L M is a straight line and O is a point on LM .Line ON us drawn not coinciding with OL or OM .If `angle MON` is three times `angleLON`, then ` angleMON` is equal to :

Options

A. `45^(@)`
B. `60^(@)`
C. `105^(@)`
D. `135^(@)`

Answer

To solve the problem step by step, we will define the angles and use the properties of straight lines and linear pairs.

### Step-by-Step Solution:

1. **Define the Angles**:
Let \( \angle LON = x \) degrees. According to the problem, \( \angle MON \) is three times \( \angle LON \). Therefore, we can express \( \angle MON \) as:
\[ \angle MON = 3x \text{ degrees} \]

2. **Use the Linear Pair Property**:
Since \( LM \) is a straight line, the angles \( \angle LON \) and \( \angle NOM \) (which is the same as \( \angle MON \)) form a linear pair. The sum of angles in a linear pair is 180 degrees. Thus, we can write:
\[ \angle LON + \angle NOM = 180 \text{ degrees} \]
Substituting the angles we defined:
\[ x + 3x = 180 \]

3. **Combine Like Terms**:
Combining the terms gives:
\[ 4x = 180 \]

4. **Solve for \( x \)**:
To find \( x \), divide both sides by 4:
\[ x = \frac{180}{4} = 45 \text{ degrees} \]

5. **Find \( \angle MON \)**:
Now that we have \( x \), we can find \( \angle MON \):
\[ \angle MON = 3x = 3 \times 45 = 135 \text{ degrees} \]

### Final Answer:
Thus, \( \angle MON \) is equal to **135 degrees**.

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