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

Can three lengths always make a triangle?

Two compass arcs find the third corner of a triangle when you know its three sides, but only if the sides fit. Then use one equilateral triangle to build a 60° angle and halve it.

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 5 (bisecting lines and angles): a segment of 12 cm is bisected. How long is each part?
    Answer6 cm. M is the midpoint, so each part is 12 ÷ 2.
  2. From Unit 4, lesson 4 (angles in a triangle): what is each angle of an equilateral triangle?
    Answer60°. The three equal angles add up to 180°, and 180 ÷ 3 = 60.
  3. From Unit 1, lesson 1 (place value and decimals): which is greater, 4.5 + 3.8 or 8.5?
    Answer8.5. 4.5 + 3.8 = 8.3, which has 3 tenths and 8.5 has 5.
Words used in this lesson
construct
to draw exactly with only a straight edge and compasses. For example: Constructing a 60° angle.
equilateral
with all sides equal; all angles are 60°. For example: A triangle with sides 6, 6 and 6 cm.
base
the side a triangle is drawn on. For example: AB in the first figure.
arc
a part of a circle. For example: The marks left by the compasses.
flat triangle
three points on one line, so the triangle has no height. For example: Sides 3, 4 and 7 cm.
inequality
a statement that one amount is more or less than another. For example: 4 + 5 is more than 8.

The third corner

A stonemason in Oaxaca has to copy a triangular floor tile without carrying the original. Three lengths are enough: draw the base, then swing an arc from each end with the length of one of the other two sides. They cross at the third corner, and the copy has the same angles as the original.

Predict first: a base of 8 cm, with arcs of 3 cm and 4 cm. Will they cross? What if the arcs are 4 cm and 4 cm? Make a guess, then slide.

6 cm
5 cm
triangle maderesult
38.6°angle at A
48.5°angle at B
92.9°angle at C

The sides are 8 cm, 6 cm and 5 cm. The arcs from A and B meet at C, and the angles are A = 38.6°, B = 48.5° and C = 92.9°, which add up to 180°.

Show me the steps
  1. Sides 8, 6 and 5 cm. The arcs from A and B cross at C. The angles are A = 38.6°, B = 48.5° and C = 92.9°, adding up to 180°.
  2. Make both sides 8 cm. The triangle is equilateral and every angle is 60.0°.
  3. Make b = 10 and a = 6. The sides are 8, 6 and 10, and the angle at B is exactly 90.0°: a right-angled triangle.
  4. Make b = 4 and a = 3. The arcs fall short: 3 + 4 = 7 is less than 8, so no triangle exists.
  5. Make b = 4 and a = 4. The arcs just touch on AB: 4 + 4 = 8, so the triangle is flat and has no height.

Three side lengths decide a triangle, if they fit. To build it, draw one side as the base, then draw an arc about each end of the base with the length of one of the other sides. The arcs cross at the third corner, and the three angles then add up to 180°.

They fit only when the two shorter sides add up to more than the longest side. If they add up to exactly the longest side the triangle is flat, and if they add up to less, the arcs cannot meet.

Building 60° and 30° with no protractor

An equilateral triangle has three equal sides and so three equal angles of 60°. A pair of compasses makes one without measuring: every arc keeps the same radius. Then bisecting the angle (lesson 5) makes 30°. Change the radius and see that the angles do not move.

4 cm
ray onlydrawn so far
none yetangle made
4 cmcompass radius

A ray from O. Press Next step to draw an arc about O that cuts the ray at U.

Show me the steps
  1. A ray from O, with the compasses set to 4 cm.
  2. Compass point on O: draw an arc, radius 4 cm, that cuts the ray at U. OU = 4 cm.
  3. Compass point on U, same radius: draw an arc that meets the first arc at W. UW = 4 cm and OW = 4 cm.
  4. Join O to W. OU = OW = UW = 4 cm, so triangle OUW is equilateral and angle UOW is 60°.
  5. Compass point on U, then on W, same radius: the arcs meet at X. Join O to X: angle UOX is 30° and angle XOW is 30°.

A common mistake: “A bigger radius makes a bigger angle.” The angle made by an equilateral triangle is 60° for every radius: only the triangle gets bigger. An angle is a turn, not a length, so changing the compass setting changes the size of the drawing and leaves every angle the same.

Two sides and the angle between them

Three sides are not the only way to fix a triangle. With a protractor you can build one from two sides and the angle between them. Draw AB = 5 cm. At A, mark 50° and draw a ray. Along the ray measure 7 cm to C, then join C to B. Now measure: BC comes out at about 5.4 cm and the angle at B at about 85°. Only one triangle fits those two sides and that angle. Two angles and the side between them work the same way: the two rays meet at one corner. A construction on paper is drawn accurately, so here you do measure, to check it: lengths to the nearest millimetre and angles to the nearest degree.

Worked example: does it fit, and how do you build it?

Can a triangle have sides of 4 cm, 5 cm and 8 cm? If it can, say how to build it.

  1. The longest side is 8. The two shorter sides add up to 4 + 5 = 9.
  2. 9 is more than 8, so the arcs will cross and the triangle exists.
  3. Draw the 8 cm side as the base AB. With the compass at A, draw an arc of 5 cm. With it at B, draw an arc of 4 cm.
  4. The arcs cross at C. Join AC and BC. The triangle exists, and it is built from two arcs.

Your go. The first two steps are given. Can a triangle have sides of 3 cm, 4 cm and 7 cm?

  1. The longest side is 7. The two shorter sides add up to 3 + 4 = 7.
  2. Compare the sum with the longest side: is it more than, equal to, or less than 7?
  3. Now say whether the arcs cross, and what happens, then check.
Check the full answer

The sum is exactly 7, which is not more than 7. The arcs only touch on the base, so the triangle is flat and has no height. No triangle exists.

Check before you go on

1. Can a triangle have sides of 4 cm, 5 cm and 10 cm?

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. A triangle is built with all three sides equal to 6 cm. What is the size of each angle, in degrees?

3. Which set of lengths, in centimetres, can be the sides of a triangle?

Level 2 · Building

4. You construct a 60° angle and bisect it. You then bisect one of the halves again. What is the size of the smallest angle, in degrees?

5. A 60° angle is built at a point O on a straight line, on one side of the line. What is the angle left on that side of the line, in degrees?

6. Mele builds a triangle OUW with compasses, using the same setting for every arc. She says: “The angle at O is 60° whatever the setting.” Why is she right?

Level 3 · Stretch

7. You can start from a 60° angle or a 90° angle and bisect it up to three times, keeping only halves. What is the smallest angle you can make, in degrees, to one decimal place?

8. Two equilateral triangles with sides of 5 cm are placed together along one side to make a four-sided shape. What is the size of the larger angles of this shape, in degrees?

Something to think about

Using only compasses and a ruler you can make 60° and 90°, and halve any angle. Which of 15°, 45°, 75° and 10° can you build by halving and adding these angles? Why does 10° give trouble?

One way to think about it

15° is half of 30°, which is half of 60°. 45° is half of 90°. 75° is 60° + 15°. But 10° is a third of 30°, and bisecting only halves, so every angle you reach is a multiple of 30° divided by 2, 4, 8 and so on. 10 is never one of these. Cutting an angle into three equal parts is a famous task that cannot be done in general with only compasses and a ruler.

Exit ticket

9. Yuki tries to construct a triangle with sides 3 cm, 4 cm and 8 cm. She draws the 8 cm base, but her two arcs do not meet. What does that tell her?