ABCD – convex quadrilateral, AB = 6cm, BC = 9cm, CD = 10cm, DA = 25, AC = 15cm. Prove that ABCD is a trapezoid.

1) For the proof, it is necessary to prove that BC ∥ AD, for which we consider triangles ABC and ACD. They are similar, since they contain sides that are proportional in pairs.

2) Prove: AB: CD = BC: AC = AC: AD, substitute the values of the sides into the formula, we get:

6 cm: 10 cm = 9 cm: 15 cm = 15 cm: 25 cm = 3: 5.

3) In such triangles, the corresponding angles are equal, which means <BCA = <CAD, (these are the angles of parallel AB and CD, since they are in similar triangles ABC and ACD against similar sides AB and CD.

4) Since BC ∥ AD, and the sides are all different, ABCD is a trapezoid.