Three angles are not enough

Meera and Rabia measured the three SIDES and that was enough. Suppose instead they had measured the three ANGLES. Would that let them recreate the frame exactly?

Say the angles are 30°, 70° and 80°. Try to draw a triangle with those three angles. You will find it easy — and then you will find it easy to draw another, and another.

Draw a short base and put 30° at one end and 80° at the other, and the arms meet to make a small triangle with 70° at the top. Now draw a base twice as long and do the same: the arms meet further away, making a bigger triangle with exactly the same three angles.

Compare the triangles the class has drawn. They all have the same SHAPE — they look like scaled photographs of one another. But they do NOT have the same SIZE.

Congruence requires both. So two triangles that have the same set of angles need NOT be congruent.

This is worth pausing over, because it is the one case where everything looks right and the answer is still no. The angles pin down the shape completely and say nothing at all about how big the triangle is.

So there is no AAA condition for congruence. Some LENGTH must appear among the measurements, or the size is left free — which is exactly what the rest of the chapter goes on to arrange.