ABCD is a cyclic quadrilateral. The side AB is extended to E in such a way that BE =BC. If `/_ADC = 70^@, /_BAD=95^@,` then `/_DCE` is equal to

Options

A. `140^@`
B. `120^@`
C. `165^@`
D. `110^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Cyclic Quadrilateral Properties
A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. The opposite angles of a cyclic quadrilateral sum up to 180 degrees.

### Step 2: Identify Given Angles
We are given:
- Angle \( \angle ADC = 70^\circ \)
- Angle \( \angle BAD = 95^\circ \)

### Step 3: Find Angle \( \angle BCD \)
Using the property of cyclic quadrilaterals, we can find angle \( \angle BCD \):
\[ \angle BAD + \angle BCD = 180^\circ \]
Substituting the known value:
\[ 95^\circ + \angle BCD = 180^\circ \]
\[ \angle BCD = 180^\circ - 95^\circ = 85^\circ \]

### Step 4: Find Angle \( \angle ACD \)
Next, we can find angle \( \angle ACD \) using the property of opposite angles:
\[ \angle ADC + \angle ACD = 180^\circ \]
Substituting the known value:
\[ 70^\circ + \angle ACD = 180^\circ \]
\[ \angle ACD = 180^\circ - 70^\circ = 110^\circ \]

### Step 5: Analyze Triangle \( BEC \)
Since \( BE = BC \), triangle \( BEC \) is isosceles with \( BE = BC \). Therefore, the angles opposite these equal sides are equal:
Let \( \angle EBC = x \). Then,
\[ \angle ECB = x \]
The sum of angles in triangle \( BEC \) is:
\[ \angle EBC + \angle ECB + \angle BEC = 180^\circ \]
Substituting the known values:
\[ x + x + \angle BEC = 180^\circ \]
\[ 2x + \angle BEC = 180^\circ \]
\[ \angle BEC = 180^\circ - 2x \]

### Step 6: Relate Angles
We know that:
\[ \angle ACD = \angle BEC \]
Thus,
\[ 110^\circ = 180^\circ - 2x \]
Solving for \( x \):
\[ 2x = 180^\circ - 110^\circ \]
\[ 2x = 70^\circ \]
\[ x = 35^\circ \]

### Step 7: Find Angle \( \angle DCE \)
Now, we can find angle \( \angle DCE \):
\[ \angle DCE = \angle ACD - \angle ECB \]
Substituting the known values:
\[ \angle DCE = 110^\circ - 35^\circ = 75^\circ \]

### Final Answer
Thus, the angle \( \angle DCE \) is equal to \( 75^\circ \).

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