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42 The Nuts and Bolts of Proof, Third Edition
Often we need to prove that two or more definitions of the same object are
equivalent. The existence of different definitions is usually generated by
different approaches that emphasize a certain property and point of view
over another.
EXAMPLE 4 The following definitions are equivalent:
i. A triangle is an isosceles triangle if it has two equal sides,
ii. A triangle is an isosceles triangle if it has two equal angles.
Proof:
Part 1. If i, then ii.
We have to prove that if a triangle has two equal sides, then it has two equal
angles.
Suppose that the two sides AC and AB are equal. Consider the two
triangles ADC and CDB, obtained by constructing the segment CD,
perpendicular to the base AB (this is the third side not mentioned in the
hypothesis).
The angles /.ADC and LBDC are equal, because they are right angles.
The two triangles have two equal sides: CD, because it is a common side,
and AC, which is equal to CB by hypothesis. Thus, AD and DB are equal (we
can use the Pythagorean theorem to reach this conclusion). This imphes that
the triangles ADC and CDB are congruent, and the two angles at the vertices
A and B are equal.
Part 2. If ii, then i.
We have to prove that if a triangle has two equal angles, then it has two
equal sides.