Angles measure turns. Use the relationship shown in the diagram: 180° on a straight line, 360° around a point, 180° inside a triangle, or equal/supplementary angles formed by parallel lines.
Foundation & Higher6 worked examples10 original questions
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Start with the meaning
What you need to know about angle rules
Imagine turning a pointer about a fixed point. A full turn brings it back to its starting direction and measures 360 degrees, written 360°. A quarter-turn is 90°: a right angle. A half-turn is 180°: the starting and finishing directions form a straight line. An angle measures this amount of turn, not the length of the lines drawn beside it.
See the idea first
A missing part of a known turn
The two adjacent angles below share a side, and their outer sides make a straight line. Together they must make 180°. The letter x stands for the missing angle in degrees.
The two adjacent angles fill one half-turn. They therefore add to 180°, even if the whole drawing is rotated.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Complete a straight line or full turn
What the problem asks: Adjacent angles on a straight line are 65° and x°.
How to solve it: Their total is 180°, so x = 180 − 65 = 115. For angles filling a complete turn around a point, use 360° instead.
Find a triangle angle
What the problem asks: A triangle has angles 48°, 67° and x°.
How to solve it: The three interior angles total 180°, so x = 180 − 48 − 67 = 65. A triangle is isosceles if two sides are equal; its opposite angles are then equal too.
Use crossing straight lines
What the problem asks: Two straight lines cross. One angle is 72°. Find the angle directly opposite.
How to solve it: It is 72°. Both it and the original angle form a straight angle with the same neighbouring angle, so they must be equal. These are vertically opposite angles.
Use a line crossing parallel lines
What the problem asks: Two parallel lines are crossed by another line. Find angles related to a given 68° angle.
How to solve it: Corresponding angles occupy matching corners at the two crossings and are equal. Alternate interior angles lie between the parallel lines on opposite sides of the crossing line and are equal. Interior angles on the same side of the crossing line total 180°, so that partner would be 112°. These rules require parallel lines.
A reliable routine
For a missing interior angle in a triangle
The three angles inside a triangle add to 180°. One way to see why is to draw a line through one vertex parallel to the opposite side: the other two angles can be matched there using alternate angles, making a straight angle with the vertex angle.
Identify the three interior angles belonging to the same triangle. In three-letter notation such as angle BAC, the middle letter A names the vertex of the angle.
Use any equal-side markings to identify equal opposite angles.
Write their sum as 180°, then subtract the known total or solve the resulting equation.
State the angle in degrees and give the reason used.
Check: Do not assume lines are parallel, sides are equal or an angle is 90° just because the sketch looks that way. Use stated information and standard markings. There is no single angle rule that applies to every diagram.
Fully worked
Angle rules GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
A straight line
2 marks
Question
Two adjacent angles lie on a straight line. They are 47° and x°. Find x and give a reason.
x+47=180x=133
So x = 133°. Reason: adjacent angles on a straight line sum to 180°.
Example 2
Around a point
2 marks
Question
Three angles fill a complete turn around a point: 90°, 125° and x°. Find x.
90+125+x=360215+x=360x=145
So x = 145°. Reason: angles around a point total 360°.
Example 3
Triangle total
2 marks
Question
A triangle has interior angles 52°, 73° and x°. Find x.
52+73+x=180125+x=180x=55
The missing angle is 55°, using the triangle angle sum.
Example 4
Isosceles triangle
3 marks
Question
AB = AC in triangle ABC. Angle BAC is 38°. Find angles ABC and ACB.
Matching ticks mark equal sides AB and AC. Their opposite angles, at C and B, are equal. In angle BAC, the middle letter A names the vertex.
The equal sides AB and AC are opposite the angles at C and B, so those angles are equal. Let each be x°.
2x+38=1802x=142x=71
Both base angles are 71°.
Example 5
Parallel lines with a named position
3 marks
Question
Two parallel lines are crossed by another straight line, as shown. Use the given 112° angle to find angles a and b. Give a reason for each answer.
