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Angles in a triangle add to 180°

Geometry · level 4, lesson 3 of 6

A question to start

Draw any triangle you like: tall and thin, short and fat, tiny, or as big as a football field. Measure its three angles and add them up. You will get 180°, or very close to it (measuring is never perfect).

Now the real question. Thousands of different triangles all give the same answer. Is that a coincidence? Is it just that we have not yet found the strange triangle that gives 181°? Or is there a reason, so that no flat triangle anywhere could ever give anything else?

The idea

The fact

The three angles inside any triangle add up to 180°.

Inside angles are also called interior angles. If a triangle has angles \(a\), \(b\) and \(c\), then

\[a + b + c = 180°\]

This lesson gives three ways to see it: two hands-on ones, then a real reason that works for every triangle at once.

Way 1: tear off the corners

Draw a triangle on paper and cut it out. Colour each corner, then tear the three corners off. Put the three points together, side by side, with no gaps and no overlaps.

the three corners make a straight line

They always make a straight line: a straight angle, 180°. Try it with a very different triangle. It still works.

This is convincing, but it is not yet a proof. Tearing is not exact, and a straight-looking line might be off by half a degree. And we have only tried a few triangles. We want a reason.

Way 2: fold the corners in

Here is a neater paper method, with no tearing.

  1. Cut out a triangle and put its longest side at the bottom.
  2. Fold the top corner straight down so that the fold line is parallel to the bottom and the corner just touches the bottom side. The corner lands exactly where a line straight down from it would meet the base (the dot).
  3. Fold the left corner in along an upright fold so that it also touches the dot.
  4. Fold the right corner in the same way.

The three corners now meet at the dot, side by side, along the bottom edge, which is straight. So again they make 180°. And the folded shape is a neat rectangle, half as wide as the base and half as tall as the triangle. The paper lies exactly two layers thick everywhere, so the triangle's area is twice the rectangle's: twice half the base times half the height, which is half of base times height. That is the usual area rule for a triangle, proved by folding (more in Measurement).

Way 3: walk round it (a reason for every triangle)

Now an argument with no paper at all. Imagine walking all the way round the edge of a triangle, like a robot on a track, starting in the middle of the bottom side and facing along it.

turn 1 turn 2 turn 3

At each corner you stop and turn to face along the next side. The amount you turn is the angle between the way you were going (the dashed line, carrying on straight) and the next side. This is called the exterior angle (outside angle) at that corner.

Two facts:

Fact A. At each corner, the turn and the interior angle sit side by side on a straight line (the dashed line continues the side you came along). So

turn \(= 180° -\) interior angle (angles on a straight line).

Fact B. When you get back to the start, you are facing exactly the way you started, and you have turned the same way (left) every time, once round. So the three turns together make one full turn: 360°.

Now put them together. With interior angles \(a\), \(b\), \(c\):

\[(180 - a) + (180 - b) + (180 - c) = 360\]

The left side is \(540 - (a + b + c)\). So \(540 - (a + b + c) = 360\), which means

\[a + b + c = 540 - 360 = 180.\]

This argument never used the size or shape of the triangle, so it works for every triangle. And it gave us a bonus fact, which you will use again in lesson 5: the exterior angles of a triangle add to 360°.

(Fact B is very believable, but a careful mathematician wants a reason for it too. In the next lesson you will meet the facts about angles and parallel lines, and they give the classic proof of the 180° fact, the one Euclid wrote down over two thousand years ago. It is in the next lesson.)

Facts that come free

Once you know the 180° fact, many other facts follow at once.

  • An equilateral triangle has angles of 60°. All three angles are equal (level 3), so each is \(180 \div 3 = 60\).
  • A right triangle's two other angles are complementary. The right angle uses 90°, so the other two share \(180 - 90 = 90\) degrees.
  • The base angles of an isosceles triangle are each half of what is left after the top angle. If the top angle is 40°, the base angles share \(180 - 40 = 140\), so each is 70°.
  • A triangle has at most one right or obtuse angle. Two angles of 90° or more would already use 180° or more, leaving nothing for the third. (You saw a picture reason for this in level 3; now you have a number reason too.)

The exterior angle fact

Extend one side of a triangle past a corner. The outside angle \(d\) that you make is equal to the sum of the two interior angles at the other two corners: \(d = a + b\).

a b d c

Why? Call the interior angle at that corner \(c\).

