Base Angles in a Proof

The base angles theorem says that in a triangle with two equal sides, the angles opposite those sides are equal. You can now see why. Draw the bisector of the apex angle: it splits the triangle into two halves with two equal sides and the angle between them, so the halves are congruent by SAS, and the base angles are matching parts.

In a proof the theorem is one line, Base angles of an isosceles triangle, citing the line with the equal sides. It is often the step that gives a congruence its angle pair.

Here KL = KJ, so ∠KLJ = ∠KJL. M lies on LJ, so ∠KLM is the same angle as ∠KLJ: the base angle of the big triangle is also an angle of the small triangle KLM. The same goes for ∠KJM in △KJM. That is how the base angles theorem hands two small triangles an angle pair.