Myriora Maths › 11+ maths › Chapter 4: Shape puzzles

Hidden angles

11+Chapter 4 · Lesson 3

In this lesson

Learn the rules: a straight line is a hundred and eighty, a triangle is a hundred and eighty, and a full turn is three hundred and sixty.
Work out one angle at a time, from what you know.
And never measure or guess from the picture.

Your turn

An isosceles triangle has a top angle of forty degrees. What are the two bottom angles?

Show the answer

The three angles add up to a hundred and eighty. Take away the forty: a hundred and forty is left. The two bottom angles are equal, so each one is half of a hundred and forty: seventy degrees!

The common mistake

A common mistake is guessing from the picture. Angle x looks like it might be a right angle. But exam diagrams are often not drawn to scale! Always use the rules: x is sixty degrees.

Try it at home

Draw a triangle, tear off its three corners and line them up: they make a straight line.

Read the whole lesson

Hello, and welcome to Myriora Maths! Can you find the angle marked x? There's no protractor allowed. You have to work it out, one step at a time!

You need just a few angle rules. Angles on a straight line add up to a hundred and eighty degrees. Angles in a triangle add up to a hundred and eighty degrees too. Angles all the way round a point add up to three hundred and sixty. And when two lines cross, the opposite angles are equal.

Start with what you can work out. The angle inside the triangle, at the bottom right, sits on a straight line with the hundred and ten. So it's a hundred and eighty take away a hundred and ten: seventy degrees.

Now the triangle has two angles we know: fifty and seventy. The three angles add up to a hundred and eighty. So x is a hundred and eighty, take away fifty, take away seventy: sixty degrees!

Why a hundred and eighty on a straight line? A full turn is three hundred and sixty degrees. A straight line is half a turn: a hundred and eighty.

Here's another rule. When two straight lines cross, the opposite angles are equal: this one is fifty degrees, so the one opposite is fifty too. And the angles next to them are a hundred and eighty take away fifty: a hundred and thirty.

One more: an isosceles triangle has two equal sides, marked with little dashes. So the two angles at the bottom of those sides are equal too.

A common mistake is guessing from the picture. Angle x looks like it might be a right angle. But exam diagrams are often not drawn to scale! Always use the rules: x is sixty degrees.

Your turn! An isosceles triangle has a top angle of forty degrees. What are the two bottom angles? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! The three angles add up to a hundred and eighty. Take away the forty: a hundred and forty is left. The two bottom angles are equal, so each one is half of a hundred and forty: seventy degrees!

Quick review. Learn the rules: a straight line is a hundred and eighty, a triangle is a hundred and eighty, and a full turn is three hundred and sixty. Work out one angle at a time, from what you know. And never measure or guess from the picture. See you next time!