Angles: revision notes and practice

What you will learn

• Choose angle facts using stated geometric information.
• Calculate missing angles on lines and in triangles.
• Distinguish alternate, corresponding and co-interior angles.
• Explain a calculation with a precise geometrical reason.

Key rules

Straight line

Adjacent angles on a straight line add to 180°.

Triangle

The three interior angles of a triangle add to 180°.

Parallel lines

Alternate and corresponding angles are equal; co-interior angles add to 180°, when the lines are parallel.

Worked examples

Example 1: A triangle has angles 46° and 71°. Find its third angle.

1. Add the known angles: 46 + 71 = 117.
2. Subtract from 180: 180 − 117 = 63.

63°

Example 2: A co-interior angle is paired with 112° between parallel lines. Find it.

1. Co-interior angles add to 180°.
2. Calculate 180 − 112 = 68.

68°

A practice question

Two adjacent angles form a straight line. One is 52°. Find the other angle x.

1. Use information rather than appearance

An angle measures the turn between two rays meeting at a vertex. GCSE questions often use degrees, written with the symbol °. A diagram may be drawn for clarity rather than to scale, so a narrow-looking angle is not automatically 30° and an almost-square corner is not automatically 90°. Use labelled values, right-angle marks, parallel arrows and facts stated in the question.

Record which angle is required before calculating. Several different angles can meet at one vertex, and a correct calculation for the wrong region is still the wrong answer. Trace the rays bounding the required angle and notice whether it is inside or outside a triangle or between two lines. This step often decides which angle fact applies and prevents an unnecessary chain of calculations.

2. Angles on a straight line

A half turn measures 180°. Adjacent angles that together form a straight line therefore add to 180°. If two such angles are 67° and x°, write 67 + x = 180, giving x = 113°. The word adjacent means they share a vertex and a boundary, while the remaining outer rays point in opposite directions along the same straight line.

There may be more than two angles in a straight-line arrangement. For three adjacent angles of 35°, 48° and x°, first add the known angles to get 83°, then subtract from 180° to obtain 97°. Do not subtract just one labelled angle and forget another angle occupying the same half-turn. A useful check is to add all the final angles and confirm that they total 180°.

Adjacent angles on a straight line total 180°

3. Angles around a point and opposite angles

A complete turn measures 360°, so angles that fill the space around one point add to 360°. If three angles around a point are 80°, 115° and 95°, their total is 290°, leaving 70° for the remaining angle. Check that your selected regions fill one complete turn without overlapping. A straight-line total of 180° applies to only half that turn.

When two straight lines cross, vertically opposite angles are equal. The opposite pair are across the intersection, not next to each other. If one angle is 58°, its vertically opposite angle is also 58°, while an adjacent angle is 122° because it completes a straight line. These facts work together, but naming the correct relationship matters when a question asks you to justify an answer.

4. Interior angles of a triangle

The three interior angles of a triangle add to 180°. For angles 46°, 71° and x°, the known total is 117°, so x = 180 − 117 = 63°. Interior means inside the triangle at its vertices. An exterior angle drawn outside the triangle is a different region and cannot simply replace an interior angle in the three-angle sum.

Special triangles give extra information. An isosceles triangle has two equal sides and equal angles opposite those sides. An equilateral triangle has three equal angles, each 60°. A right-angled triangle contains a 90° angle, leaving 90° for the other two interior angles together. Only use these properties when the triangle type, equal-side markings or right-angle mark is actually given.

A + B + C = 180° for a triangle

5. Recognise alternate and corresponding angles

A transversal is a line crossing two other lines. When those two lines are parallel, alternate interior angles are equal. They lie between the parallel lines on opposite sides of the transversal. A marked alternate interior angle of 74° therefore gives 74° for its partner. The parallel condition is essential: a similar-looking arrangement with non-parallel lines does not guarantee equality.

Corresponding angles are also equal for parallel lines. They occupy matching positions at the two crossings, such as the upper-right region at each intersection. Sketching or tracing the transversal can help you compare positions. Avoid relying solely on remembered letter shapes, which can look different when a diagram is rotated. State “alternate angles between parallel lines” or “corresponding angles” as the reason, rather than saying that they simply look equal.

6. Use co-interior angles correctly

Co-interior angles lie between parallel lines on the same side of a transversal. They add to 180° rather than being equal. If one is 112°, its co-interior partner is 68°. The pair is sometimes called allied or same-side interior angles. Whatever terminology your course uses, the key relationship is a supplementary total, supported by the given parallel lines.

Compare this with alternate interior angles: those are on opposite sides of the transversal and are equal. Confusing the two facts produces a common incorrect answer of 112° instead of 68°. Before calculating, identify whether the two angles are between the lines and whether they lie on the same side or opposite sides of the crossing line. Those positions determine the appropriate rule.

Co-interior angles between parallel lines total 180°

7. Link several facts in one solution

An angle question may require an intermediate angle. For example, a triangle with angles 39° and 84° has third interior angle 57°. If a side is extended at that vertex, the adjacent exterior angle is 180 − 57 = 123°. Show the triangle calculation first, then the straight-line calculation. Using an unexplained final subtraction makes it harder to follow which region was found.

The same approach works with parallel lines and triangles together. Find a corresponding or alternate angle using the parallel condition, then use that value in a triangle sum. Write a reason beside each step and keep the intermediate angle labelled. If an algebraic expression such as 2x + 10 describes an angle, form an equation from the appropriate total before solving and substitute back to find the angle itself.

8. Explain and check your angle answer

A complete answer includes the numerical angle, the degree symbol and any requested geometrical reason. “Angles in a triangle add to 180°” justifies a triangle subtraction. “Co-interior angles between parallel lines add to 180°” justifies the same-side calculation. A vague statement such as “because of the shape” does not identify a mathematical fact and may not earn the reasoning credit.

Check that every calculated triangle angle is positive and below 180°, and that the three angles total 180°. Check a co-interior pair by adding it, and an alternate pair by comparing it. These tests can reveal a missed label or use of the wrong angle region. This guide covers core two-dimensional angle facts; circle theorems, bearings and all polygon applications require further topic coverage.

Check your understanding

How to earn the marks

• Use the given geometric marks and facts rather than measuring the picture.
• Name the angle relationship when reasoning is requested.
• Show intermediate angles in a chain of calculations.
• Check the relevant total or equality and include °.

A common mistake

Alternate angles are equal, but co-interior angles add to 180°. The line positions and parallel condition decide which rule applies.

Course context and official specifications

This guide covers selected skills. Some extensions are Higher content; choose your official tier separately from practice difficulty in the app. These are original questions and explanations.

• AQA GCSE Mathematics — G3, G4

  This bank uses two-dimensional angle facts, parallel lines and polygons, available at both tiers. It does not cover the full geometry specification.

• Edexcel GCSE Mathematics — G3, G4

  This bank uses two-dimensional angle facts, parallel lines and polygons, available at both tiers. It does not cover the full geometry specification.

• OCR GCSE Mathematics — 8.03, 8.04

  This bank uses two-dimensional angle facts, parallel lines and polygons, available at both tiers. It does not cover the full geometry specification.

Keep your next step small

Try five questions, read the feedback, and revisit one thing you found difficult. The app tracks a 20-answer practice goal separately from accuracy.

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