Year 7 · Unit 4 Angles and constructions · Lesson 4 of 6

Why do the three angles of every triangle add up to 180°?

Change a triangle's angles and the third one answers. Line the three corners up on a straight line, then prove with a parallel line why it always works.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Unit 4, lesson 3 (parallel lines and transversals): parallel lines are cut by a transversal and one angle is 55°. What is its alternate angle?
    Answer55°. Alternate angles are equal when the lines are parallel.
  2. From Unit 4, lesson 2 (angles on a line and at a point): angles on a straight line are 123° and x. Find x.
    Answer57°. Angles on a straight line add up to 180°, and 180 − 123 = 57.
  3. From Unit 1, lesson 6 (order of operations): what is 180 − (50 + 60)?
    Answer70. Brackets first: 50 + 60 = 110, then 180 − 110 = 70.
Words used in this lesson
triangle
a shape with three straight sides and three angles. For example: A roof frame.
scalene
a triangle with no equal sides. For example: Sides 3, 4 and 5 cm.
isosceles
a triangle with two equal sides and two equal angles. For example: The gable end of a house, with two equal rafters.
equilateral
a triangle with three equal sides and three 60° angles. For example: A road sign.
interior angle
an angle inside a shape at a corner. For example: The three angles of a triangle.
exterior angle
the angle outside a shape between one side and the extension of the next. For example: PRS in question 7.

Tear the corners

In an origami class in Kyoto, students fold a paper triangle, tear off its three corners and place them point to point. Whatever triangle they started with, the three corners make a straight edge. That is a half turn, 180°. Change the angles and see what the third one has to be.

Predict first: A = 50° and B = 60°. What must C be? Then try A = 90° and B = 30°, and a triangle with two equal angles. What do you notice about its sides?

50°
60°
70°the third angle C
acutetype by its angles
scalenetype by its sides

A = 50° and B = 60°, so C = 180° − 50° − 60° = 70°. The three angles fill the straight line exactly. All three are below 90° and all different, so the triangle is acute and scalene. Here they are placed side by side; the proof below shows why they always fit.

Show me the steps
  1. A = 50° and B = 60°. C = 180 − 110 = 70°. All angles are below 90° and all differ: an acute scalene triangle.
  2. Set A = 90° and B = 30°. C = 180 − 120 = 60°. One angle is 90°, so the triangle is right-angled; all angles differ, so it is scalene.
  3. Set A = 50° and B = 50°. C = 80°. Two angles are equal, so two sides are equal: an acute isosceles triangle.
  4. Set A = 60° and B = 60°. C = 60°. All three angles are equal: an equilateral triangle, each angle 60°.
  5. Set A = 30° and B = 30°. C = 120°, which is more than 90°: an obtuse isosceles triangle.

The three angles of a triangle always add up to 180°. So when two angles are known, the third is 180° minus their sum.

Triangles are sorted in two ways, and a triangle gets one name from each.

  • By angles: acute (all three below 90°), right-angled (one angle of 90°) or obtuse (one angle above 90°).
  • By sides: scalene (no sides equal), isosceles (two equal sides, and the angles opposite them are equal) or equilateral (three equal sides and three angles of 60°).

A triangle can have only one obtuse or right angle, because two would already use 180° or more and leave nothing for the third.

Why it is always 180°

Tearing corners shows it works for one triangle. A proof says it works for every triangle, and it needs only one new line, drawn through C parallel to the base. Step through it, then change A and B and run it again.

50°
60°
triangle onlydrawn so far
70°angle C
not shown yetA + C + B

Triangle ABC has A = 50°, B = 60° and C = 70°. Press Next step to draw a line through C parallel to AB.

Show me the steps
  1. Start with any triangle ABC. Its angles are A = 50°, B = 60° and C = 70° at the opening values.
  2. Draw a line through C parallel to the base AB.
  3. The angle on the left, between the parallel and CA, is alternate to angle A, so it is also 50°.
  4. The angle on the right, between the parallel and CB, is alternate to angle B, so it is also 60°.
  5. The three angles at C, 50°, 70° and 60°, lie on a straight line, so they add up to 180°. Therefore A + B + C = 180°.

A common mistake: “My triangle's angles add up to 178°, so this triangle is different.” A drawn angle can be off by a degree or two when it is measured by hand. The proof shows that the sum is exactly 180° for every triangle, so a total of 178° means a measuring error, not a new kind of triangle. The same goes for a diagram marked “not drawn to scale”: find angles from the facts, not by measuring the picture.

Worked example: a missing angle and a name

Two angles of a triangle are 38° and 64°. Find the third angle and name the triangle by its angles and by its sides.

  1. The three angles add up to 180°, so the third angle is 180 − (38 + 64).
  2. 38 + 64 = 102, and 180 − 102 = 78°.
  3. The angles are 38°, 64° and 78°. All are below 90°, so the triangle is acute.
  4. All three angles are different, so no sides are equal. It is acute and scalene, with third angle 78°.

Your go. The first two steps are given. An isosceles triangle has an angle of 50° between its two equal sides. Find the other two angles and name the triangle.

  1. The two equal sides have equal angles opposite them, so the other two angles are equal.
  2. All three add up to 180°, so the other two add up to 180 − 50 = 130°.
  3. Now find each angle yourself, then name the triangle by its angles, then check.
Check the full answer

130 ÷ 2 = 65°, so the angles are 50°, 65° and 65°. All three are below 90°: the triangle is acute and isosceles.

Check before you go on

1. Which set of three angles could be the angles of a triangle?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Two angles of a triangle are 52° and 67°. What is the third angle, in degrees?

3. A triangle has angles of 40°, 70° and 70°. Which name fits it best?

Level 2 · Building

4. A roof frame is an isosceles triangle. The angle at the top, between the two equal rafters, is 40°. What is each of the other two angles, in degrees?

5. The angles of a triangle are x°, 2x° and 3x°. How big is the largest angle, in degrees?

6. A right-angled triangle has one angle of 35° besides the right angle. What is its third angle, in degrees?

Level 3 · Stretch

7. In triangle PQR, angle P is 48° and angle Q is 59°. The side QR is extended beyond R to a point S. What is angle PRS, in degrees? 59°48°?PQRSnot drawn to scale

8. An isosceles triangle has two equal angles of (2x + 5)° each and a third angle of (x + 10)°. Find x.

Something to think about

Extend one side of a triangle to make an angle outside it, called an exterior angle. Use the angle facts you know to show that it equals the sum of the two interior angles at the far corners. Then check it with 50° and 65°.

One way to think about it

The exterior angle and the interior angle next to it are on a straight line, so they add to 180°. The three interior angles also add to 180°. So the exterior angle is 180° minus the interior angle at that corner, which is the sum of the other two. Check: 50 + 65 = 115, and the interior angle is 180 − 115 = 65, so the exterior is 180 − 65 = 115.

Exit ticket

9. Jabu says a triangle can have two obtuse angles, because 100° + 100° is less than 360°. Which reply is right?