6.3 Angles in Triangles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: types of triangle, the angle sum, isosceles triangles and the exterior angle of a triangle. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Name and describe the different types of triangle
• Use the fact that the angles in a triangle add up to 180°
• Use the equal base angles of an isosceles triangle
• Use the exterior angle of a triangle, and solve problems with algebra

Key words

• Equilateral: 3 equal sides and 3 equal angles (each 60°).
• Isosceles: 2 equal sides. The 2 angles opposite them (the base angles) are equal.
• Scalene: No equal sides and no equal angles.
• Acute / obtuse: An acute triangle has all angles less than 90°. An obtuse triangle has one angle more than 90°.
• Exterior angle: The angle between one side of a shape and the next side made longer.

Types of triangle

A triangle can have two names, e.g. a right-angled isosceles triangle has angles 90°, 45° and 45°.

The angle sum of a triangle

Key fact: The three angles in any triangle add up to 180°.

Worked example: two angles known

A triangle has angles 48° and 67°. Find the third angle.
48 + 67 = 115. 180 − 115 = 65

Answer: 65°

Worked example: a right-angled triangle

One angle is 90° and another is 34°. 180 − 90 − 34 = 56

Answer: 56°

Isosceles and equilateral triangles

In an isosceles triangle the two base angles are equal. Look for the dashes on the equal sides: the equal angles are at the ends of the third side.

Worked example: the top angle is known

An isosceles triangle has a top (apex) angle of 40°. Find each base angle.
180 − 40 = 140. 140 ÷ 2 = 70

Answer: 70° each

Worked example: a base angle is known

One base angle is 52°. The other base angle is also 52°.
Apex angle = 180 − 52 − 52 = 76

Answer: 76°

Key fact: Each angle of an equilateral triangle is 180° ÷ 3 = 60°.

The exterior angle of a triangle

Make one side of a triangle longer. The exterior angle equals the sum of the two interior opposite angles.

Worked example: why it works

Angles on a straight line: exterior angle = 180 − c.
Angles in a triangle: a + b = 180 − c. So the exterior angle = a + b.
If a = 55° and b = 70°, the exterior angle is 55° + 70° = 125°.

Answer: 125°

Using algebra

Worked example: angles in terms of x

The angles of a triangle are x, 2x and 3x. Find each angle.
6x = 180, so x = 30. Angles: 30°, 60°, 90°.

Answer: 30°, 60°, 90° (a right-angled triangle)

Common mistakes

• Halving the wrong angle in an isosceles triangle: only the two base angles are equal.
• Assuming there is only one answer when you are told “one angle of an isosceles triangle is 70°”: it could be the apex or a base angle.
• Adding the exterior angle to the interior angles. Exterior angle = sum of the two interior opposite angles.
• Forgetting the right angle when it is shown only by a small square.

Revision questions

Try these, then check your answers below each one.

Question 1. Two angles of a triangle are 46° and 79°. Work out the third angle.

Answer: 55 °

Working: 180 - 46 - 79 = 55°

Question 2. The top (apex) angle of an isosceles triangle is 34°. Work out the size of each base angle.

Answer: 73 °

Working: (180 - 34) ÷ 2 = 146 ÷ 2 = 73°

Question 3. One side of a triangle is extended to make an exterior angle. The two interior opposite angles are 38° and 77°. Work out the exterior angle.

Answer: 115 °

Working: 38 + 77 = 115°

Related lessons

• Previous lesson: 6.2 Angles in Parallel Lines
• Next lesson: 6.4 Exterior and Interior Angles
• In this chapter: 6.1 Properties of Shapes
• In this chapter: 6.5 More Exterior and Interior Angles
• In this chapter: 6.6 Geometrical Problems

Continue in the full course

This lesson is section 6.3 (EGCMF0603) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.