Two sides and the angle between them

The idea

Given two sides and the angle included between them, draw one side as the base, construct the given angle at one end of it, mark the other side's length along the new arm, and join the endpoints.

Where is C?

We want a triangle ABC with AB = 5 cm, AC = 4 cm and the angle at A, ∠A = 45°. The base AB is drawn.

How can we find the third corner C?

• Draw the 45° arm from A and mark 4 cm along it
• Guess a point and check it with a ruler
• Draw two arcs, one from A and one from B

Build the triangle

Use the protractor to swing the arm from A. Then slide the pencil along the arm with the ruler marks.

Angle first, then length

The angle at A sits between the sides AB and AC. We call it the included angle of those two sides.

The angle says which way AC goes. The length says how far. Together they fix C. You did not choose BC: it comes out by itself.

Given two sides and the angle included between them, draw one side as the base, construct the given angle at one end of it, mark the other side’s length along the new arm, and join the endpoints.

• Step | What to do
• 1 | Draw the base AB = 5 cm.
• 2 | At A, construct ∠A = 45° and draw the other arm.
• 3 | On that arm, mark C so that AC = 4 cm.
• 4 | Join BC.

Notes

Given two sides and the angle included between them, draw one side as the base, construct the given angle at one end of it, mark the other side’s length along the new arm, and join the endpoints.

Check yourself

In ∆PQR you know the sides PQ and QR. Which angle must be given to build the triangle with the steps you learned?

In ∆PQR, PQ = 7 cm, PR = 5 cm and ∠P = 60°. You draw PQ as the base and the 60° arm at P. What do you mark on the arm?

To build a triangle from two sides and the angle between them, how many measurements do you need in all (lengths and angles together)?

Answer: 3

Two sides and the angle between them: 2 + 1 = 3 measurements.

Tara draws AB = 5 cm and the 45° arm at A. Then she marks C on the arm so that BC = 4 cm, not AC. What went wrong?

• ∠Q — correct. Yes! ∠Q is between the sides QP and QR. It is their included angle.
• ∠P. ∠P is between PQ and PR. PR is not one of the two given sides.
• ∠R. ∠R is between RQ and RP. RP is not one of the two given sides.
• The point R, 7 cm from P. PQ = 7 cm is the base and is already drawn. The arm gets the other side, PR.
• The point R, 5 cm from P — correct. Yes! PQ is the base. The arm gets the other side, PR = 5 cm, measured from P.
• The point Q, 5 cm from P. Q is already the other end of the base. It is R that is marked on the arm.
• The 4 cm is the length of AC. It must be measured from A, along the arm. — correct. Yes! The given side AC starts at the vertex A of the angle, so the 4 cm is marked from A.
• Nothing. BC and AC mean the same thing.. BC joins B and C. AC joins A and C. They are different sides with different lengths.
• The angle should have been drawn at B, not at A.. The angle ∠A is included between AB and AC, so it is drawn at A.