The problem you uploaded asks to "Find the value of $x$ so that line $l \parallel m$" and involves angles at the intersection of two lines.

From the image:

• There are two expressions for angles formed by the intersecting lines:
  • One angle is $(4x + 7)^\circ$
  • The other angle is $(6x - 63)^\circ$

Since the lines $l$ and $m$ must be parallel, these angles are alternate interior angles, which are equal when the lines are parallel. Therefore, we can set the angles equal to each other and solve for $x$:

$4x + 7 = 6x - 63$

Let's solve this equation:

$4x + 7 = 6x - 63$ Subtract $4x$ from both sides: $7 = 2x - 63$ Add 63 to both sides: $70 = 2x$ Divide by 2: $x = 35$

Thus, the value of $x$ is 35.

Converse:

The converse of the statement refers to proving that if the angles are equal, then the lines are parallel. This is a basic result from geometry known as the converse of the Alternate Interior Angles Theorem: If two lines are cut by a transversal, and a pair of alternate interior angles are equal, then the lines must be parallel.

Would you like more details, or do you have any questions?

Related Questions:

1. What is the definition of alternate interior angles?
2. How do we prove lines are parallel using corresponding angles?
3. Can two lines that intersect be parallel? Why or why not?
4. What is the converse of the Corresponding Angles Postulate?
5. How do supplementary angles relate to parallel lines?

Tip:

When working with parallel lines, remember that alternate interior angles, corresponding angles, and consecutive interior angles have special relationships that make problem-solving easier.