If A, B, C, D and E are points in a plane such that line CD bisects `/_ACB` and line CB bisects right angle `/_ACE`, then `/_DCE ` =

Options

A. `22.5^@`
B. `45^@`
C. `57/5^@`
D. `67.5^@`

Answer

To solve the problem, we need to analyze the given information about the angles and the points in the plane.

### Step 1: Draw the Configuration
We start by placing the points A, B, C, D, and E in a plane. We can visualize the points as follows:
- Place point C at the origin (0,0).
- Point A is located such that angle ACB is formed.
- Point B is positioned such that angle ACB is formed.
- Point E is positioned such that angle ACE is a right angle (90 degrees).
- Point D lies on line segment CB.

### Step 2: Identify the Angles
From the problem, we know:
- Line CD bisects angle ACB.
- Line CB bisects angle ACE, which is a right angle (90 degrees).

### Step 3: Calculate Angles
1. Since line CB bisects angle ACE, we have:
\[ \angle ACB = \angle BCE = \frac{1}{2} \times 90^\circ = 45^\circ \]

2. Since line CD bisects angle ACB, we have:
\[ \angle ACD = \angle DCB = \frac{1}{2} \times \angle ACB = \frac{1}{2} \times 45^\circ = 22.5^\circ \]

### Step 4: Find Angle DCE
Now we need to find angle DCE:
\[ \angle DCE = \angle DCB + \angle BCE \]
From our calculations:
- \(\angle DCB = 22.5^\circ\)
- \(\angle BCE = 45^\circ\)

Thus,
\[ \angle DCE = 22.5^\circ + 45^\circ = 67.5^\circ \]

### Final Answer
The value of angle DCE is:
\[ \angle DCE = 67.5^\circ \]