Two angles and the side between them

We have used two sides and an angle. Now try two ANGLES and a side — starting with the side BETWEEN them.

Take ∆ABC and ∆XYZ with BC = YZ = 5 cm, ∠B = ∠Y = 50° and ∠C = ∠Z = 30°. Are they congruent?

You know this construction from Part 1: draw the base, set the two angles at its ends, and let the arms meet. Every step is forced — the base length is given, and each arm's direction is given — so the meeting point is fixed and every such triangle is the same.

So ∆ABC ≅ ∆XYZ, and we have the ASA — Angle Side Angle — condition for congruence.

Now use these tools on a real figure. Point O is the midpoint of both AD and BC, which cross at O. What can we say about the lengths AB and CD?

List what is known. AO = OD, since O is the midpoint of AD. BO = OC, since O is the midpoint of BC. Anything else? Yes: ∠AOB = ∠DOC, because they are vertically opposite angles at the crossing.

That is two sides and the angle included between them, so the SAS condition is satisfied and the triangles are congruent. Which vertices correspond? For AO to lie over OD and BO over CO, the pairs are A with D, O with O (the common vertex) and B with C. So ∆AOB ≅ ∆DOC — and AB and DC are corresponding sides, so their lengths are equal.