Properties of triangles is a foundational GCSE Maths topic tested at both tiers on AQA, Edexcel, and OCR papers. You need to classify triangles by their sides and angles, recall the properties of each type, and use these properties to solve problems. This guide covers every triangle type with clear definitions, worked examples, and practice questions.
What Are the Types of Triangle?
A triangle is a 2D shape with three straight sides and three angles that sum to 180°. Triangles are classified in two ways: by their sides and by their angles.
Classification by Sides
| Type | Sides | Angles |
|---|---|---|
| Equilateral | All 3 sides equal | All 3 angles = 60° |
| Isosceles | 2 sides equal | 2 base angles equal |
| Scalene | No sides equal | No angles equal |
Classification by Angles
| Type | Largest angle |
|---|---|
| Acute-angled | All angles less than 90° |
| Right-angled | One angle exactly 90° |
| Obtuse-angled | One angle greater than 90° |
A triangle can be described by both classifications — for example, a "right-angled isosceles triangle" has a 90° angle and two equal sides (with the two equal angles each being 45°).
Key Properties
Step-by-Step Method
Identifying a Triangle Type
- Look at the side lengths — are any equal? Are all equal?
- Look at the angles — is there a 90° angle? An angle greater than 90°?
- Use both pieces of information to classify fully (e.g. "obtuse-angled scalene").
Using Properties to Find Missing Angles or Sides
- If the triangle is equilateral, all angles are 60° and all sides are equal.
- If isosceles, identify the two equal sides — the angles opposite them are equal.
- Use the angle sum of 180° to find any missing angle.
Worked Example 1 — Foundation Level
Question: A triangle has sides of 5 cm, 5 cm, and 7 cm. Classify the triangle and find all its angles given that the angle opposite the 7 cm side is 88.9°.
Working:
Step 1 — Two sides are equal (5 cm and 5 cm), so the triangle is isosceles.
Step 2 — The angle opposite the 7 cm side is 88.9° (this is the angle between the two equal sides).
Step 3 — The base angles are equal. Each = (180° − 88.9°) ÷ 2 = 91.1° ÷ 2 = 45.55°.
Step 4 — All angles are less than 90°, so it is acute-angled.
Answer: The triangle is an acute-angled isosceles triangle with angles 88.9°, 45.55°, and 45.55°.
Worked Example 2 — Higher Level
Question: Prove that an equilateral triangle has rotational symmetry of order 3.
Working:
Step 1 — All three sides of an equilateral triangle are equal, and all three angles are 60°.
Step 2 — Rotating the triangle 120° about its centre maps each vertex to the position of the next vertex. Since all sides and angles are identical, the rotated triangle looks exactly the same.
Step 3 — This mapping works at 120°, 240°, and 360°, giving three positions in a full turn.
Answer: The equilateral triangle maps onto itself at 120°, 240°, and 360°, so it has rotational symmetry of order 3.
Worked Example 3 — Exam Style
Question: Triangle ABC has AB = AC. Angle BAC = (2x + 30)° and angle ABC = (3x − 5)°. Find x and all three angles.
Working:
Step 1 — Since AB = AC, the triangle is isosceles with base angles ABC and ACB equal. So angle ACB = (3x − 5)°.
Step 2 — Angle sum: (2x + 30) + (3x − 5) + (3x − 5) = 180.
Step 3 — 8x + 20 = 180. So 8x = 160, giving x = 20.
Step 4 — Angle BAC = 2(20) + 30 = 70°. Angle ABC = 3(20) − 5 = 55°. Angle ACB = 55°.
Step 5 — Check: 70 + 55 + 55 = 180°. Correct.
Answer: x = 20. Angles are 70°, 55°, and 55°.
Common Mistakes
- Assuming isosceles means two angles of 45°. The base angles are only 45° in a right-angled isosceles triangle. In general, isosceles means two angles are equal — their value depends on the triangle.
- Confusing the equal sides with the equal angles. The equal angles are opposite the equal sides, not adjacent to them.
- Forgetting that a right-angled triangle can also be isosceles. A triangle with angles 90°, 45°, 45° is both right-angled and isosceles.
Exam Tips
- When justifying your answer in an exam, name the triangle type and state the property you used (e.g. "The triangle is isosceles because AB = AC, so angle ABC = angle ACB").
- Mark equal sides with tick marks and equal angles with arcs on your diagram.
- In circle geometry, two radii form an isosceles triangle — this is a very common setup.
- Always verify your angles sum to 180°.
Practice Questions
Q1 (Foundation): Classify a triangle with sides 6 cm, 8 cm, and 10 cm.
Q2 (Foundation): An isosceles triangle has a base angle of 72°. Find the angle at the top.
Q3 (Higher): A triangle has angles in the ratio 2 : 3 : 4. Find each angle and classify the triangle.
Practise triangle property questions with instant feedback free on GCSEMathsAI.
Related Topics
- Angles in a Triangle — angle sum and exterior angle theorem.
- Properties of Quadrilaterals — side and angle properties of four-sided shapes.
- Congruence and Similarity — comparing triangles.
Summary
Properties of triangles covers classifying triangles by both sides (equilateral, isosceles, scalene) and angles (acute, right-angled, obtuse). The angle sum is always 180°. In an equilateral triangle all angles are 60°; in an isosceles triangle the base angles are equal. These properties are used constantly in GCSE geometry — from simple angle calculations to algebraic problems and circle theorem questions. Always name the triangle type, state the property you are using, and verify that your angles sum to 180°.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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