We will complete the proof by providing the reasons for each step. Let's first review what we are given and what we want to prove.
Given:& l ∥ m, ∠ 2 ≅ ∠ 4
Prove:& n ∥ p
Now, we can complete the blanks!
Blank a.
The first missing information is the reason for why the lines l and m are parallel. Notice that this is a part of the given information. Therefore, we can complete the first blank as shown below.
1)& l ∥ m
1)& a. Given
Blank b.
Now, we want to find the reason why ∠ 1 and ∠ 2 are congruent. Notice that they lie on the same side of the transversal n that intersects the lines l and m. Therefore, ∠ 1 and ∠ 2 are corresponding angles.
The lines l and m are parallel, so by the Corresponding Angles Theorem we can conclude that ∠ 1 ≅ ∠ 2.
2)& ∠ 1 ≅ ∠ 2
2)& b. Corresponding Angle Theorem
Blank c.
The next statement says that ∠ 2 is congruent to ∠ 4. We can tell that this is the second piece of information that is given. Therefore, we can provide the reason for the third step as shown below.
3)& ∠ 2 ≅ ∠ 4
3)& c. Given
Blank d.
From the second and third step we have the following information.
∠ 1 ≅ ∠ 2 and ∠ 2 ≅ ∠ 4
Therefore, by the Transitive Property of Congruence we can conclude that ∠ 1 ≅ ∠ 4.
4)& ∠ 1 ≅ ∠ 4
4)& d. Transitive Property of Congruence
Blank e.
Notice that ∠ 1 and ∠ 4 lie on the same side of the transversal l that intersects the lines n and p. Therefore, they are corresponding angles.
We know that ∠ 1 ≅ ∠ 4. Thus, by the Converse of the Corresponding Angles Postulate, we can conclude that the lines n and p are parallel, which is what we wanted to prove!
5)& n ∥ p
5)& e. Converse of the & Corresponding Angles Postulate
