Angles around a point and on a straight line

​​In a nutshell

Angles on a straight line add up to $$180\degree$$ and angles around a point add up to $$360\degree$$. You can use this knowledge to work out missing angles.

Angles on a straight line

Angles on a straight line add up to $$180\degree$$. One way to see this is to look at two right angles lined up side by side. 

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A right-angle is an angle that is exactly $$90\degree$$. When two right angles are lined up together, a straight line is formed. Therefore, angles on a straight line add up to $$180\degree$$ because $$90 + 90 = 180$$. 

Example 1

Find the missing angle.

The angle given is $$100\degree$$. Angles on a straight line add up to $$180\degree$$, so the missing angle can be worked out by:

$$180 - 100 = 80$$

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

Angles around a point 

Angles around a point add up to $$360\degree$$. Right-angles can be used to prove this again.

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As right-angles are angles which are exactly $$90\degree$$​, angles around a point must be $$90 + 90 + 90 + 90 = 360\degree$$.

Example 2

Find the missing angle.

The angle given is $$140\degree$$. Angles around a point add up to $$360\degree$$.​

Therefore, the missing angle can be worked out by:

$$360-140=220$$​​

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

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Vertically opposite angles

When two lines intersect, they create a cross. When this happens, the opposite angles they create are equal. 

In the case of this diagram, due to the cross these lines create, the angles A and B are equal. 

Example 3

If angle $$A$$​ is $$60\degree$$​, work out angle $$B$$.

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Angles that are vertically opposite are equal, so angle A = C = $$60\degree$$.

This means that:

$$A + C = 120\degree$$​​​

Angles around a point add up to $$360\degree$$. This means that:

​$$B + D = 360 - 120 = 240$$​​

B and D are opposite, and therefore equal, so:

​$$B = \dfrac{240}{2} = 120$$

$$B=\underline{120\degree}$$​​​​