Understanding Parallel Lines and Transversals in Geometry

This problem requires understanding the properties of angles formed when a transversal line intersects two parallel lines. We are given information about two angles, ∠APE and ∠PQD, and need to find the measure of ∠APQ.

Given Information Summary

• Line AB is parallel to Line CD (AB || CD).
• Line EF acts as a transversal, intersecting AB at point P and CD at point Q.
• The measure of angle APE is given as $6x$ (∠APE = $6x$).
• The measure of angle PQD is given as $4x$ (∠PQD = $4x$).

Goal

Determine the measure of angle APQ (∠APQ).

Key Geometric Concepts

To solve this, we utilize the angle relationships established when parallel lines are intersected by a transversal:

• Vertically Opposite Angles: When two lines intersect, the angles opposite each other are equal. In this case, ∠APE and ∠BPQ are vertically opposite angles, meaning ∠BPQ = ∠APE.
• Consecutive Interior Angles: These are angles on the same side of the transversal and between the parallel lines. They are supplementary, meaning their sum is $180^\circ$. Here, ∠BPQ and ∠PQD are consecutive interior angles.
• Linear Pair: Angles that form a straight line sum to $180^\circ$. ∠APQ and ∠APE form a linear pair along the line AB.

Step-by-Step Calculation

1. Relate Angle APE to Angle BPQ

   Since ∠APE and ∠BPQ are vertically opposite angles:

   ∠BPQ = ∠APE = $6x$

2. Use the Consecutive Interior Angles Property

   Because AB || CD, the sum of the consecutive interior angles ∠BPQ and ∠PQD must be $180^\circ$.

   Setting up the equation:

   ∠BPQ + ∠PQD = $180^\circ$

   Substituting the given expressions:

   $6x + 4x = 180^\circ$

3. Solve for 'x'

   Combine the terms with $x$:

   $10x = 180^\circ$

   Isolate $x$ by dividing both sides by 10:

   $x = \frac{180^\circ}{10}$

   $x = 18^\circ$

4. Calculate the Measure of Angle APQ

   Angles ∠APQ and ∠APE form a linear pair on the straight line AB. Therefore, their sum is $180^\circ$.

   ∠APQ + ∠APE = $180^\circ$

   Substitute the expression for ∠APE:

   ∠APQ + $6x = 180^\circ$

   Now substitute the value of $x = 18^\circ$ into the equation:

   ∠APQ = $180^\circ - 6(18^\circ)$

   ∠APQ = $180^\circ - 108^\circ$

   ∠APQ = $72^\circ$

Conclusion

By applying the properties of vertically opposite angles and consecutive interior angles formed by a transversal intersecting parallel lines, we found the value of $x$ to be $18^\circ$. Using this value, we calculated ∠APQ, which forms a linear pair with ∠APE, to be $72^\circ$.