CHAPTER VI. PARALLELOGRAMS AND QUADRILATERALS. 30. If a parallelogram have one angle a right angle, all its angles are right angles. D A Let A be a right angle, then the opposite angle C is also a right angle. C B ZA+B= two right angles; therefore B is a right angle. LA+ D= two right angles; therefore D is a right angle. 31. A quadrilateral which has its opposite angles equal is a parallelogram. Let A B C D be a quadrilateral in which A = B = D, then Therefore A+ ≤ B = half the sum of the angles of the quadrilateral, that is to say, ≤ A+ Consequently A D and B C are parallel. Similarly, since ≤ A+ 2 D= < C + 2 B ; Therefore A+ ZD = two right angles. Consequently A B and C D are parallel. 32. A quadrilateral which has its opposite sides equal is a parallelogram. Let A B C D be a quadrilateral in which A B = DC and AD = BC, then shall A B C D be a parallelo gram. A B Draw the diagonal D B. Then the triangles A B D, CDB have the three sides A D, A B, B D equal respectively to the three sides CB, CD, BD. Hence the triangles are equal in all respects, and ▲ ADB = 4 C B D and ▲ ABD = 4CDB; But A D B and C BD are alternate angles, therefore A D and C B are parallel, and C D B and A B D are alternate angles, therefore CD and AB are parallel. Consequently A B C D is a parallelogram. 33. When two sides of a quadrilateral are equal and parallel the quadrilateral is a parallelogram. Let CD and A B be equal and parallel, then A B C D shall be a parallelo gram. Draw the diagonal D B. Then the triangles C D B, B ABD have CD, D B, and CDB respectively equal to A B, B D, and A B D, therefore they are equal in all respects, and AD = BC. Also LADB=LC B D, and since these are alternate angles, AD and BC are parallel. Consequently ABCD is a parallelogram. 34. The diagonals of a rectangle are equal. Let A C and BD be diagonals of the rectangle ABCD; then A C B D. = -B In the triangles ABC, DC B, the sides A B, BC and the included angle ABC are respectively equal to the two sides DC, CB, and the included angle DCB. Consequently the triangles are equal in all respects, and AC = BD. 35. When the diagonals of a parallelogram are equal the parallelogram is a rectangle. Let A B C D be a parallelogram in which A C BD; then A B C D shall be a rectangle. = The triangles A B C, DCB have the three sides of one respectively equal to the three sides of the other; therefore they are equal in all respects and LABC = 4 DC B. But ABC + DCB two right angles; therefore each of these angles is a right angle. The angles BAD, CDA opposite to D C B and A B C are therefore also right angles, and consequently A B C D is a rectangle. 36. The diagonals of a square are at right angles. Let ABCD be a square, and let the diagonals A C and B D intersect in E; then shall the angles at E be right angles. In the right-angled ▲ BAD, ZABD+Z ADB a right angle. = But because AD AB, therefore ABD = ZADB. Hence ABD is half a right angle. Similarly C A B is half a right angle. But AED = LABD + CAB. Consequently ▲ AED is a right angle. 37. When the diagonals of a rectangle are at right angles the rectangle is a square. Let A B C D be a rectangle in which the diagonals A C and BD are at right angles; then shall ABCD be a square. Because ABCD is a rectangle the diagonals. bisect one another and are equal. In the triangles A E D, A E B, BE = ED and AE is common; also the ▲ AED = / AEB, and the triangles are equal in all respects; .. ABAD; consequently the other sides are equal, and the figure A B C D is a square. 38. The sum of the four sides of a quadrilateral is greater than the sum and less than twice the sum of the diagonals. A B AD + DC > AC AB+ BC > AC DA + AB > DB DC + CB > D B. By addition ... twice the sum of the sides is greater than twice the sum of the By addition .. the sum of the sides is less than twice the sum of the diagonals. 39. Any straight line, drawn through the point of intersection of the diagonals of a parallelogram and terminating in the sides, is bisected at the point. Let the diagonals A D and C B of the parallelc |