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

Consider triangles ABD and BCD formed by the diagonal BD.

Let us prove that these triangles are similar.

Let’s pay attention to the ratio of the sides of the triangles.

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

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

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

Since the aspect ratio is equal, the triangles are similar in three proportional sides.

In such triangles, similar angles are equal, the angle ADB = CBD, and since these are cross-lying angles, the straight lines AD and BC are parallel, therefore ABCD is a trapezoid, in which AD and BC are bases, which was required to be proved.

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