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

**Solution:**

We know that, the sum of all angles in a triangle is 180°.

If the sum of pair of opposite angles in a quadrilateral is 180°, then it is a cyclic quadrilateral.

Consider ΔABC,

∠ABC + ∠BCA + ∠CAB = 180° (Angle sum property of a triangle)

90° + ∠BCA + ∠CAB = 180°

∠BCA + ∠CAB = 90°.....(1)

Consider ΔADC,

∠CDA + ∠ACD + ∠DAC = 180° (Angle sum property of a triangle)

90° + ∠ACD + ∠DAC = 180°

∠ACD + ∠DAC = 90°.....(2)

Adding Equations (1) and (2), we obtain

∠BCA + ∠CAB + ∠ACD + ∠DAC = 180°

(∠BCA +∠ACD) + (∠CAB + ∠DAC) = 180°

∠BCD + ∠DAB = 180°.....(3)

However, it is given that

∠B + ∠D = 90° + 90° = 180°.....(4)

From Equations (3) and (4), it can be observed that the sum of the measures of opposite angles of quadrilateral ABCD is 180°. Therefore, it is a cyclic quadrilateral.

Since it is a cylclic quadrilateral the below figure can be drawn.

Consider chord CD. ∠CAD and ∠CBD are formed on the same segment CD.

∠CAD = ∠CBD (Angles in the same segment are equal)

**Video Solution:**

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

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.5 Question 11:

**Summary:**

ABC and ADC are two right triangles with common hypotenuse AC. We have found that ∠CAD = ∠CBD as they are lying on the same segment, and we know that they will always be equal