The arrow marks show that the horizontal lines are parallel. The 112° angle and a are interior angles on the same side of the crossing line; 112° and b are alternate interior angles.
(a) The two interior angles on the same side total 180°.
x=180−112=68
So (a) is 68°. (b) is alternate to the original 112° angle and therefore equals 112°. It also forms a straight angle with 68°. The parallel-line condition is essential.
Example 6
Angles given by algebra
Harder4 marks
Question
A triangle has interior angles x°, (2x + 10)° and (3x − 4)°. Find all three angles.
Add the three interior angles.
x+(2x+10)+(3x−4)=1806x+6=1806x=174x=29
The angles are 29°, 68° and 83°. Check: 29 + 68 + 83 = 180.
10 original questions · total 20 marks
Angle rules GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1
Half-turn
1 mark
An angle of 128° and an adjacent angle fill a straight angle. Find the missing angle.
Show worked answer
180−128=52
The missing angle is 52°. Straight-line angles sum to 180°.
2
Full turn
2 marks
Angles of 80°, 145° and x° fill a turn around a point. Find x.
Show worked answer
x=360−80−145x=135
The full turn is 360°.
3
Opposite angle
1 mark
Two straight lines cross. One angle is 63°. Find its vertically opposite angle.
Show worked answer
It is 63°, because vertically opposite angles are equal. The adjacent angle would instead be 117°.
4
Triangle
2 marks
A triangle has angles 41°, 86° and x°. Find x.
Show worked answer
x=180−41−86x=53
The interior angles total 180°.
5
Right triangle
2 marks
A right-angled triangle has another angle of 34°. Find its third angle.
Show worked answer
x=180−90−34x=56
A right angle contributes 90° to the total.
6
Isosceles base angles
3 marks
Triangle PQR has PQ = PR. Its angle at P is 46°. Find the other two angles.
Show worked answer
The angles opposite PQ and PR are equal.
2x=180−462x=134x=67
The angles at Q and R are both 67°.
7
Isosceles vertex
2 marks
An isosceles triangle has two equal base angles of 72°. Find its vertex angle.
Show worked answer
x=180−72−72x=36
Count both equal angles before subtracting.
8
Corresponding angles
1 mark
A line crosses two parallel lines. An angle at one crossing is 57°. Find the corresponding angle at the other crossing.
Show worked answer
It is 57°. Corresponding angles are in matching positions at the two crossings and are equal when the lines are parallel.
9
Same-side interior angles
2 marks
Two interior angles on the same side of a line crossing parallel lines are 104° and x°. Find x.
Show worked answer
x+104=180x=76
These angles are supplementary: their sum is 180°, not their individual values.
10
Algebraic triangle
Harder4 marks
A triangle's angles are x°, (x + 20)° and 2x°. Find them.
Show worked answer
x+(x+20)+2x=1804x+20=1804x=160x=40
The angles are 40°, 60° and 80°. Their total is 180°.
Examiner-style feedback
Common angle rules mistakes
Using 360° for a triangle
360° describes a complete turn around a point; a triangle's three interior angles total 180°. Identify the shape before choosing a total.
Assuming parallel lines
Corresponding and alternate angles are guaranteed equal only with parallel lines.
Giving a number without a reason
When asked, name the relationship used: for example, angles in a triangle sum to 180°. The calculation alone does not explain the geometry.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
An angle measures a turn.
A straight angle is 180° and a full turn 360°.
Triangle interior angles total 180°.
Use markings and state the relevant reason.
Quick answers
Angle rules FAQ
Are co-interior angles equal?
For parallel lines, co-interior or same-side interior angles add to 180°. They are equal only in the special case where both are 90°.
What do F and Z shapes mean?
They are memory aids for corresponding and alternate angles. The actual condition is a line crossing parallel lines. That crossing line is called a transversal; the whole diagram can be rotated.
G1 and G3: angle notation, straight lines, points, triangles, isosceles triangles and parallel-line relationships across tiers. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
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