Question

step reason

1. ( l parallel k ) 1. given
2. ( angle 2 cong angle 4 ) ( angle 1 cong angle 3 ) 2. ?
3. ( angle 2 cong angle 5 ) ( angle 1 cong angle 6 ) 3. if two angles are vertical angles, then they have equal measures.
4. ( angle 5 cong angle 4 ) ( angle 6 cong angle 3 ) 4. ?

select two reasons that could replace the question mark in step 2 and step 4
□ alternate angles are equal to each other
□ substitution property
□ linear pair postulate
□ if two lines are cut by a transversal, then corresponding angles are congruent
□ congruent angles are equal in measure

Explanation:

For Step 2: We know that \( l \parallel k \), and \( \angle 2 \cong \angle 4 \), \( \angle 1 \cong \angle 3 \). The reason for this is that when two parallel lines are cut by a transversal, corresponding angles are congruent. So the reason "If two lines are cut by a transversal, then corresponding angles are congruent" applies here.

For Step 4: We have \( \angle 5 \cong \angle 4 \) and \( \angle 6 \cong \angle 3 \). We already know from Step 2 that \( \angle 2 \cong \angle 4 \) and \( \angle 1 \cong \angle 3 \), and from Step 3 that \( \angle 2 \cong \angle 5 \) and \( \angle 1 \cong \angle 6 \). So we can use the substitution property (since \( \angle 2 \cong \angle 5 \) and \( \angle 2 \cong \angle 4 \), we can substitute to get \( \angle 5 \cong \angle 4 \); similarly for the other pair) or we can also think in terms of congruent angles being equal in measure (but the substitution property is more about replacing equal/congruent quantities). Wait, actually, another way: since we have \( \angle 2 \cong \angle 5 \) (from vertical angles) and \( \angle 2 \cong \angle 4 \) (from corresponding angles), then by substitution (substituting \( \angle 2 \) with \( \angle 5 \) in \( \angle 2 \cong \angle 4 \)) we get \( \angle 5 \cong \angle 4 \). Similarly for \( \angle 6 \cong \angle 3 \). But also, the property "Congruent angles are equal in measure" is a general property, but the more specific reason for Step 4 could be substitution property, and for Step 2 it's the corresponding angles postulate. Wait, let's re - evaluate:

Step 2: When two parallel lines (\( l \parallel k \)) are cut by a transversal, corresponding angles are congruent. So the reason "If two lines are cut by a transversal, then corresponding angles are congruent" is correct for Step 2.

Step 4: We have \( \angle 5 \cong \angle 2 \) (vertical angles) and \( \angle 2 \cong \angle 4 \) (corresponding angles), so by substitution (replacing \( \angle 2 \) with \( \angle 5 \) in \( \angle 2 \cong \angle 4 \)) we get \( \angle 5 \cong \angle 4 \). So the substitution property applies here. Also, the property "Congruent angles are equal in measure" is a basic property, but the substitution property is about replacing congruent quantities.

Wait, the options are:

1. Alternate angles are equal to each other - alternate angles are different from corresponding or vertical, so not for Step 2 or 4.

1. Substitution property - applies to Step 4 (since we can substitute congruent angles).

1. Linear pair postulate - linear pairs are supplementary, not about congruence here, so no.

1. If two lines are cut by a transversal, then corresponding angles are congruent - applies to Step 2.

1. Congruent angles are equal in measure - this is a general property, but the substitution property is more specific for Step 4.

So the two reasons are:

For Step 2: "If two lines are cut by a transversal, then corresponding angles are congruent"

For Step 4: "Substitution property"

Answer:

Step 2: If two lines are cut by a transversal, then corresponding angles are congruent

Step 4: Substitution property