Let's solve problem 6 from the image.

Given:

We have two parallel lines $l \parallel m$ and a transversal crossing them, forming two angles:

• One angle is $(8x - 9)^\circ$,
• The other angle is $53^\circ$.

Since $l \parallel m$, the angles are corresponding angles. For corresponding angles, the two angles are equal.

Step-by-Step Solution:

$8x - 9 = 53$

1. Add 9 to both sides: $8x = 62$

2. Divide both sides by 8: $x = \frac{62}{8} = 7.75$

So, the value of $x$ is $7.75$.

Converse Used:

The Corresponding Angles Converse states that if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. In this case, the angles were set equal to ensure that $l \parallel m$.

Would you like me to walk through question 7 or explain any part of this solution? Here are 5 questions to deepen your understanding:

1. What are corresponding angles, and why do they help prove lines are parallel?
2. What would happen if the angles were not equal? Could the lines still be parallel?
3. How can you apply the Corresponding Angles Converse in real-world problems?
4. Could other angle relationships (like alternate interior angles) also be used here?
5. How would you check if the solution is correct if $x$ was given a different value?

Tip: When solving angle problems with parallel lines, always think about angle pairs like corresponding, alternate interior, and alternate exterior angles—they provide shortcuts for finding missing values!