In a convex quadrilateral ABCD AB = 9 BC = 8 CD = 16 AD = 6 BD = 12. prove that ABCD is a trapezoid.

Consider triangle DAB and triangle CBD. Let’s find the ratio of their respective sides: DA / CB = AB / BD = DB / CD 6/8 = 9/12 = 12/16, we will reduce the fractions: 3/4 = 3/4 = 3/4. We got that the sides of these triangles are proportional, which means that the triangles are similar. For similar triangles, the corresponding angles are equal, which means the angle ADB is equal to the angle DBC. But for lines AD, BC and the secant BD, these are cross-lying angles, which means that AD is parallel to BC. AB is not parallel to CD, since if they were parallel, then we would get a parallelogram, and its opposite sides are equal, which contradicts the condition of the problem. So our quadrilateral is a trapezoid. Q.E.D.