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 rules, worked examples at both tiers, and practice questions for exam preparation.

What Are the Angle Rules in a Triangle?

Every triangle has three interior angles that always add up to 180°. This is the single most important fact for this topic.

Key Rules

• 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°.

Step-by-Step Method

Finding a Missing Angle in a Triangle

1. Add up the known angles.
2. Subtract the total from 180°.
3. If the triangle is isosceles, use the base-angle rule to set two angles equal before solving.

Using the Exterior Angle Theorem

1. Identify the exterior angle — it is formed by extending one side of the triangle.
2. The exterior angle equals the sum of the two non-adjacent interior angles.
3. Use this relationship to find missing angles.

Worked Example 1 — Foundation Level

Question: A triangle has angles of 72° and 53°. Find the third angle.

Working:

Step 1 — Add the known angles: 72° + 53° = 125°.

Step 2 — Subtract from 180°: 180° − 125° = 55°.

Answer: The third angle is 55°.

Worked Example 2 — Higher Level

Question: In an isosceles triangle, the angle between the two equal sides is 34°. Find the base angles.

Working:

Step 1 — The angle between the equal sides is the "top" angle = 34°.

Step 2 — The two base angles are equal. Let each base angle = x.

Step 3 — 34° + x + x = 180°, so 34° + 2x = 180°.

Step 4 — 2x = 146°, giving x = 73°.

Answer: Each base angle is 73°.

Worked Example 3 — Exam Style

Question: In triangle PQR, angle P = (2x + 10)°, angle Q = (3x − 5)°, and angle R = (x + 25)°. Find the value of x and each angle.

Working:

Step 1 — Angles sum to 180°: (2x + 10) + (3x − 5) + (x + 25) = 180.

Step 2 — Simplify: 6x + 30 = 180.

Step 3 — 6x = 150, so x = 25.

Step 4 — Angle P = 2(25) + 10 = 60°. Angle Q = 3(25) − 5 = 70°. Angle R = 25 + 25 = 50°.

Step 5 — Check: 60 + 70 + 50 = 180°. Correct.

Answer: x = 25. Angles are 60°, 70°, and 50°.

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).

Practice Questions

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.

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Related Topics

• Angles: Basic Rules and Parallel Lines — foundational angle facts.
• Properties of Triangles — classifying triangles by sides and angles.
• Exterior Angle Theorem — deeper look at exterior angles.

Summary

Angles in a triangle always sum to 180°. Equilateral triangles have three 60° angles, isosceles triangles have two equal base angles, and the exterior angle of any triangle equals the sum of the two opposite interior angles. These rules underpin many multi-step geometry questions at GCSE. Always state the angle rule you are using, check your answers add to 180°, and combine with other angle facts for more complex problems.