AB is a straight line and O is a point lying on AB .A line OC is drawn from O such that `angleCOA=36^(@)`. OD is a line within `angleCOA` such that `angleDOA=(1)/(3)`
`angleCOA` . If OE is a line within the `angleBOC,angleBOE=(1)/(4)angleBOC` , then `angleDOE` must be

Options

A. `60^(@)`
B. `132^(@)`
C. `144^(@)`
D. `108^(@)`

Answer

To solve the problem step by step, we will follow the given information and derive the angles systematically.

### Step 1: Identify the angles
We are given that:
- \( \angle COA = 36^\circ \)
- \( \angle DOA = \frac{1}{3} \times \angle COA \)

### Step 2: Calculate \( \angle DOA \)
Using the information from Step 1:
\[ \angle DOA = \frac{1}{3} \times 36^\circ = 12^\circ \]

### Step 3: Find \( \angle COB \)
Since \( AB \) is a straight line, we know that:
\[ \angle AOB = 180^\circ \]
Thus, we can find \( \angle COB \):
\[ \angle COB = 180^\circ - \angle COA = 180^\circ - 36^\circ = 144^\circ \]

### Step 4: Calculate \( \angle BOE \)
We are told that:
\[ \angle BOE = \frac{1}{4} \times \angle BOC \]
Substituting \( \angle BOC = \angle COB = 144^\circ \):
\[ \angle BOE = \frac{1}{4} \times 144^\circ = 36^\circ \]

### Step 5: Find \( \angle COE \)
Since \( \angle COE \) is the remaining angle in \( \angle BOC \):
\[ \angle COE = \angle BOC - \angle BOE = 144^\circ - 36^\circ = 108^\circ \]

### Step 6: Calculate \( \angle DOE \)
Now, we can find \( \angle DOE \):
\[ \angle DOE = \angle DOA + \angle COE \]
Substituting the values from previous steps:
\[ \angle DOE = 12^\circ + 108^\circ = 120^\circ \]

### Final Answer
Thus, the measure of \( \angle DOE \) is:
\[ \angle DOE = 120^\circ \]

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