The diagram shows two parallel lines with a transversal, and you're given the following information:

• $m\angle 3 = 97^\circ$
• $m\angle 6 = (x + 7)^\circ$

We need to determine the value of $x$.

Key observations:

• Angles $\angle 3$ and $\angle 6$ are corresponding angles because they are on the same side of the transversal and occupy the same relative position at the intersection points.
• Corresponding angles are congruent when the lines are parallel, meaning $\angle 3 = \angle 6$.

Set up the equation:

Since corresponding angles are equal: $m\angle 3 = m\angle 6$ $97^\circ = (x + 7)^\circ$

Solve for $x$:

$97 = x + 7$ Subtract 7 from both sides: $x = 90$

Thus, the value of $x$ is $90^\circ$.

Would you like further details or have any additional questions on this? Here are some related questions:

1. What are corresponding angles, and why are they equal when lines are parallel?
2. Can you describe how to identify alternate interior angles in a diagram like this?
3. How do you solve for $x$ if $m\angle 6$ was given differently?
4. What happens if the lines aren't parallel? Would the angles still be equal?
5. How would you calculate $x$ if $m\angle 6 = (x - 7)^\circ$ instead?

Tip: Always look for angle relationships such as corresponding, alternate interior, and consecutive angles when dealing with parallel lines and a transversal.