In a convex quadrilateral ABCD, AB = 9cm, BC = 8cm, CD = 16cm, AD = 6cm

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

Consider triangles ABD and BCD and find the ratio of the sides of these triangles.

AD / BC = 6/8 = 3/4.

AB / BD = 9/12 = 3/4.

BD / CD = 12/16 = 3/4.

Since the ratio of the lengths of the sides of triangles ABD and BCD are equal, these triangles are similar in three proportional sides.

Then the angle ABD of the triangle ABD is equal to the angle CDB of the triangle BCD as similar angles of similar triangles, and since these are cross-lying angles at the intersection of the straight lines AB and CD of the secant BD, the straight lines AB and CD are parallel, and then ABCD is a trapezoid, which was required to be proved.