Two sides and the angle between them

Take two triangles ∆ABC and ∆XYZ with AB = XY = 6 cm, AC = XZ = 5 cm, and ∠A = ∠X = 30°. Are they congruent?

To decide, ask the question that settles every case in this chapter: can there exist NON-congruent triangles with these measurements? If not, congruence is forced.

These measurements are two sides and the angle BETWEEN them — the included angle. You already know how to construct a triangle from them, from Part 1.

Construct it. Draw AB = 6 cm. At A, set off an arm at 30°. Measure 5 cm along that arm and mark C. Join BC.

Now look back over the construction and notice how little freedom there was. The base was fixed. The direction of the second arm was fixed by the angle. The position of C on that arm was fixed by the length. At no point was there a choice to make.

So every triangle built from these measurements must come out the same. Compare yours with your classmates' and you will find them all congruent.

Thus, when two sides and the included angle of two triangles are equal, the two triangles are congruent. This is the SAS — Side Angle Side — condition, and the word INCLUDED is doing important work in it, as the next lesson shows.