Isosceles Triangles

One of Euclid's theorems is that the base angles of an isosceles triangle are congruent (equal). An isosceles triangle is a triangle in which two of the sides are conguent (equal). The base angles are the angles opposite the two congruent sides. The theorem is simple to prove:

• AB=AC
• AC=AB
• angle BAC=angle CAB (the angle is congruent to itself)
• triangle BAC=triangle CAB (Side-Angle-Side, see Congruence Of Triangles, Part I)
• Therefore, angle B=angle C

Did we pull a fast one, there? We proved the triangle congruent to itself, with the parts in a different order. Mathematicians are fairly happy with this proof. They seem to feel that this way of proving the triangle congruent to itself is a smooth and valid trick.

Euclid was not so happy with it. He proved the theorem in a similar, but longer way, by extending the sides, and proving other triangles congruent. See the diagram on the left. In this proof, we make BD=CE. Then we show that triangle DAC=triangle EAB. Then we show that triangle CBD=triangle BCE. So angle CBD=triangle BCE. And angle CBA=angle BCA by subtracting from the congruent straight angles.

Slightly simpler is a proof in which the sides are not extended, but again shows two pairs of triangles congruent. See the diagram on the right. The proof is almost identical to the above proof. In this proof, we make BD=CE. Then we show that triangle DAC=triangle EAB. Then we show that triangle CBD=triangle BCE. So angle CBA=angle BCA.

The converse of this theorem (If two angles of a triangle are congruent, the sides opposite these angles are congruent) is proven in a similar way. Conguence of the triangles is shown by Angle-Side-Angle instead of Side-Angle-Side. In the first diagram, angle B=angle C, and angle C=angle B. BC=CB. Then triangle BAC=triangle CAB (by Angle-Side-Angle). And AB=AC.

The converse of a theorem is its opposite. If the original theorem is A implies B, then the converse is B implies A. Sometimes the converse is true, sometimes it is false, sometimes it doesn't even make sense. In the case of The Pythagorean Theorem, the converse would be: If the sides (a, b, and c) of a triangle are related by c²=b²+a² (c^2=b^2+a^2 without special characters), then the angle opposite side c is a right angle. This converse is another true theorem. The Pythagorean Theorem is used to prove it.