Given: ABCD is a quadrilateral, BC║AD, m∠ A = 90°

Prove: BD^2 − AC^2 = AD^2 − BC^2

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  Rearrange the equations that result from use of the Pythagorean theorem.

Step-by-step explanation:

Transversal AB crossing parallel lines AD and BC makes supplementary interior same-side angles A and B. Since A = 90°, B must be 90°. The Pythagorean theorem then applies in the right triangles ABC and ABD.

We can use that theorem to write two expressions for AB^2:

  BD^2 -AD^2 = AB^2 = AC^2 -BC^2

The middle expression, AB^2, isn't needed beyond this point. Adding (AD^2 -AC^2) to both sides of the equation gives the desired result:

  BD^2 -AC^2 = AD^2 -BC^2

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