Isosceles triangle and the ASA criterion

The equal sides of an isosceles triangle are the legs, the third side is the base, and the vertex opposite the base is the apex. Theorem: the base angles of an isosceles triangle are equal. Proof: let AB = AC and draw the bisector AL; in △BAL and △CAL, AL is common, AB = AC and ∠BAL = ∠CAL, so by SAS △BAL = △CAL and ∠B = ∠C. Theorem: the bisector drawn to the base of an isosceles triangle is also its median and altitude (since BL = LC and ∠ALB = ∠ALC are adjacent and equal, hence 90°). Conclusion: from the apex of an isosceles triangle the bisector, median and altitude coincide. In an equilateral triangle any side can be taken as base, so all its angles are equal.