Go through each point of the proof and analyze why the given statement is true or what the given reason tells us. Also, use the Alternate Interior Angles Theorem.
Statements
Reasons
a. Lines m and n are parallel. Line l is a transversal.
a. Given
b. ∠ 1 and ∠ 3 form a linear pair; ∠ 2 and ∠ 4 form a linear pair.
b. Definition of a linear pair
c. ∠ 1 and ∠ 3 are supplementary; ∠ 2 and ∠ 4 are supplementary.
c. If two angles form a linear pair, then they are supplementary.
d. ∠ 1 ≅ ∠ 4, ∠ 2 ≅ ∠ 3
d. Alternate Interior Angles Theorem
e. m∠ 1 = m∠ 4, m∠ 2= m∠ 3
e. Definition of Congruence
f. ∠ 1 and ∠ 2 are supplementary; ∠ 3 and ∠ 4 are supplementary.
If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is congruent.
Now, we can go through each point one at a time.
Blank a.
From the exercise, we are given two parallel lines m and n, and the transversal l. The diagram shown in the book next to the written theorem illustrates this situation.
Blank b.
The second statements says that ∠ 1 and ∠ 3 as well as ∠ 2 and ∠ 4 form linear pairs.
Why can we say that? Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. In our case, ∠ 1 and ∠ 3 share a common side, so they are indeed adjacent angles. They are formed by the intersecting lines m and l. Thus, by the defintion they form a linear pair. This is the same for angles ∠ 2 and ∠ 4.
Blank c.
The explanation states: "if two angles form a linear pair, then they are supplementary." In the previous point, we concluded that two pairs of angles are linear angles. Using this, we can also tell that they are supplementary angles.
Blank d.
In this point, we are told that ∠ 1 is congruent to ∠ 4 and ∠ 2 is congruent to ∠ 3. How do we know that? Let's analyze the angles using the diagram.
The angle pairs ∠ 1 and ∠ 4, and ∠ 2 with ∠ 3, are called alternate interior angles. We can use the Alternate Interior Angles Theorem.
If two parallel lines are cut by a transversal,
then each pair of alternate interior angles
is congruent.
Therefore, angles ∠ 1 and ∠ 4, and ∠ 2 and ∠ 3, are indeed congruent angles by the above theorem.
Blank f.
There is no statement or reason written. However, since it's the last point, we know that it should end the proof. The Consecutive Interior Angles Theorem states that each pair of consecutive interior angles is supplementary, and this should be the statement of this point. Why can we conclude that? From points c. and e. we know the following.
l m∠ 1 + m∠ 3=180^(∘) m∠ 1= m∠ 4 } ⇒ m∠ 4 + m∠ 3=180^(∘)=180^(∘)
Using the substitution property we arrive at the above equality. Thus, angles ∠ 4 and ∠ 3 are supplementary. In the case with ∠ 2 and ∠ 4, we can substitute m∠ 4 with m∠ 1 and get that ∠ 2 and ∠ 1 are also supplementary.
Let's now copy and complete the two-column proof of this theorem.
Statements
Reasons
a. Lines m and n are parallel. Line l is a transversal.
a. Given
b. ∠ 1 and ∠ 3 form a linear pair; ∠ 2 and ∠ 4 form a linear pair.
b. Definition of linear pair
c. ∠ 1 and ∠ 3 are supplementary; ∠ 2 and ∠ 4 are supplementary.
c. If two angles form a linear pair, then they are supplementary.
d. ∠ 1 ≅ ∠ 4 (∠ 2 ≅ ∠ 3)
d. Alternate Interior Angles Theorem
e. m∠ 1 = m∠ 4 (m∠ 2= m∠ 3)
e. Definition of Congruence
f. ∠ 1 and ∠ 2 are supplementary; ∠ 3 and ∠ 4 are supplementary.
