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

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

Let’s find the ratio of the lengths of the sides of the triangles ABD and BCD.

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

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

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

Then triangles ABD and BCD are similar in three proportional sides.

Angle ABD = CDB as similar angles of similar triangles, and since these are criss-cross angles at the intersection of lines AB and CD of secant BD, then lines AB and CD are parallel, and then ABCD is a trapezoid, which was required to be proved.