Basic angle rules

​​In a nutshell

There are five rules involving angles on lines and shapes. These rules refer to angles in a triangle, angles on a straight line, angles in a quadrilateral, angles around a point and angles in an isosceles triangle. The rules help to find missing angles on lines and in shapes.

Angles in a triangle

The angles in a triangle add up to $$180\degree$$.

Example 1

Find the size of the missing angle in the triangle:

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Angles in a triangle add up to $$180\degree$$. 

$$\begin{aligned}70\degree+30\degree & =100\degree \\180\degree-100\degree &=80\degree\end{aligned}$$​   

The missing angle is $$\underline{80\degree}$$.

Angles on a straight line

Angles on a straight line add up to $$180\degree$$.

Example 2

Find the size of the missing angle on the straight line:

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Angles on a straight line add up to $$180\degree$$. 

$$180\degree - 100\degree = 80\degree$$

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The missing angle is $$\underline{80\degree}$$.

Angles in a quadrilateral

Angles in a quadrilateral add up to $$360\degree$$.

Example 3

Find the size of the missing angle in the quadrilateral shown:

Angles in a quadrilateral add up to $$360\degree$$. Subtract the sum of the known angles from $$360\degree$$. 

$$\begin{aligned}80\degree+120\degree+40\degree &= 240\degree \\ 360\degree- 240\degree &= 120\degree\end{aligned}$$

The missing angle is $$\underline{120\degree}$$.

Angles around a point

Angles around a point add up to $$360\degree$$.

Example 4

Find the size of the missing angle:

Angles around a point add up to $$360\degree$$, so take $$140\degree$$ away from $$360\degree$$.

$$360\degree- 140\degree= 220\degree$$​​

The missing angle is $$\underline{220\degree}$$.

Angles in an isosceles triangle

Isosceles triangles have two sides equal in length and two equal angles. The two equal angles are sometimes referred to as the 'base angles' in an isosceles triangle. If you know one of the angles in an isosceles triangle, the other two angles can be found easily.

Example 5

The isosceles triangle has equal sides $$AC$$ and $$BC$$​. If $$\angle BAC$$​ is $$50\degree$$, find the size of $$\angle ACB$$​.

$$\angle BAC$$​ and $$\angle ABC$$​ are both $$50\degree$$, as they are the base angles of the triangle. As angles in a triangle add up to $$180\degree$$, you can find $$\angle ACB$$​ by subtracting the sum of $$\angle ABC$$​ and $$\angle BAC$$​ from $$180\degree$$​.

$$\begin{aligned}50\degree+50\degree &= 100\degree \\180\degree - 100\degree &= 80\degree\end{aligned}$$​​

$$\underline{\angle\ ACB\ = 80\degree}$$

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