at $A$ is a different angle with a different size.

Mistake: Using corresponding, alternate or co-interior angles when the lines are not parallel.

Instead: Look for arrowheads on the lines, or the word 'parallel' in the question, before you use any of the three facts.

The facts come from the transversal crossing both lines at the same slant. Lines that are not parallel are crossed at different slants, so the angle pairs are not equal.

Mistake: Thinking co-interior angles are equal.

Instead: Co-interior angles add up to $180 °$ . One of the pair is acute and the other obtuse, unless both are $90 °$ .

Corresponding and alternate angles are the equal pairs. A co-interior angle is on a straight line with the angle that corresponds to its partner, so the pair makes $180 °$ .

Mistake: Taking $360 ° \div n$ as the interior angle of a regular polygon.

Instead: $360 ° \div n$ is the exterior angle. The interior angle is $180 ° - 360 ° \div n$ .

For a regular hexagon, $360 ° \div 6 = 60 °$ is the turn at each corner. The angle inside the corner is $180 ° - 60 ° = 120 °$ .

Mistake: Giving 'Z angles' or 'F angles' as the reason.

Instead: Write the property in full: 'alternate angles are equal', 'corresponding angles are equal'.

The syllabus expects correct geometrical terminology in reasons. A letter shape describes how the angles look on the page; it does not name a property.

Check yourself

Angle $X Y Z$ is marked on a diagram. At which point is the angle, and which two lines are its arms?
Answer
At $Y$ , the middle letter. Its arms are $Y X$ and $Y Z$ .
Four angles fit together round a point. Three of them are $95 °$ , $140 °$ and $47 °$ . Find the fourth angle, and give a reason.
Answer
$360 ° - 95 ° - 140 ° - 47 ° = 78 °$ (angles at a point add up to $360 °$ ).
The angles of a quadrilateral are $x °$ , $2 x °$ , $(3 x - 20) °$ and $110 °$ . Find $x$ , then find each angle.
Answer
$x + 2 x + 3 x - 20 + 110 = 360$ , so $6 x + 90 = 360$ , $6 x = 270$ and $x = 45$ . The angles are $45 °$ , $90 °$ , $115 °$ and $110 °$ , which add up to $360 °$ .
In triangle $A B C$ , angle $B A C = 38 °$ and angle $A B C = 71 °$ . The side $B C$ is extended beyond $C$ to $D$ . Find angle $A C D$ .
Answer
Angle $A C B = 180 ° - 38 ° - 71 ° = 71 °$ (angles in a triangle), so angle $A C D = 180 ° - 71 ° = 109 °$ (angles on a straight line). Or directly: the exterior angle is $38 ° + 71 ° = 109 °$ .
Two parallel lines are crossed by a transversal. One angle between the parallel lines is $72 °$ . Write down the size of the angle co-interior to it and the size of the angle alternate to it.
Answer
Co-interior: $180 ° - 72 ° = 108 °$ . Alternate: $72 °$ , because alternate angles are equal.
A transversal crosses two lines. A pair of co-interior angles are $75 °$ and $110 °$ . Explain whether the two lines are parallel.
Answer
$75 ° + 110 ° = 185 °$ . Co-interior angles on parallel lines add up to $180 °$ , and these do not, so the lines are not parallel.
Find the sum of the interior angles of an octagon, and the size of each interior angle of a regular octagon.
Answer
Sum $= (8 - 2) \times 180 ° = 1080 °$ . Each angle $= 1080 ° \div 8 = 135 °$ . Check with the exterior angle: $360 ° \div 8 = 45 °$ and $180 ° - 45 ° = 135 °$ .
A regular hexagon and a square share one side and do not overlap. At one end of the shared side, find the angle between the hexagon and the square that is outside both shapes.
Answer
The hexagon's interior angle is $120 °$ and the square's is $90 °$ . The three angles fill the space round the point, so the gap is $360 ° - 120 ° - 90 ° = 150 °$ (angles at a point add up to $360 °$ ).

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. The straight lines $P Q$ and $R S$ cross at the point $O$ . Angle $P O R = 3 x °$ and angle $Q O S = (x + 50) °$ .

(a) Explain why $3 x = x + 50$ . 1 mark No calculator

Mark scheme

B1 angles POR and QOS are vertically opposite, and vertically opposite angles are equal oe

(b) Find the value of $x$ . 2 marks No calculator

Mark scheme

M1 2x = 50 oe
A1 x = 25

(c) Calculate angle $R O Q$ . Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 105; FT 180 − 3 × their x or 180 − (their x + 50)
B1 angles on a straight line [add up to 180°] oe

Question 2. $A B$ and $C D$ are parallel straight lines. $A B$ is above $C D$ , with $A$ and $C$ on the left. $P$ is a point on $A B$ . $Q$ and $R$ are points on $C D$ , with $Q$ to the left of $R$ , so that $P Q R$ is a triangle. Angle $A P Q = 52 °$ and angle $B P R = 63 °$ .