  1. \(a + b + c = 180°\) (angles in a triangle).
  2. \(c + d = 180°\) (angles on a straight line).
  3. Both \(a + b\) and \(d\) are what you add to \(c\) to make 180°, so \(d = a + b\).

This is a shortcut: you can find the exterior angle without ever finding \(c\).

Worked examples

1. The third angle. A triangle has angles 47° and 68°. Find the third.

\(180 - 47 - 68 = 65°\) (angles in a triangle add to 180°).

2. An isosceles triangle. In an isosceles triangle one base angle is 52°. Find the other two angles.

  • The other base angle is also 52° (base angles of an isosceles triangle are equal).
  • The top angle is \(180 - 52 - 52 =\) 76° (angles in a triangle).

3. Angles in a ratio, with algebra. The angles of a triangle are \(x\), \(2x\) and \(3x\). Find them.

\(x + 2x + 3x = 180\), so \(6x = 180\) and \(x = 30\). The angles are 30°, 60°, 90°: it is a right triangle.

4. Exterior angle. One side of a triangle is extended, making an exterior angle of 125°. One of the two interior angles at the other corners is 70°. Find the other one.

Exterior angle = sum of the two opposite interior angles: \(125 = 70 + ?\), so \(? = 55°\). (Check the long way: the interior angle next to the 125° is \(180 - 125 = 55°\), and \(70 + 55 + 55 = 180\). ✓)

5. Two triangles sharing a side. A quadrilateral \(ABCD\) is cut by the diagonal \(AC\) into two triangles. The angles of the quadrilateral are made of the angles of the two triangles. So the angles of any quadrilateral add to \(180 + 180 = 360°\). That is the fact you previewed in level 3, now with a proper reason. (For a quadrilateral with a corner poking inwards, cut along the diagonal that stays inside.)

Try it

  1. Find the third angle: (a) 35° and 90°; (b) 100° and 25°; (c) 60° and 60°.
  2. An isosceles triangle has a top angle of 30°. Find the base angles.
  3. An isosceles triangle has a base angle of 30°. Find the top angle.
  4. A right triangle has one angle of 28°. Find the third angle.
  5. The angles of a triangle are \(x\), \(x + 20\) and \(x + 40\). Find them.
  6. An exterior angle of a triangle is 140°, and the two interior angles at the other corners are equal. Find them.
  7. Can a triangle have angles of 50°, 60° and 80°? Why or why not?
  8. Walking round a triangle, you turn 120° at the first corner and 110° at the second. How much do you turn at the third? What are the triangle's three interior angles?
Answers
  1. (a) \(180 - 35 - 90 = 55°\). (b) \(180 - 100 - 25 = 55°\). (c) \(180 - 60 - 60 = 60°\) (an equilateral triangle).
  2. The base angles share \(180 - 30 = 150\), so each is 75°.
  3. The two base angles are both 30°, so the top angle is \(180 - 30 - 30 =\) 120° (an obtuse isosceles triangle).
  4. \(180 - 90 - 28 = 62°\).
  5. \(x + (x + 20) + (x + 40) = 3x + 60 = 180\), so \(3x = 120\) and \(x = 40\). The angles are 40°, 60°, 80°.
  6. The exterior angle equals the sum of the two: \(140 = y + y\), so each is 70°. (The third interior angle is \(180 - 140 = 40°\), and \(70 + 70 + 40 = 180\) ✓.)
  7. No. \(50 + 60 + 80 = 190\), not 180.
  8. The three turns make 360°, so the third turn is \(360 - 120 - 110 = 130°\). Each interior angle is 180° minus the turn: \(180 - 120 = 60°\), \(180 - 110 = 70°\), \(180 - 130 = 50°\). Check: \(60 + 70 + 50 = 180\). ✓

Common mistake

Believing a fact because it worked on a few drawings. Measuring ten triangles and getting about 180° is evidence, not proof. The walking argument (and the parallel-line proof in the next lesson) shows it must work for every flat triangle. Mathematicians always ask "but why?"

Mixing up the exterior angle with the angle next to it. The exterior angle equals the sum of the two far interior angles, not the one right next to it. The one next to it is its supplement: together they make 180°.

Where it leads