If an exterior angle of a cyclic quadrilateral be `50^(@)`, then the opposite interior angle is :

Options

A. `130^(@)`
B. `40^(@)`
C. `50^(@)`
D. `90^(@)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the properties of a cyclic quadrilateral
A cyclic quadrilateral is a four-sided figure where all vertices lie on the circumference of a circle. One important property of cyclic quadrilaterals is that the sum of the opposite angles is equal to 180 degrees.

### Step 2: Identify the given information
We are given that the exterior angle of the cyclic quadrilateral is 50 degrees. Let's denote the cyclic quadrilateral as ABCD, and the exterior angle at vertex D (angle CDE) is given as 50 degrees.

### Step 3: Relate the exterior angle to the interior angle
Since angle CDE is an exterior angle, it can be expressed in terms of the interior angle ADC. The relationship is given by:
\[ \text{Angle ADC} + \text{Angle CDE} = 180^\circ \]
Substituting the value of angle CDE:
\[ \text{Angle ADC} + 50^\circ = 180^\circ \]

### Step 4: Calculate the interior angle ADC
To find angle ADC, we rearrange the equation:
\[ \text{Angle ADC} = 180^\circ - 50^\circ \]
\[ \text{Angle ADC} = 130^\circ \]

### Step 5: Use the property of cyclic quadrilaterals
Now that we have angle ADC, we can use the property of cyclic quadrilaterals that states:
\[ \text{Angle ADC} + \text{Angle ABC} = 180^\circ \]
Substituting the value of angle ADC:
\[ 130^\circ + \text{Angle ABC} = 180^\circ \]

### Step 6: Calculate the opposite interior angle ABC
Rearranging the equation to find angle ABC:
\[ \text{Angle ABC} = 180^\circ - 130^\circ \]
\[ \text{Angle ABC} = 50^\circ \]

### Conclusion
Thus, the opposite interior angle ABC is 50 degrees.