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© Copyright 1999, Jim Loy
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:
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.
Addendum:
A couple of readers of this page did not like my definition of an isosceles triangle. Their definition is: "An isosceles triangle is a triangle that has at least two sides which are congruent (equal)." This is the same as my definition, but the emphasis is different. I did not say that only two sides are congruent. But my definition can be misunderstood, although I got it straight out of a mathematics encyclopedia. Saying "at least" clears up the possible misunderstanding. My definition implies that we don't care about the third side, at all. We can point out two sides which are congruent, and that is good enough. Certainly, the third side may also be congruent.