Question

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

Answer 1

We are given two right triangles \( \Delta ABC \) and \( \Delta ADC \) with a common hypotenuse \( AC \).

Step 1: Understanding the given information

• Both \( \Delta ABC \) and \( \Delta ADC \) are right-angled triangles.
• Since both triangles share the hypotenuse \( AC \), they are inscribed in the same circle with the diameter \( AC \).
• In right-angled triangles inscribed in a circle, the angle subtended by the diameter is a right angle.

Step 2: Identifying angles

- In triangle \( \Delta ABC \), \( \angle ABC = 90^\circ \). - In triangle \( \Delta ADC \), \( \angle ADC = 90^\circ \). Now, \( AC \) is common to both triangles. In \( \Delta ABC \), the angle \( \angle CAD \) and in \( \Delta ADC \), the angle \( \angle CBD \) are the angles we need to prove equal.

Step 3: Using the property of cyclic quadrilaterals

Since both triangles are right-angled and share the common hypotenuse \( AC \), we have the property of cyclic quadrilaterals: angles subtended by the same chord (hypotenuse \( AC \)) at the circumference of the circle are equal. Therefore, \( \angle CAD = \angle CBD \).

Conclusion:

We have proved that \( \angle CAD = \angle CBD \), as required.