In cyclic quadrilateral ABCD,`/_CAB = 30^(@)` and `/_ABC = 100^(@)` Then,`/_ADB=`

Options

A. `50^(@)`
B. `100^(@)`
C. `75^(@)`
D. `60^(@)`

Answer

To find the angle \( \angle ADB \) in the cyclic quadrilateral ABCD, we can follow these steps:

### Step 1: Identify the Given Angles
We are given:
- \( \angle CAB = 30^\circ \)
- \( \angle ABC = 100^\circ \)

### Step 2: Use the Triangle Angle Sum Property
In triangle ABC, the sum of the angles is \( 180^\circ \). Therefore, we can write the equation:
\[ \angle CAB + \angle ABC + \angle ACB = 180^\circ \]
Substituting the known angles:
\[ 30^\circ + 100^\circ + \angle ACB = 180^\circ \]

### Step 3: Solve for \( \angle ACB \)
Now, simplify the equation:
\[ 130^\circ + \angle ACB = 180^\circ \]
Subtract \( 130^\circ \) from both sides:
\[ \angle ACB = 180^\circ - 130^\circ = 50^\circ \]

### Step 4: Use the Cyclic Quadrilateral Property
In a cyclic quadrilateral, the angles subtended by the same chord are equal. Therefore, we have:
\[ \angle ACB = \angle ADB \]
Since we found \( \angle ACB = 50^\circ \), we can conclude:
\[ \angle ADB = 50^\circ \]

### Final Answer
Thus, the measure of \( \angle ADB \) is \( 50^\circ \).

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