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Hi, I'm Miss Davies.

In this lesson we're going to find missing angles when two or more regular polygons are joined together.

In this diagram we have an equilateral triangle and a square and an angle x in between them.

We know that interior angles in triangles sum to 180 degrees.

This means that each angle in an equilateral triangle is 60 degrees.

Interior angles in quadrilaterals sum up to 360 degrees, so each angle in a square is 90 degrees.

Our x angle is located on a straight line with the 60 degree and 90 degree angles on either side of it.

Angles on a straight line sum to 180 degrees.

To work out angle x, we're going to add together 60 and 90, which gives us 150.

We can then subtract this from 180 to give an answer of 30 degrees.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

We know that each interior angle in a regular pentagon is 108 degrees.

Around the point we have 108 degrees, 108 degrees, and the angle x.

Because angles around a point sum to 360, the calculation we need to do is 360 take away two times 108.

This gives us 360 subtract 216, which gives us an answer of 144 degrees.

In question two, we've got the two lots of 108, a 90 degree, which is the interior angle of a square, and the angle x.

The calculation we need to do is 360 subtract two lots of 108 and subtract 90.

This gives us 360 subtract 306, which gives an answer of 54 degrees.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

In question three, we've got the interior angle of a regular pentagon, which is 108 degrees, the interior angle of a regular hexagon, which is 120 degrees.

To work out the angle x, we need to find the sum of 108 and 120 and subtract this from 360.

This gives an answer of 132 degrees.

In question four, we've got the interior angle of a regular pentagon, which is 108, and the interior angle of a square, which is 90.

To find the angle x, we need to do 108 subtract 90, which gives an answer of 18 degrees.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here's the answer.

We have got an equilateral triangle, which has got an interior angle of 60 degrees, a square, which has an interior angle of 90 degrees, and a regular hexagon, which has an interior angle of 120 degrees.

These are all meeting at the point along with angle x.

These three angles sum to 270 degrees.

If we take this from 360, as we know that there are 360 degrees around a point, it gives us an answer of 90 degrees.

This image shows two congruent regular octagons.

This means that both of the octagons have all equal sides.

When the two octagons have crosses, they have formed a hexagon.

This hexagon has all equal sides, but not equal angles.

We have been asked to find the size of the angle labelled x.

To do this, we're going to start off by stating that the interior angles in an octagon sum to 1,080 degrees.

This means that each interior angle in a regular octagon is 135 degrees.

The four angles that I've labelled in the hexagon are 135 degrees, as they are also interior angles of the octagons.

Because the interior angles in a hexagon sum to 720, we're going to add together the four angles that we've already got, which gives us 540.

We're then going to subtract this from 720.

This means that the top and the bottom angles in our hexagon sum to 180 degrees.

These two angles are equal to each other, as the hexagon has a line of symmetry, going both across and down.

To work out the size of angle x, we're going to divide 180 by two.

This means that angle x is 90 degrees.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here's the answer.

The two hexagons crossing forms a rhombus.

A rhombus has all sides equal and two sets of two equal angles.

The left and the right angle are both equal to 120 degrees, as they are the interior angles of the hexagons.

These sum to 240 degrees.

If we subtract this from 360, as any quadrilateral has angles that sum to 360 degrees, this leaves us with 120, meaning the top and the bottom angles of the rhombus sum to 120 degrees.

These two angles are equal to each other.

So to work out the size of angle x, we need to divide 120 by two.

This gives us an answer of 60 degrees.

That's all for this lesson.

Thanks for watching.