On Parallelograms. On Translations
45. Among quadrilaterals, we consider in particular trapezoids and parallelo-
grams. A quadrilateral is called a trapezoid (Fig. 42) if it has two parallel sides.
These parallel sides are called the bases of the trapezoid. A quadrilateral with two
pairs of parallel sides is called a parallelogram. (Fig. 43).
Figure 42
Figure 43
Theorem. In a parallelogram, opposite angles are equal, and angles adjacent
to the same side are supplementary.
Indeed, in parallelogram ABCD (Fig. 43), angles A and C, adjacent to side
AB, are interior angles on the same side formed by parallels AD, BC cut by
transversal AB; therefore they are supplementary. The opposite angles A, C are
equal because they have parallel sides in opposite directions.
Remark. We see that it suffices to know one angle of a parallelogram in order
to know all of them.
Converse. If, in a quadrilateral, the opposite angles are equal, then the quadri-
lateral is a parallelogram.
Indeed, the sum of the four angles of a quadrilateral equals four right angles
(44b). Since A = C, B = D, the sum of the four angles A + B + C + D can be
writen as 2A + 2B. We then have A + B = two right angles, so the lines AD, BC
are parallel, since they form two supplementary interior angles on the same side of
transversal AB. Similarly, we can show that AB is parallel to CD.
46. Theorem. In any parallelogram, the opposite sides are equal.
In parallelogram ABCD (Fig. 44), we draw diagonal AC. This diagonal divides
the parallelogram into two triangles ABC, CDA, which are congruent because they
have the common side AC between equal angles: A1 = C1 are alternate interior
angles formed by parallels AB, CD, and A2 = C2 are alternate interior angles
formed by parallels AD, BC.
These congruent triangles yield AB = CD, AD = BC.
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