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Geometry - Sum of angles in a triangle

Grade 6IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The sum of the interior angles in any triangle is always 180∘180^\circ.

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An equilateral triangle has three equal sides and three equal angles, each measuring 60∘60^\circ.

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An isosceles triangle has two equal sides and two equal base angles.

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A right-angled triangle contains one angle of 90∘90^\circ.

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A scalene triangle has no equal sides and no equal angles, but the sum remains 180∘180^\circ.

📐Formulae

Angle A+Angle B+Angle C=180∘\text{Angle } A + \text{Angle } B + \text{Angle } C = 180^\circ

Missing Angle=180∘−(Sum of two known angles)\text{Missing Angle} = 180^\circ - (\text{Sum of two known angles})

Base Angle of Isosceles Triangle=(180∘−Vertex Angle)÷2\text{Base Angle of Isosceles Triangle} = (180^\circ - \text{Vertex Angle}) \div 2

💡Examples

Problem 1:

In a triangle, two angles are 45∘45^\circ and 85∘85^\circ. Find the third angle xx.

Solution:

x=50∘x = 50^\circ

Explanation:

To find the missing angle, add the known angles: 45∘+85∘=130∘45^\circ + 85^\circ = 130^\circ. Subtract this sum from 180∘180^\circ: 180∘−130∘=50∘180^\circ - 130^\circ = 50^\circ.

Problem 2:

An isosceles triangle has a vertex angle (the angle between the two equal sides) of 40∘40^\circ. Find the size of the two base angles.

Solution:

70∘70^\circ each

Explanation:

The sum of all angles is 180∘180^\circ. First, subtract the vertex angle: 180∘−40∘=140∘180^\circ - 40^\circ = 140^\circ. Since the two base angles are equal in an isosceles triangle, divide the remainder by 2: 140∘÷2=70∘140^\circ \div 2 = 70^\circ.

Problem 3:

One angle of a right-angled triangle is 32∘32^\circ. Calculate the third angle.

Solution:

58∘58^\circ

Explanation:

A right-angled triangle always contains a 90∘90^\circ angle. The sum of the two non-right angles must be 90∘90^\circ. Therefore, 90∘−32∘=58∘90^\circ - 32^\circ = 58^\circ.