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Properties of triangles

Use the angle sum, the exterior angle rule and Pythagoras' theorem in a triangle.

Every triangle, however it is drawn, has three angles that add up to 180°. That one fact does most of the work: given two angles you can always find the third, and being told a triangle is equilateral or isosceles tells you about its angles before you measure anything at all.

Have a play

A right-angled triangle, with the corner that makes it one marked. Find out what each of its three sides is called.

Tap any part of the picture.

The exterior angle shortcut

Extend one side of a triangle and the angle that opens up outside it is an exterior angle. It equals the two interior angles at the other two corners, added together. You can always reach it the long way instead — 180° minus the interior angle beside it — and the two routes agree, which makes each one a check on the other.

Worked example

A right-angled triangle has shorter sides of 7 cm and 24 cm. How long is the hypotenuse?

  1. The hypotenuse is the side opposite the right angle, and it is always the longest of the three.

    Naming it before calculating stops it being mixed up with one of the shorter sides, which is where most of these answers go wrong.

Pythagoras needs a right angle

With no right angle in the triangle, squaring and adding two sides tells you nothing about the third. Run the rule backwards and it becomes a test instead: if the two shorter sides squared add up to the longest side squared, the triangle is right-angled, and if they do not, it is not.

Try it together

Now one together. A triangle has angles of 55° and 65°, and one of its sides is extended at the third corner.

Find the third angle first. The exterior angle then has two routes to it, and they ought to agree.

    1.The three angles of a triangle add up to 180°. What is the third angle here, in degrees?

    Have a go

    Have a go on your own. An exterior angle of a triangle is 145°, and one of the two opposite interior angles is 60°. How big is the other one, in degrees?

    Hint: The exterior angle is those two interior angles added together.

    Ready to practise?

    Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.

    Print a worksheet