Copy an angle

The idea

To copy an angle, draw an arc from its vertex cutting both arms, draw an equal-radius arc from the new vertex, and transfer the distance between the cut points onto it; by SSS the angles are equal.

Exact copies of one angle

To draw a design made of the same unit turned in different directions, every unit needs the same arm lengths and the same angle between the arms. A compass makes equal arm lengths easily.

How can we make sure the angles are equal, without measuring them?

• Build a triangle with the same three sides
• Estimate it by eye
• Make the arms longer

Copy the angle

The angle at A has to be copied at X, on the new line. Follow the steps. Watch what is carried from the old angle to the new one.

Carry two lengths

The triangle ABC has AB = AC (one arc) and a third side BC. The new triangle XYZ has XY = XZ (the same arc) and YZ = BC, which the compass carried across. Three equal sides: SSS. So the angles are equal.

To copy an angle, draw an arc from its vertex cutting both arms. Draw an equal-radius arc from the new vertex, and transfer the distance between the cut points onto it. By SSS the angles are equal.

• ∆ABC | ∆XYZ
• AB = AC = the arc radius | XY = XZ = the same radius
• BC | YZ = BC, measured with the compass
• ∠A | ∠X = ∠A

Notes

To copy an angle, draw an arc from its vertex cutting both arms, draw an equal-radius arc from the new vertex, and transfer the distance between the cut points onto it. By SSS the angles are equal.

Check yourself

Which two lengths does the compass carry from the old angle to the new one?

The copied angle ∠YXZ equals ∠BAC = 72°. In ∆XYZ, XY = XZ. How many degrees is ∠XYZ? (The angles of a triangle add up to 180°.)

Answer: 54

∠XYZ + ∠XZY = 180° − 72° = 108°, and they are equal, so ∠XYZ = 108° ÷ 2 = 54°.

Rahul opens the compass a little wider than BC when he transfers the distance. The new angle will be:

∆ABC has AB = AC = 3 cm, and its perimeter (the sum of its three sides) is 8 cm. After copying the angle, how many cm is YZ?

Answer: 2

BC = 8 − 3 − 3 = 2 cm. YZ is the distance BC carried over with the compass, so YZ = 2 cm.

• The lengths of the two arms of the old angle. The arms can be any length. The arcs and the distance BC are what count.
• The size of the angle in degrees. The compass cannot measure degrees. It carries lengths.
• The radius of the first arc, and the distance BC between the cut points — correct. Yes! The radius gives XY = AB. The distance BC gives YZ = BC.
• Bigger than the old angle — correct. Yes! A longer distance between the two cut points means the arms are further apart.
• Equal to the old angle. It would be equal only if the compass were opened exactly as wide as BC.
• Smaller than the old angle. A wider opening puts Z further round the arc, so the angle grows.