The Angles of a Triangle

Draw a triangle on a piece of paper, any triangle you like: tall and thin, short and wide, with a right angle or without. Measure its three angles with a protractor and add them up. Your neighbour draws a completely different triangle and does the same. You both get the same answer. What is it, and why?

Tear off the corners

Here is a way to see it without measuring at all.

Cut out a paper triangle and colour its three corners in three colours.
Tear off the three corners.
Lay them side by side so that their points meet at one place, with no gap and no overlap between them.

The three corners always fit together into a straight line. In the Measuring Angles in Degrees lesson we saw that a straight line is half a turn, a straight angle of 180°. So the three angles of the triangle add up to 180°.

A paper triangle with its three corners coloured red, green and blue. Next to it, the three torn-off corners placed point to point, together making a straight line

Try it with a different triangle. The three angles change, but they still make a straight line. This is true for every triangle: its angles add up to 180°.

Checking with numbers

An equilateral triangle has three equal angles. 180° ÷ 3 = 60°, so each is 60°.
Fold a square piece of paper corner to corner. The triangle has a right angle and two angles of 45°: 90° + 45° + 45° = 180°.
A set square, the triangle ruler in a school pencil case, often has angles of 90°, 60° and 30°: 90° + 60° + 30° = 180°.

Three triangles with their angles written inside: an equilateral triangle with 60°, 60° and 60°; a right-angled triangle with 90°, 45° and 45°; and a right-angled triangle with 90°, 60° and 30°

This also lets you find a missing angle without a protractor. If two angles of a triangle are 50° and 70°, they make 120° together, and 180° - 120° = 60°. The third angle is 60°.

Why not two right angles?

Could a triangle have two right angles? Try to draw one. Start with a side along the bottom of the page and draw a line straight up from each end, at a right angle. The two lines are parallel: they go up side by side and never meet. There is no third corner, so there is no triangle.

A line segment along the bottom with a line going straight up from each end, both at right angles. The two lines run side by side, parallel, and never meet, so they make no third corner

The numbers say the same: two right angles already make 90° + 90° = 180°, leaving nothing for the third angle. For the same reason, a triangle can't have two obtuse angles, or even one right angle and one obtuse angle.

Try it at home

Do the tearing experiment with two very different paper triangles.
Two angles of a triangle are 40° and 100°. How big is the third?
A right-angled triangle has one angle of 35°. How big is the other one that isn't the right angle?

Answers: 2. 40° + 100° = 140°, and 180° - 140° = 40°. 3. 90° + 35° = 125°, and 180° - 125° = 55°.

Words to know

angle sum - all the angles of a shape added together; for every triangle it is 180°

Big idea

The three corners of any triangle fit together into a straight line, so the angles of every triangle add up to 180°. That is why a triangle has at most one right angle or one obtuse angle.