Let's go through the questions one by one.

1. Name the figure below in two different ways.

The figure is a line segment with points $H$, $A$, and $D$ on it.

• The figure can be named as Line segment $\overline{HA}$ or $\overline{AH}$.
• It can also be named as Line segment $\overline{AD}$ or $\overline{DA}$.
• Additionally, it can be referred to as Line segment $\overline{HD}$ or $\overline{DH}$.

So the two different ways to name the figure could be:

1. $\overline{HA}$ and $\overline{AD}$
2. $\overline{HD}$ and $\overline{DA}$

2. Name the plane that is highlighted in the diagram below.

The plane highlighted in the diagram is defined by the points $P$, $O$, $N$, and $M$.

The plane can be named as Plane $PONM$.

3. Given $m \parallel n$, find the value of $x$.

In the diagram, $m$ and $n$ are parallel lines, and there is a transversal that creates an angle of $44^\circ$ and an unknown angle $x$.

Since the angles are alternate interior angles, they are equal. Therefore: $x = 44^\circ$

4. Given $m \parallel n$, find the value of $x$.

In this problem, the parallel lines $m$ and $n$ are cut by a transversal, creating an angle of $115^\circ$ and an unknown angle $x$.

Again, these are alternate interior angles, so: $x = 115^\circ$

5. Given $m \parallel n$, find the value of $x$.

Here, $m$ and $n$ are parallel, and the angles formed are a $19^\circ$ angle and $x$.

Since $x$ and the $19^\circ$ angle are supplementary (they form a straight line), you can calculate: $x = 180^\circ - 19^\circ = 161^\circ$

If you need more details or have any questions, feel free to ask!

Here are some additional questions to consider:

1. What are alternate interior angles, and why are they equal when the lines are parallel?
2. How do you determine the name of a plane from a given figure?
3. How would you calculate an angle if it were complementary instead of supplementary?
4. What are corresponding angles, and how do they relate to parallel lines?
5. Can a line segment have more than two names? Why or why not?

Tip: When dealing with angles formed by a transversal cutting through parallel lines, remember the different types of angle pairs (corresponding, alternate interior, alternate exterior, and consecutive interior) and their properties to solve for unknown angles efficiently.