A cyclic quadrailatral ABCD is such that AB=BC, AD=Dc `AC _|_ BD, angle CAD = theta` Then the angle `angle ABC`

Options

A. `theta`
B. `(theta)/(2)`
C. `2 theta`
D. `3 theta`

Answer

To solve the problem, we need to find the angle \( \angle ABC \) in the cyclic quadrilateral \( ABCD \) with the given conditions. Here’s a step-by-step breakdown of the solution:

### 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 sum of the opposite angles is equal to \( 180^\circ \).

**Hint:** Remember that in a cyclic quadrilateral, opposite angles add up to \( 180^\circ \).

### Step 2: Identify the given information
We have:
- \( AB = BC \) (which means triangle \( ABC \) is isosceles)
- \( AD = DC \) (which means triangle \( ADC \) is also isosceles)
- \( AC \perp BD \) (the diagonals are perpendicular)
- \( \angle CAD = \theta \)

**Hint:** Write down all the given information clearly to visualize the problem.

### Step 3: Analyze triangle \( ADC \)
Since \( AD = DC \), the angles opposite these sides are equal. Therefore, we can say:
\[ \angle ACD = \angle CAD = \theta \]

**Hint:** Use the property of isosceles triangles to find equal angles.

### Step 4: Find angle \( A + D + C \) in triangle \( ADC \)
The sum of angles in triangle \( ADC \) is \( 180^\circ \):
\[ \angle A + \angle D + \angle C = 180^\circ \]
Substituting the known angles:
\[ \theta + (180^\circ - 2\theta) + \theta = 180^\circ \]
This simplifies to:
\[ 180^\circ = 180^\circ \]
This confirms that our angles are consistent.

**Hint:** Use the triangle angle sum property to relate the angles.

### Step 5: Use the property of cyclic quadrilaterals
Since \( ABCD \) is a cyclic quadrilateral, we can use the property that the sum of opposite angles is \( 180^\circ \):
\[ \angle ADC + \angle ABC = 180^\circ \]
Substituting \( \angle ADC = 180^\circ - 2\theta \):
\[ (180^\circ - 2\theta) + \angle ABC = 180^\circ \]

**Hint:** Set up the equation using the cyclic quadrilateral property.

### Step 6: Solve for \( \angle ABC \)
Rearranging the equation gives:
\[ \angle ABC = 2\theta \]

**Hint:** Isolate \( \angle ABC \) to find its value.

### Conclusion
Thus, the angle \( \angle ABC \) is \( 2\theta \).

**Final Answer:** \( \angle ABC = 2\theta \)