Chapter 3 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

M3.5.3 Chapter 3

Triangles

Do the three angles of every triangle add up to the same number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Show that the angles of a triangle add up to 180°
  • Find a missing angle in a triangle
  • Name equilateral, isosceles, scalene and right angled triangles

A triangle has three corners and three angles. Add the three angles of any triangle. The answer is always 180°. This is the angle sum of a triangle.

Draw it first

Tear off the corners

  1. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°

    1 Draw a triangle. Its angles are 70°, 50° and 60°.

  2. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°60°70°50°

    2 Tear off the three corners. Put them side by side with their points together.

  3. A triangle with angles 70°, 50° and 60°. Below it the three angles sit side by side at one point on a straight line. From the right they are 60°, then 70°, then 50°, each on the side where it began. Under the picture it says 60° + 70° + 50° = 180°.70°50°60°60°70°50°60° + 70° + 50° = 180°

    3 They fill a straight line. So the three angles add to 180°.

Chapter 2 shows why. Draw a line through the top corner, parallel to the base. The alternate angles match the two bottom corners.

The same works for every triangle. So you can find a missing angle. Add the two angles you know, and take the answer from 180°.

Worked example

A triangle has angles of 65° and 48°. What is the third angle, x?

  1. The three angles add to 180°.
  2. The two known angles add to 65° + 48° = 113°.
  3. x = 180° − 113° = 67°.

Answer: 67°

A triangle with corners A, B and C. The angle at A is x, which is 67°. The angle at B is 65° and the angle at C is 48°.ABCx65°48°
The angle x is 67°: 67° + 65° + 48° = 180°.

Triangles have names. A name tells you how many sides are equal. Small tick marks on sides show which sides are equal.

Four triangles with their names: equilateral, isosceles, scalene and right angled. The equilateral triangle has one tick on each of its three equal sides and one arc on each of its three equal angles of 60°. The isosceles triangle has one tick on each of its two equal sides and one arc on each of its two equal angles of 70°. The scalene triangle has no equal sides and no ticks. The right angled triangle has a small square at its right angle of 90°.equilateralisoscelesscaleneright angled
Ticks show equal sides. Matching arcs show equal angles. The little square shows a right angle.

An equilateral triangle has three equal sides and three angles of 60°. An isosceles triangle has two equal sides and two equal angles. The equal angles are opposite the equal sides.

A scalene triangle has no equal sides and no equal angles. A right angled triangle has one angle of 90°. A right angled triangle can also be isosceles.

Worked example

An isosceles triangle has an angle of 40° between its two equal sides. What are the other two angles?

  1. The three angles add to 180°.
  2. The other two angles add to 180° − 40° = 140°.
  3. The two equal sides make two equal angles.
  4. Each of them is 140° ÷ 2 = 70°.

Answer: 70° and 70°

An isosceles triangle with corners A, B and C. Tick marks show that sides AB and AC are equal. The angle at A is 40°. The angles at B and C are both 70°.ABC40°70°70°
The angles at B and C are equal because the sides AB and AC are equal.

Worked example

A right angled triangle has an angle of 35°. What is the third angle?

  1. One angle is 90°.
  2. The three angles add to 180°.
  3. The other two angles add to 180° − 90° = 90°.
  4. One of them is 35°.
  5. The third angle is 90° − 35° = 55°.

Answer: 55°

A right angled triangle with corners A, B and C. A small square marks the right angle at A. The angle at B is 35° and the angle at C is x, which is 55°.ABC35°x
The two angles that are not the right angle add to 90°.

See it three ways

With things

Draw a triangle on paper. Tear off its three corners. Put them side by side with their points together.

As a picture

Three angles side by side at one point on a straight line. From the right they are 60°, 70° and 50°.60°70°50°

In symbols

a + b + c = 180°. For example 70° + 50° + 60° = 180°.

Try it

Tear the corners off a triangle

You need: a sheet of paper, a pencil.

  1. Draw a triangle on the paper. Make it any shape you like.
  2. Mark each corner with a small curved line.
  3. Tear off the three corners.
  4. Put the corners side by side with their points together.

What you will see: The three corners fill a straight line. That is 180°. It works for every triangle you try.

If you cannot try it, this is what happens.
Three torn corners side by side at one point on a straight line. From the right they are 60°, 70° and 50°.60°70°50°

A wrong idea many people have

A bigger triangle has a bigger angle sum.

Why it feels right: A big triangle has longer sides, so it seems to have more of everything.

The test that breaks it: Draw a small triangle and a big triangle. Measure the angles of each and add them up. Both add to about 180°.

What is true: Every triangle has an angle sum of 180°. Longer sides do not make the angles add to more.

Key words

angle sum
the total of the angles in a shape. For a triangle it is 180°.
equilateral triangle
a triangle with three equal sides and three angles of 60°.
isosceles triangle
a triangle with two equal sides and two equal angles.
scalene triangle
a triangle with no equal sides.
right angled triangle
a triangle with one angle of 90°.

Back to your guess

The answer is A: Yes. They always add up to 180°.

Tear off the three corners of a paper triangle and put them side by side. They fill a straight line, which is 180°. This works for every triangle, big or small.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A triangle has angles of 55° and 72°. What is the third angle x?

