Let's go over each question in the image.

Problem 1

Goal: Show that $p \parallel q$.

• Given:
  • $\angle 4 \cong \angle 5$
  • $m\angle 1 = (4x + 16)^\circ$, $m\angle 8 = (5x - 12)^\circ$, and $x = 28$
  • $m\angle 2 = (6x - 19)^\circ$, $m\angle 5 = (3x + 14)^\circ$, and $x = 11$

Solution Strategy: Apply the Converse of the Corresponding Angles Postulate. We check if pairs of corresponding angles are congruent.

Problem 2

Goal: Show that $r \parallel s$.

• Given:
  • $\angle 1 \cong \angle 5$
  • $m\angle 3 + m\angle 4 = 180^\circ$
  • $\angle 3 \cong \angle 7$
  • Equations involving angles based on $x$ values:
    • $m\angle 4 = (13x - 4)^\circ$ and $m\angle 8 = (9x + 16)^\circ$, $x = 5$
    • $m\angle 8 = (17x + 37)^\circ$ and $m\angle 7 = (9x - 13)^\circ$, $x = 6$
    • $m\angle 2 = (25x + 7)^\circ$ and $m\angle 6 = (24x + 12)^\circ$, $x = 5$

Solution Strategy: Use properties such as the Alternate Interior Angles Theorem and Consecutive Interior Angles Theorem to prove parallel lines.

Problem 3

Goal: Complete a two-column proof to show $XY \parallel WV$.

• Given:
  • $\angle 1 \cong \angle 2$, $\angle 3 \cong \angle 1$
  • We must show $XY \parallel WV$.

Solution Strategy: Use the properties of congruent angles and the Converse of the Corresponding Angles Postulate.

Would you like detailed solutions for any of these steps, or help on a specific part?

Practice Questions

1. How do you apply the Converse of the Corresponding Angles Postulate to prove lines are parallel?
2. What is the role of alternate interior angles in proving lines parallel?
3. How do you solve for variables when given angle equations?
4. What properties of angles help in a two-column proof for parallel lines?
5. Can you explain the difference between corresponding angles and alternate interior angles?

Tip

When proving lines are parallel, focus on angle relationships like corresponding, alternate interior, and consecutive angles.