ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

Given: Triangle ABC and ACD are two right triangle on common base AC.
To prove: ∠CAD = ∠CBD.
Proof: Triangle ABC and ADC are on common base BC and ∠BAC = ∠BDC.
Hence, points A, B, C and D lie on the same circle.
[∵ if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle.]
Therefore,
∠CAD = ∠CBD
[∵ Angles in the some segment are equal]