Two parallel lines AB and CD are intersected by a transversal EF at M and N respectively.
The lines MP and NP are the bisectors of interior angles `angle BMN` and `angle DNM` on the same side of the transversal. Then `angle MPN` is equal to :

Options

A. `60^@`
B. `90^@`
C. `45^@`
D. `120^@`

Answer

To solve the problem, we need to find the measure of angle MPN given the conditions of the parallel lines and the angle bisectors. Here's a step-by-step solution:

### Step 1: Understand the Setup
We have two parallel lines AB and CD intersected by a transversal EF at points M and N. The lines MP and NP are the angle bisectors of angles BMN and DNM, respectively.

### Step 2: Identify Angles
Let:
- Angle BMN = x
- Angle DNM = y

Since AB is parallel to CD, we know that:
- Angle BMN + Angle DNM = 180° (because they are corresponding angles formed by the transversal EF).

### Step 3: Set Up the Equation
From the above relationship, we can write:
\[ x + y = 180° \]

### Step 4: Use the Angle Bisector Property
Since MP is the bisector of angle BMN, it divides angle BMN into two equal parts:
\[ \text{Angle BMP} = \frac{x}{2} \]
\[ \text{Angle PMN} = \frac{x}{2} \]

Similarly, since NP is the bisector of angle DNM:
\[ \text{Angle DNP} = \frac{y}{2} \]
\[ \text{Angle MNP} = \frac{y}{2} \]

### Step 5: Find Angle MPN
Now, we need to find angle MPN. In triangle MPN, the sum of the angles is 180°:
\[ \text{Angle MPN} + \text{Angle PMN} + \text{Angle MNP} = 180° \]

Substituting the values we have:
\[ \text{Angle MPN} + \frac{x}{2} + \frac{y}{2} = 180° \]

### Step 6: Simplify the Equation
Now, we can simplify this equation:
\[ \text{Angle MPN} + \frac{x + y}{2} = 180° \]

Since we know from Step 3 that \( x + y = 180° \), we can substitute this into our equation:
\[ \text{Angle MPN} + \frac{180°}{2} = 180° \]
\[ \text{Angle MPN} + 90° = 180° \]

### Step 7: Solve for Angle MPN
Now, solving for angle MPN:
\[ \text{Angle MPN} = 180° - 90° = 90° \]

Thus, the measure of angle MPN is:
\[ \text{Angle MPN} = 90° \]

### Final Answer
Angle MPN is equal to 90°.

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