Geometry Angles: Parallel Lines and Transversal Problems

This problem involves finding an unknown angle ($\angle EGF$) given information about parallel lines, a transversal, and other angles.

Understanding Parallel Lines and Transversals

Key concepts used:

• When a transversal intersects two parallel lines, alternate interior angles are equal.
• Vertically opposite angles are equal.
• Angles on a straight line add up to 180°.
• The sum of the angles in a triangle is always 180°.

Step-by-Step Calculation for Angle EGF

We are given:

• Line AB is parallel to line CD ($AB \parallel CD$).
• PQ is a transversal intersecting AB at E and CD at F.
• $\angle PEB = 49^\circ$.
• G is a point between AB and CD.
• $\angle BEG = 35^\circ$.
• $\angle GFE = 16^\circ$.

Let's find the measure of $\angle EGF$:

1. Find $\angle AEF$: The angle $\angle AEF$ is vertically opposite to $\angle PEB$. Therefore, $\angle AEF = \angle PEB = 49^\circ$.
2. Find $\angle BEF$: Angles $\angle AEF$ and $\angle BEF$ form a linear pair along the straight line AB. Thus, $\angle BEF + \angle AEF = 180^\circ$.
   $\angle BEF = 180^\circ - \angle AEF$
   $\angle BEF = 180^\circ - 49^\circ = 131^\circ$.
3. Find $\angle FEG$: We are given $\angle BEG = 35^\circ$. Since G is a point between the parallel lines AB and CD, the ray EG must lie within the angle $\angle BEF$. Therefore, we can write $\angle BEF$ as the sum of $\angle BEG$ and $\angle FEG$.
   $\angle BEF = \angle BEG + \angle FEG$
   $131^\circ = 35^\circ + \angle FEG$
   $\angle FEG = 131^\circ - 35^\circ = 96^\circ$.
4. Find $\angle EGF$ using triangle EFG: Now consider the triangle $\triangle EFG$. The sum of angles in a triangle is $180^\circ$. We know $\angle EFG$ and $\angle FEG$.
   $\angle EGF + \angle FEG + \angle EFG = 180^\circ$
   We have $\angle EFG = 16^\circ$ (given) and $\angle FEG = 96^\circ$ (calculated).
   $\angle EGF + 96^\circ + 16^\circ = 180^\circ$
   $\angle EGF + 112^\circ = 180^\circ$
   $\angle EGF = 180^\circ - 112^\circ$
   $\angle EGF = 68^\circ$.

Conclusion

The measure of angle $\angle EGF$ is $68^\circ$. This result is obtained by applying the properties of angles formed by parallel lines and transversals, specifically vertically opposite angles and angles on a straight line, and finally using the angle sum property of a triangle.