Let's analyze the given diagram step by step. We are asked to find the measure of angle $\angle FHD$, given that the measure of $\angle DHE = 71^\circ$.

Step 1: Identify relationships

From the diagram:

• $\angle DHE = 71^\circ$ is given. This angle is labeled as angle 6.
• $\angle FHD$ is the angle we need to determine, labeled as angle 5.

Step 2: Use vertical and linear pair relationships

• Angles 5 and 6 (i.e., $\angle FHD$ and $\angle DHE$) are vertical angles because they are opposite each other at point $H$, where the two lines cross.
• Vertical angles are congruent, meaning they have equal measures.

Step 3: Conclude the solution

Since $\angle DHE = 71^\circ$ and $\angle FHD$ is a vertical angle to it, we can conclude:

$m \angle FHD = 71^\circ$

So, $m \angle FHD = 71^\circ$.

Would you like a deeper explanation, or do you have any further questions on this?

Here are 5 related questions you might find useful:

1. How are vertical angles defined geometrically?
2. What other types of angle pairs are formed when two lines intersect?
3. Can linear pairs also help in solving for unknown angles in such problems?
4. What are supplementary angles, and how do they relate to linear pairs?
5. How would the solution change if we were given a different angle?

Tip: Vertical angles are always congruent, so they are useful for quickly solving problems involving intersecting lines!