ABCD is a cyclic quadrilateral and AB is the diameter of the circle. If `angleCAB = 48^@`, then what is the value (in degrees) of `angleADC`?

Options

A. 52
B. 77
C. 138
D. 142

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the 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 opposite angles sum up to 180 degrees.

### Step 2: Identify the given information
We are given that:
- ABCD is a cyclic quadrilateral.
- AB is the diameter of the circle.
- Angle CAB = 48 degrees.

### Step 3: Use the inscribed angle theorem
Since AB is the diameter, angle ACB (the angle opposite to the diameter) is a right angle (90 degrees). This is due to the inscribed angle theorem, which states that an angle inscribed in a semicircle is a right angle.

### Step 4: Calculate angle CBA
In triangle ABC, we can use the fact that the sum of angles in a triangle is 180 degrees:
\[ \text{Angle CAB} + \text{Angle ACB} + \text{Angle CBA} = 180^\circ \]
Substituting the known values:
\[ 48^\circ + 90^\circ + \text{Angle CBA} = 180^\circ \]
This simplifies to:
\[ \text{Angle CBA} = 180^\circ - 48^\circ - 90^\circ = 42^\circ \]

### Step 5: Use the property of cyclic quadrilaterals
Now, since ABCD is a cyclic quadrilateral, we know that:
\[ \text{Angle CBA} + \text{Angle ADC} = 180^\circ \]
Substituting the value of angle CBA:
\[ 42^\circ + \text{Angle ADC} = 180^\circ \]
Solving for angle ADC:
\[ \text{Angle ADC} = 180^\circ - 42^\circ = 138^\circ \]

### Final Answer
The value of angle ADC is **138 degrees**.

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