In this problem, you are asked to find the measure of $\angle EFI$ given that:

• $EG \parallel HJ$ (lines $EG$ and $HJ$ are parallel),
• $m \angle JIF = 62^\circ$.

Step-by-step solution:

1. Identify the relationship between the angles: Since $EG \parallel HJ$ and $IF$ is a transversal, angles $\angle JIF$ and $\angle EFI$ are corresponding angles. Corresponding angles are equal when a transversal crosses two parallel lines.

2. Apply the corresponding angle theorem: Therefore, $m \angle EFI = m \angle JIF$.

3. Substitute the value of $m \angle JIF$: We know $m \angle JIF = 62^\circ$.

Thus, $m \angle EFI = 62^\circ$.

Would you like further details, or do you have any questions?

Here are 5 related questions for practice:

1. What is the sum of the interior angles formed by two parallel lines and a transversal?
2. If $m \angle EFI = 62^\circ$, what is $m \angle IFJ$?
3. Can you identify another pair of corresponding angles in this diagram?
4. What is the difference between alternate interior angles and corresponding angles?
5. What are the properties of angles formed by parallel lines and a transversal?

Tip: In geometry, it's helpful to remember that corresponding angles formed by a transversal crossing parallel lines are always congruent!