To prove that ∠CAD = ∠CBD in two right triangles ABC and ADC with a common hypotenuse AC, we use the properties of similar triangles and angles in a circle.
Since ABC and ADC are right triangles with a common hypotenuse AC, they are both inscribed in a semicircle with AC as the diameter. By the property of a circle, an angle subtended by a diameter at the circumference is a right angle. Therefore, ∠ABC and ∠ADC are both right angles.
In triangle ABC, ∠CAB + ∠ABC + ∠BCA = 180°. Since ∠ABC is a right angle, ∠CAB + 90° + ∠BCA = 180°. Thus, ∠CAB + ∠BCA = 90°.
Since ∠BCA and ∠ACD are angles at the circumference subtended by the same arc BC, they are equal. Therefore, ∠CAB + ∠BCA = ∠CAD + ∠ACD.
Given that ∠BCA = ∠ACD, it follows that ∠CAB = ∠CAD. Therefore, ∠CAD = ∠CBD, as required.

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## Geometric Relationships in Right Triangles

In the study of geometry, the exploration of angles within right triangles, especially those sharing a common hypotenuse, reveals fascinating relationships. In this scenario, we have two right triangles, ABC and ADC, sharing a common hypotenuse AC. Our objective is to prove that ∠CAD is equal to ∠CBD. This proof involves understanding the properties of right triangles and the subtleties of angles within a circle.

### The Significance of the Common Hypotenuse

The common hypotenuse AC in triangles ABC and ADC plays a crucial role. In a circle, the diameter (or hypotenuse, in this case) subtends a right angle to any point on the circle’s circumference. This means that both triangles ABC and ADC can be inscribed in a semicircle with AC as the diameter, making ∠ABC and ∠ADC right angles by definition.

#### Analyzing Triangle ABC

In triangle ABC, the sum of the angles must equal 180°. Given that ∠ABC is a right angle (90°), the sum of the other two angles, ∠CAB and ∠BCA, must be 90° to satisfy the angle sum property of a triangle. This relationship is key to understanding the angle measures within the triangle and their relation to the circle.

Similarly, in triangle ADC, the angle sum property holds true. With ∠ADC also being a right angle (90°), the sum of ∠CAD and ∠ACD must be 90°. This parallel between the two triangles in terms of their angle sums provides a foundation for proving the equality of ∠CAD and ∠CBD.

##### Establishing the Equality of Angles

Since ∠BCA and ∠ACD are angles subtended by the same arc BC in the semicircle, they are equal. This equality leads to the conclusion that ∠CAB (in triangle ABC) and ∠CAD (in triangle ADC) must also be equal, as both pairs of angles add up to 90°. Therefore, ∠CAD = ∠CAB.