Sheet № 215 · Foundation + Higher · AQA · Edexcel · OCR
Angles in a Triangle –
Angles in a triangle is one of the most fundamental topics in GCSE Maths, tested at both Foundation and Higher tiers on every exam board. You must know that the angles in any triangle sum to 180°, recognise the special properties of isosceles and equilateral triangles, and apply the exterior angle theorem. This guide covers all the key ru
§Key definitions
Question:
A triangle has angles of 72° and 53°. Find the third angle.
Answer:
The third angle is 55°.
Q1 (Foundation):
An isosceles triangle has a base angle of 68°. Find the other angles.
Q2 (Foundation):
An exterior angle of a triangle is 130°. One of the opposite interior angles is 55°. Find the other opposite interior angle.
Q3 (Higher):
In triangle ABC, angle A = (3x + 10)°, angle B = (2x + 20)°, and angle C = (x + 30)°. Find x and determine whether the triangle is scalene, isosceles, or equilateral.
§Formulas to memorise
Angle sum of a triangle = 180°
Exterior angle of a triangle = sum of the two opposite interior angles
Equilateral triangle: — All three sides are equal and all three angles are 60°.
Isosceles triangle: — Two sides are equal and the base angles (the angles opposite the equal sides) are equal.
Scalene triangle: — No sides are equal and no angles are equal.
Right-angled triangle: — One angle is exactly 90°, so the other two must sum to 90°.
If the triangle is isosceles, use the base-angle rule to set two angles equal before solving.
The exterior angle equals the sum of the two non-adjacent interior angles.
Worked example
A triangle has angles of 72° and 53°. Find the third angle.
Working:
⚠ Common mistakes
- ✗Using 360° instead of 180°. The angle sum of a triangle is 180°, not 360° (which is the sum for a quadrilateral).
- ✗Assuming all isosceles triangles have the equal angles at the base. The two equal angles are always opposite the two equal sides — make sure you identify which angles they are.
- ✗Forgetting to check answers. Always verify that your three angles sum to exactly 180°.
✦ Exam tips
- →State the angle rule you are using (e.g. "angles in a triangle sum to 180°") — this earns a reasoning mark in "show that" or "give a reason" questions.
- →In multi-step problems, combine angle-in-a-triangle rules with angles on a straight line (180°), vertically opposite angles, and angles in parallel lines.
- →When angles are given as algebraic expressions, form an equation, solve for the variable, then substitute back to find each angle.
- →Look for isosceles triangles hidden in diagrams — they often appear inside circles (two radii form an isosceles triangle).