Geometry Problem: Parallel Lines and Transversal

The question asks us to find the measure of angle $\angle AMN$ given that lines AB and CD are parallel, PQ is a transversal intersecting AB at M and CD at N, and $\angle MND = 47^\circ$. We need to use the properties of angles formed when a transversal intersects parallel lines.

Identifying Key Angles

• Given: Line AB is parallel to line CD (AB || CD).
• Transversal PQ intersects AB at M and CD at N.
• Given angle: $\angle MND = 47^\circ$.
• Angle to find: $\angle AMN$.

Calculating Angle AMN using Corresponding Angles

We can find $\angle AMN$ by relating it to the given angle $\angle MND$ using angle properties.

1. Vertically Opposite Angles: Angle $\angle MND$ and angle $\angle CNP$ are vertically opposite angles. Vertically opposite angles are equal.
   Therefore, $\angle CNP = \angle MND = 47^\circ$.
2. Corresponding Angles: Angle $\angle AMN$ and angle $\angle CNP$ are corresponding angles. Corresponding angles are in the same relative position at each intersection where a transversal crosses two lines. Since lines AB and CD are parallel (AB || CD), their corresponding angles are equal.
   Therefore, $\angle AMN = \angle CNP$.
3. Result: Substituting the value from step 1 into step 2, we get:
   $\angle AMN = 47^\circ$.

Thus, the measure of angle $\angle AMN$ is $47^\circ$. This corresponds to Option C.

Alternative Method: Consecutive Interior Angles

We can also solve this using consecutive interior angles.

1. Linear Pair: Angle $\angle MND$ and angle $\angle MNC$ form a linear pair on the straight line CD. The sum of angles in a linear pair is $180^\circ$.
   $\angle MNC + \angle MND = 180^\circ$
   $\angle MNC + 47^\circ = 180^\circ$
   $\angle MNC = 180^\circ - 47^\circ = 133^\circ$.
2. Consecutive Interior Angles: Angle $\angle AMN$ and angle $\angle MNC$ are consecutive interior angles (or same-side interior angles) formed by the transversal PQ intersecting parallel lines AB and CD. Consecutive interior angles are supplementary, meaning their sum is $180^\circ$.
   $\angle AMN + \angle MNC = 180^\circ$
   $\angle AMN + 133^\circ = 180^\circ$
   $\angle AMN = 180^\circ - 133^\circ = 47^\circ$.

Both methods confirm that $\angle AMN = 47^\circ$.