Two angles and the side between them

The idea

Given two angles and the included side, the side that is part of both angles, draw that side as the base and construct the two angles at its ends; the point where their arms meet is the third vertex.

Two arms, one corner

We want a triangle ABC with AB = 5 cm, ∠A = 45° and ∠B = 80°. The base AB is drawn.

How can we find the third corner C?

• Draw a 45° arm from A and an 80° arm from B
• Draw an arc of 5 cm from A and an arc from B
• Draw one arm only and guess where C is

Cross the two arms

Use the protractor to swing the arm from A. Then swing the arm from B. Where do they cross?

The arms meet at C

The side AB is part of both angles: ∠A and ∠B both have AB as an arm. We call AB the included side of the two angles.

C is on the arm from A and on the arm from B. So it makes the given angle at A and the given angle at B. It can only be the crossing point.

Given two angles and the included side, the side that is part of both angles, draw that side as the base and construct the two angles at its ends. The point where their arms meet is the third vertex.

• Step | What to do
• 1 | Draw the base AB = 5 cm.
• 2 | At A, construct ∠A = 45°. At B, construct ∠B = 80°.
• 3 | The two new arms cross at the third vertex C.

Notes

Given two angles and the included side, the side that is part of both angles, draw that side as the base and construct the two angles at its ends. The point where their arms meet is the third vertex.

Check yourself

In ∆PQR you know ∠P and ∠Q. Which side must be given to build the triangle with the steps you learned?

Rita has drawn the base AB = 5 cm and the 45° arm at A above AB. Where does she draw the 80° arm?

Why is the point where the two arms cross the third vertex?

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

Answer: 3

Two angles and the side between them: 2 + 1 = 3 measurements.

• QR. QR is part of ∠Q but not of ∠P. The included side is part of both angles.
• PR. PR is part of ∠P but not of ∠Q. The included side is part of both angles.
• PQ — correct. Yes! PQ is part of both ∠P and ∠Q. It is the included side.
• At B, above AB, measured from BA — correct. Yes! Both arms go on the same side of AB, so that they can cross above it.
• At A, on top of the 45° arm. The second angle belongs at the other end of the base, at B.
• At B, below AB. Below AB the arm leans away from the other arm. Both arms must be on the same side of AB.
• It is on both arms, so it makes the given angle at A and the given angle at B. — correct. Yes! The crossing is the only point on both arms.
• It is the middle point of AB.. The crossing is far above AB, not the middle of it. It is the only point that lies on both arms.
• Both arms are 5 cm long.. The arms can be any length. What matters is that C is on both of them.