In the diagram, we are dealing with a pair of intersecting lines, forming angles labeled 1 through 8. We are given:

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

Now let's analyze the relationships between the angles:

Step 1: Identifying the Relationship

Angle 3 and Angle 6 are corresponding angles since they are on the same side of the transversal and in corresponding positions relative to the two parallel lines.

For corresponding angles, we know that they are congruent, which means:

$m \angle 3 = m \angle 6$

Step 2: Setting Up the Equation

Since corresponding angles are equal, we can equate the given measures of the angles:

$97^\circ = (x + 7)^\circ$

Step 3: Solving for $x$

To solve for $x$, subtract 7 from both sides of the equation:

$97 - 7 = x$

$x = 90$

Final Answer:

The value of $x$ is $90$.

Would you like more details or have any further questions on this problem?

Here are 5 related questions to expand on this topic:

1. How do you determine when angles are corresponding?
2. What other angle pairs can be congruent in a similar diagram?
3. If $m \angle 5 = 83^\circ$, what is the measure of $m \angle 7$?
4. Can alternate interior angles also be used to solve for $x$ in this case?
5. How would the answer change if the lines were not parallel?

Tip: Remember, in parallel line diagrams, corresponding angles, alternate interior angles, and vertical angles are key to solving for unknowns.