Question

given: ( x parallel y ) ( w ) is a transversal of ( x ) and ( y ). prove: ( angle 4 cong angle 5 ) complete the two - column proof. 1. ( x parallel y ) 1. given 2. ( w ) is a transversal 2. given 3. ( angle 4 cong angle 1 ) 3. a 4. ( angle 1 cong angle 5 ) 4. b 5. ( angle 4 cong angle 5 ) 5. c

Explanation:

Step1: Analyze Statement 3

We know that \( x \parallel y \) (given) and \( w \) is a transversal. For \( \angle 4 \cong \angle 1 \), the reason should be vertical angles are congruent (since \( \angle 4 \) and \( \angle 1 \) are vertical angles formed by the intersection of two lines). Wait, no, actually, when \( x \parallel y \) and \( w \) is a transversal, \( \angle 4 \) and \( \angle 1 \) - wait, maybe I misread. Wait, \( \angle 4 \) and \( \angle 1 \) are vertical angles? Wait, no, the lines \( x \) and \( y \) are parallel, and \( w \) is a transversal, and the other line (let's say the one with angles 1,2,3,4) is also a transversal? Wait, maybe the first transversal is the one with \( x \) and \( y \) (the one with the right angles), and \( w \) is another transversal. Wait, the key here is:

• Statement 1: \( x \parallel y \) (given)
• Statement 2: \( w \) is a transversal (given)
• Statement 3: \( \angle 4 \cong \angle 1 \) - Wait, no, maybe \( \angle 4 \cong \angle 1 \) is vertical angles? Wait, no, when two lines are parallel, corresponding angles are congruent. Wait, maybe the first transversal (the one with \( x \) and \( y \)) and \( w \) intersect. Wait, let's correct:

Actually, the standard proof for alternate interior angles:

1. \( x \parallel y \) (given)
2. \( w \) is a transversal (given)
3. \( \angle 4 \cong \angle 1 \) - Wait, no, maybe \( \angle 4 \) and \( \angle 1 \) are vertical angles? No, vertical angles are equal, but here, since \( x \parallel y \), \( \angle 4 \) and \( \angle 1 \) - Wait, maybe \( \angle 4 \) and \( \angle 1 \) are corresponding angles? Wait, no, let's look at the diagram. The lines \( x \) and \( y \) are parallel (both have right angles, so they are parallel, same slope). The transversal \( w \) intersects them, and another line (the one with angles 1,2,3,4) also intersects. Wait, maybe \( \angle 4 \) and \( \angle 1 \) are vertical angles? No, vertical angles are opposite each other. Wait, maybe the correct reason for \( \angle 4 \cong \angle 1 \) is vertical angles congruent. Then for \( \angle 1 \cong \angle 5 \) (since \( x \parallel y \) and \( w \) is a transversal, \( \angle 1 \) and \( \angle 5 \) are corresponding angles, so corresponding angles are congruent). Then by transitive property, \( \angle 4 \cong \angle 5 \).

Wait, let's re - structure:

• Step 3: \( \angle 4 \cong \angle 1 \) (vertical angles are congruent)
• Step 4: \( \angle 1 \cong \angle 5 \) (corresponding angles are congruent, since \( x \parallel y \) and \( w \) is a transversal)
• Step 5: \( \angle 4 \cong \angle 5 \) (transitive property of congruence, since \( \angle 4 \cong \angle 1 \) and \( \angle 1 \cong \angle 5 \))

So:

• For Statement 3 (\( \angle 4 \cong \angle 1 \)): Reason A should be "Vertical angles are congruent"
• For Statement 4 (\( \angle 1 \cong \angle 5 \)): Reason B should be "Corresponding angles are congruent (since \( x \parallel y \) and \( w \) is a transversal)"
• For Statement 5 (\( \angle 4 \cong \angle 5 \)): Reason C should be "Transitive property of congruence"

So:

• A (Reason for \( \angle 4 \cong \angle 1 \)): Vertical angles are congruent
• B (Reason for \( \angle 1 \cong \angle 5 \)): Corresponding angles are congruent (by parallel lines and transversal)
• C (Reason for \( \angle 4 \cong \angle 5 \)): Transitive property of congruence

Answer:

• A: Vertical angles are congruent
• B: Corresponding angles are congruent (since \( x \parallel y \) and \( w \) is a transversal)
• C: Transitive property of congruence