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Geometry - Sum of angles in a triangle

Grade 6Cambridge (IGCSE)

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

🔑Concepts

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The fundamental property of any triangle is that the three interior angles always add up to 180∘180^\circ. This rule applies regardless of the triangle's shape or size.

Triangle with angles labeled a, b, and c illustrating that a + b + c = 180 degrees.
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In an Equilateral Triangle, all three sides are equal, which means all three angles are also equal. Each angle measures exactly 60∘60^\circ because 180∘÷3=60∘180^\circ \div 3 = 60^\circ.

Equilateral triangle with all three interior angles labeled as 60 degrees.
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An Isosceles Triangle has two equal sides and two equal 'base' angles. If you know the vertex angle, you can subtract it from 180∘180^\circ and divide the remainder by 22 to find the base angles.

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A Right-Angled Triangle contains one angle that is exactly 90∘90^\circ. The other two acute angles must sum to 90∘90^\circ to make the total 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.

Problem 4:

In the triangle shown, find the value of the missing angle yy. The given angles are 115∘115^\circ and 35∘35^\circ.

An obtuse triangle with angles 35, 115, and y.

Solution:

  1. Sum of known angles: 115∘+35∘=150∘115^\circ + 35^\circ = 150^\circ
  2. Subtract from total: y=180∘−150∘y = 180^\circ - 150^\circ
  3. y=30∘y = 30^\circ

Explanation:

Since the sum of angles in a triangle is always 180∘180^\circ, we subtract the sum of the two given angles from 180∘180^\circ to find the remaining angle.

Problem 5:

In an isosceles triangle, the two equal base angles are each 72∘72^\circ. Calculate the size of the vertex angle xx.

An isosceles triangle with base angles of 72 degrees and a vertex angle x.

Solution:

  1. Sum of base angles: 72∘×2=144∘72^\circ \times 2 = 144^\circ
  2. Subtract from total: x=180∘−144∘x = 180^\circ - 144^\circ
  3. x=36∘x = 36^\circ

Explanation:

In an isosceles triangle, two angles are identical. We add the two equal base angles together and subtract that total from 180∘180^\circ to find the vertex angle.