O is point on the line LM A line ON is drawn which is neither coincident with OL nor with OM if `angleMON` is one third of `angle LON` then `angleMON` equal to

Options

A. A)`45^(@)`
B. B)`60^(@)`
C. C)`75^(@)`
D. D)`80^(@)`

Answer

To solve the problem step by step, we will analyze the given information and use some basic properties of angles.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a line LM with points L, O, and M on it.
- Point O is between points L and M.
- A line ON is drawn from point O, which is not coincident with OL or OM.

2. **Identifying the Angles**:
- We denote:
- \( \angle LON \) as the angle formed between line OL and line ON.
- \( \angle MON \) as the angle formed between line OM and line ON.
- According to the problem, \( \angle MON \) is one-third of \( \angle LON \).

3. **Setting Up the Relationship**:
- Let \( \angle LON = 3k \) (where k is a variable representing the angle).
- Therefore, \( \angle MON = k \).

4. **Using the Straight Line Property**:
- Since points L, O, and M are on a straight line, the sum of angles \( \angle LON \) and \( \angle MON \) must equal 180 degrees:
\[ \angle LON + \angle MON = 180^\circ \]
- Substituting the expressions for the angles:
\[ 3k + k = 180^\circ \]

5. **Solving for k**:
- Combine like terms:
\[ 4k = 180^\circ \]
- Now, divide both sides by 4:
\[ k = \frac{180^\circ}{4} = 45^\circ \]

6. **Finding the Value of \( \angle MON \)**:
- Since \( \angle MON = k \), we have:
\[ \angle MON = 45^\circ \]

### Conclusion:
The value of \( \angle MON \) is \( 45^\circ \).