    A triangle with corners A, B and C. The angle at A is x. The angle at B is 55° and the angle at C is 72°.ABCx55°72°
    Tap your answer.

    1. Not this one. That is 55° + 72°, the two angles you know. Take it away from 180° to find x. Go to M3.5.3. Try another.

    2. Yes, that is right. 55° + 72° = 127°. Then 180° − 127° = 53°.

    3. Not this one. You took 127° from 360°. The angles of a triangle add to 180°, not 360°. Go to M3.5.3. Try another.

    See the answer

    53°

    55° + 72° = 127°. Then 180° − 127° = 53°.

    • If you chose A: That is 55° + 72°, the two angles you know. Take it away from 180° to find x. Go to M3.5.3.
    • If you chose C: You took 127° from 360°. The angles of a triangle add to 180°, not 360°. Go to M3.5.3.

  2. This isosceles triangle has an angle of 50° between its equal sides. What is the angle x?

    An isosceles triangle with corners A, B and C. Ticks show that AB and AC are equal. The angle at A is 50°. The angles at B and C are both x.ABC50°xx
    Tap your answer.

    1. Not this one. That is 180° − 50°. The two equal angles share it, so each is half of 130°. Go to M3.5.3. Try another.

    2. Not this one. The angle of 50° is at the top. The two base angles are equal to each other, but they are not 50°. Go to M3.5.3. Try another.

    3. Yes, that is right. The other two angles add to 180° − 50° = 130°. They are equal, so each is 130° ÷ 2 = 65°.

    See the answer

    65°

    The other two angles add to 180° − 50° = 130°. They are equal, so each is 130° ÷ 2 = 65°.

    • If you chose A: That is 180° − 50°. The two equal angles share it, so each is half of 130°. Go to M3.5.3.
    • If you chose B: The angle of 50° is at the top. The two base angles are equal to each other, but they are not 50°. Go to M3.5.3.

  3. The sides of this triangle are 4 cm, 5 cm and 6 cm. What kind of triangle is it?

    A triangle with corners A, B and C. Side BC is 5 cm, side CA is 4 cm and side AB is 6 cm. No two sides are equal.ABC5 cm4 cm6 cm
    Tap your answer.

    1. Yes, that is right. All three sides have a different length, so the triangle is scalene.

    2. Not this one. An isosceles triangle has two sides of the same length. These three sides are all different. Go to M3.5.3. Try another.

    3. Not this one. An equilateral triangle has three equal sides. These sides are 4 cm, 5 cm and 6 cm. Go to M3.5.3. Try another.

    See the answer

    Scalene triangle

    All three sides have a different length, so the triangle is scalene.

    • If you chose B: An isosceles triangle has two sides of the same length. These three sides are all different. Go to M3.5.3.
    • If you chose C: An equilateral triangle has three equal sides. These sides are 4 cm, 5 cm and 6 cm. Go to M3.5.3.

  4. A triangle has a right angle and an angle of 38°. What is the third angle?

    A right angled triangle with corners A, B and C. A small square marks the right angle at A. The angle at B is 38°. The angle at C is x.ABC38°x
    Tap your answer.

    1. Not this one. You took only 38° from 180°. Take away the right angle, 90°, as well. Go to M3.5.3. Try another.

    2. Yes, that is right. The three angles add to 180°. 90° + 38° = 128°. Then 180° − 128° = 52°.

    3. Not this one. The triangle already has one angle of 90°. The other two add to 90°, so the third is smaller. Go to M3.5.3. Try another.

    See the answer

    52°

    The three angles add to 180°. 90° + 38° = 128°. Then 180° − 128° = 52°.

    • If you chose A: You took only 38° from 180°. Take away the right angle, 90°, as well. Go to M3.5.3.
    • If you chose C: The triangle already has one angle of 90°. The other two add to 90°, so the third is smaller. Go to M3.5.3.

  5. Which three angles can be the angles of a triangle?

    Tap your answer.

    1. Not this one. These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3. Try another.

    2. Not this one. These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3. Try another.

    3. Yes, that is right. 90° + 50° + 40° = 180°. The other sets add to 190°.

    See the answer

    90°, 50° and 40°

    90° + 50° + 40° = 180°. The other sets add to 190°.

    • If you chose A: These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3.
    • If you chose B: These add to 190°. The angles of a triangle add to 180°. Go to M3.5.3.

  6. Two angles of a triangle are 48° and 77°. What is the third angle?

    See the answer

    55°

    48° + 77° = 125°. Then 180° − 125° = 55°.

  7. From an earlier chapter

    Two lines are parallel. An angle at the top is 70°. The angle y is next to the corresponding angle at the bottom, on a straight line. What is y?

    See the answer

    110°

    The corresponding angle at the bottom is 70°. The angle y fills a straight line with it, so y = 180° − 70° = 110°.

Keep it

Copy this into your exercise book.

The three angles of a triangle add to 180°. Equilateral: 3 equal sides, all angles 60°. Isosceles: 2 equal sides, 2 equal angles. Scalene: none equal. Right angled: one 90° angle.

Draw: A triangle with its three corners drawn as angles side by side on a straight line. Write 180° under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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