CHAPTER VI 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. 45

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