(a) Find angle $P Q R$ . Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 52
B1 alternate angles [are equal] oe

(b) Calculate angle $Q P R$ . Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 65
B1 angles on a straight line [add up to 180°], or angles in a triangle [add up to 180°] with angle PRQ = 63 seen oe

(c) $S$ is a point on $C D$ to the right of $R$ . Calculate angle $P R S$ . Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 117
B1 co-interior angles [add up to 180°] using angle BPR, or angles on a straight line using angle PRQ = 63, or alternate angles using angle APR = 117, or exterior angle of a triangle = sum of the two opposite interior angles oe

Question 3. This question is about the angles of polygons.

(a) Show that each interior angle of a regular decagon (10 sides) is $144 °$ . 2 marks No calculator

Mark scheme

M1 360 ÷ 10 [= 36] or (10 − 2) × 180 [= 1440]
A1 180 − 36 = 144 or 1440 ÷ 10 = 144, with no errors seen

(b) Each interior angle of a different regular polygon is $165 °$ . Calculate the number of sides of this polygon. 2 marks No calculator

Mark scheme

M1 360 ÷ (180 − 165) oe, or (n − 2) × 180 = 165n
A1 24

(c) A hexagon has interior angles of $x °$ , $2 x °$ , $2 x °$ , $(x + 40) °$ , $110 °$ and $150 °$ . Find the value of $x$ . 3 marks No calculator

Mark scheme

B1 angle sum 720 or (6 − 2) × 180 seen
M1 6x + 300 = their 720 oe
A1 x = 70

(d) Explain why no regular polygon has an interior angle of $130 °$ . 2 marks No calculator

Mark scheme

B1 exterior angle 50 and 360 ÷ 50 = 7.2, or 50 is not a factor of 360, or (n − 2) × 180 = 130n giving n = 7.2
B1 the number of sides must be a whole number oe

How this is examined

Reaching an angle through two or three facts in a parallel-lines diagram, with a reason for each step: A Find or Calculate part that ends 'Give a reason for your answer' or 'Give reasons', where the value and the reason are marked separately Each angle given its value, and each step given the property in full: 'alternate angles are equal', 'corresponding angles are equal', 'co-interior angles add up to $180 °$ ', 'angles on a straight line add up to $180 °$ '. The syllabus asks for correct geometrical terms, so 'Z angles' or just 'parallel lines' is not the reason.
Forming and solving an equation from an angle fact: Angles written as expressions such as $(2 x + 10) °$ on a straight line, round a point, at a crossing, in a triangle, quadrilateral or polygon, or as a parallel-line pair; Find the value of $x$ , then an angle An equation that matches the fact is the method, e.g. $(2 x + 10) + (3 x - 5) = 180$ for co-interior angles, or $5 x - 20 = 3 x + 12$ for vertically opposite angles. Solving gives $x$ ; substituting back gives the angle. $x$ is the angle only when the angle is $x °$ .
Finding the number of sides of a regular polygon, or one of its angles: A Calculate or Show (that) part that starts from one interior or exterior angle, or from the number of sides Exterior angle $= 180 ° -$ interior angle, then $n = 360 ° \div$ exterior angle; or the sum $(n - 2) \times 180 °$ , divided by $n$ for one angle. In a Show (that), the given value has to come out of your working, e.g. $360 \div 10 = 36$ then $180 - 36 = 144$ ; writing $144$ alone shows nothing.
Explaining why an angle is impossible, or why two lines are not parallel: A 1- or 2-mark Explain after a polygon or parallel-lines part A calculation and a conclusion that names the fact: ' $360 \div 50 = 7.2$ , which is not a whole number, so there is no regular polygon with this angle', or 'the co-interior angles add up to $185 °$ , not $180 °$ , so the lines are not parallel'.
Reading the three-letter name of the angle you have to find: The question names the angle, e.g. 'Calculate angle $P R S$ ', and your reasons use the same notation The answer is for the angle at the middle letter, between the two named arms. Naming each angle you find with three letters makes clear which angle each value belongs to.

Words to know

Vertex: The point where the two arms of an angle meet, or a corner of a polygon (plural vertices). In angle $A B C$ the vertex is $B$ .
Angles at a point: Angles that fit together all the way round one point. They add up to $360 °$ .
Vertically opposite angles: The pairs of angles directly across from each other where two straight lines cross. They are equal.
Transversal: A line that crosses two or more other lines.
Corresponding angles: Angles in the same position at each place where a transversal crosses two lines. On parallel lines they are equal.
Alternate angles: Angles between two lines, on opposite sides of a transversal, one at each crossing. On parallel lines they are equal.
Co-interior angles: Angles between two lines, on the same side of a transversal. On parallel lines they add up to $180 °$ .
Supplementary: Describes two angles that add up to $180 °$ .
Interior angle: An angle inside a polygon, at one of its vertices.
Exterior angle: The angle between one side of a polygon and the extension of the next side. Interior angle plus exterior angle $= 180 °$ .
Regular polygon: A polygon with all sides equal and all interior angles equal.
Irregular polygon: A polygon whose sides or angles are not all equal.

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