Dem.—Join BC. Now since AB is parallel to CD, and BC intersects them, the angle ABC is equal to the alternate angle DCB [xxix.]. Again, since AB is equal to CD, and BC common, the triangles ABC, DCB have the sides AB, BC in one respectively equal to the sides DC, CB in the other, and the angles ABC, DCB contained by those sides equal; therefore [iv.] the base AC is equal to the base BD, and the angle ACB is equal to the angle CBD; but these are alternate angles; hence [xxvii.] AC is parallel to BD, and it has been proved equal to it. Therefore AC is both equal and parallel to BD. Exercises. 1. If two right lines AB, BC be respectively equal and parallel to two other right lines DE, EF, the right line AC joining the extremities of the former pair is equal to the right line DF joining the extremities of the latter. 2. Right lines that are equal and parallel have equal projections on any other right line; and conversely, parallel right lines that have equal projections on another right line are equal. 3. Equal right lines that have equal projections on another right line are parallel. 4. The right lines which join transversely the extremities of two equal and parallel right lines bisect each other. PROP. XXXIV.—Theorem. The opposite sides (AB, CD; AC, BD) and the opposite angles (A, D; B, C) of a parallelogram are equal to one another, and either diagonal bisects the parallelogram. Dem.—Join BC. Since AB is parallel to CD, and BC intersects them, the angle ABC is equal to the angle BCD [xxix.]. Again, since BC intersects the parallels AC, BD, the angle ACB is equal to the angle CBD; hence the triangles ABC, DCB have the two angles ABC, ACB in one respectively equal to the two angles BCD, CBD in the other, and the side BC common. Therefore [xxvi.] AB is equal to CD, and AC to BD; the angle BAC to the angle BDC, and the triangle ABC to the triangle BDC. Again, because the angle ACB is equal to CBD, and DCB equal to ABC, the whole angle ACD is equal to the whole angle ABD. Cor. 1.—If one angle of a parallelogram be a right angle, all its angles are right angles. Cor. 2.—If two adjacent sides of a parallelogram be equal, it is a lozenge. Cor. 3.—If both pairs of opposite sides of a quadrilateral be equal, it is a parallelogram. Cor. 4.—If both pairs of opposite angles of a quadrilateral be equal, it is a parallelogram. Cor. 5.—If the diagonals of a quadrilateral bisect each other, it is a parallelogram. Cor. 6.—If both diagonals of a quadrilateral bisect the quadrilateral, it is a parallelogram. Cor. 7.—If the adjacent sides of a parallelogram be equal, its diagonals bisect its angles. Cor. 8.—If the adjacent sides of a parallelogram be equal, its diagonals intersect at right angles. Cor. 9.—In a right-angled parallelogram the diagonals are equal. Cor. 10.—If the diagonals of a parallelogram be perpendicular to each other, it is a lozenge.