Interior and Exterior Angles Worksheets | Questions and Revision | MME

Interior and Exterior Angles Worksheets, Questions and Revision

Level 4 Level 5

What you need to know

Interior and Exterior Angles

The interior anglesof a shape are the angles inside the shape.

The exterior anglesare the angles formed between a side-length and an extension.

The interior angle and its adjacent exterior angle add up to $180\degree$.

Exterior Angles

The exterior angles = $\dfrac{360\degree}{\textcolor{red}{n}}$

where $\textcolor{red}{n}$ is the number of sides.

The sum of all the exterior angles will equal $360\degree$

For the triangle shown, we can see it has $\textcolor{red}{3}$ sides.

$\dfrac{360\degree}{\textcolor{red}{3}} = 120\degree$

Interior Angles

For interior angles in a shape we normally want to find the sum of the interior angles.

Sum of interior angles = $(\textcolor{red}{n} - 2) \times 180\degree$

Where $\textcolor{red}{n}$ is the number of sides.

To find the sum of the interior angles for the triangle shown we do the following:

$(\textcolor{red}{3} - 2) \times 180\degree = 180\degree$

This means that

$\textcolor{limegreen}{a} + \textcolor{limegreen}{b} + \textcolor{limegreen}{c} = 180\degree$

Example:

$ABCD$ is a quadrilateral.

Find the missing angle marked $x$

[2 marks]

This is a $4$-sided shape, to work out the interior angles we calculate the following:

$(n-2)\times 180 =360\degree$.

Next we can work out the size of $\angle CDB$ as angles on a straight line add up to $180\degree$

$180 - 121 = 59\degree$

Now we know all 4 interior angles, we get that

$x = 360 - 84 - 100 - 59 = 117\degree$.

You will find algebra involved more in questions on this topic, but as long as you know what the interior and exterior angles add up to then you can write the statement “these angles add up to ___” as an equation which you can then solve. Have a go at the questions below to see.

Example Questions

This shape has 5 sides, so its interior angles add up to,

$180 \times (5 - 2) = 540\degree$

Hence each interior angle is,

$x\degree=540\degree \div 5 = 108\degree$

This shape has 8 sides, so its interior angles add up to,

$180 \times (8 - 2) = 1080\degree$

Hence each interior angle is,

$x\degree=1080\degree \div 8 = 135\degree$

An isosceles triangle has two sides the same length but also two angles the same size. Specifically, the two angles at the base of the triangle, which in this case is on the corners B and C given that the markings on the triangle show that AB and AC are the equal sides.

So, we have that angle ACB is also $34\degree$. Angles in a triangle add up to 180, so we get

$x = 180 - 34 - 34 = 112\degree$.

This shape has 5 sides, so its interior angles must add up to

$180 \times (5 - 2) = 540\degree$.

We can’t find this solution with one calculation as we did previously, but we can express the statement “the interior angles add up to 540” as an equation. This looks like

$33 + 140 + 2x + x + (x + 75) = 540$

Now, this is a linear equation we can solve. Collecting like terms on the left-hand side, we get

$4x + 248 = 540$.

Subtract 248 from both sides to get

$4x = 292$.

Finally, divide by 4 to get the answer:

$x = 292 \div 4 = 73\degree$

This shape has 4 sides, so its interior angles add up to

$180 \times (4 - 2) = 360\degree$.

We don’t have any way of expression two of the interior angles at the moment, but we do have their associated exterior angles, and we know that interior plus exterior equals 180. So, we get

$\text{interior angle CDB } = 180 - (y + 48) = 132 - y$

Furthermore, we get

$\text{interior angle CAB } = 180 - 68 = 112$

Now we have figures/expressions for each interior angle, so we write the sum of them equal to 360 in equation form:

$112 + 90 + 2y + (132 - y) = 360$

Collecting like terms on the left-hand side, we get

$y + 334 = 360$

Then, if we subtract 334 from both sides we get the answer to be

$y = 360 - 334 = 26\degree$.

Level 4-5

Level 4-5

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