Question

proof:
statements

1. ∠1 and ∠2 form a linear pair

∠3 and ∠4 form a linear pair
2.

1. ∠1 ≅ ∠3

4.
reasons
1.

1. supplement thm.

3.
4.

• supp. thm

∠1 and ∠2 are supp.
∠3 and ∠4 are supp.
∠2 ≅ ∠4
def. of linear pair
given

Explanation:

Step1: Fill in Reason 1

The statement "∠1 and ∠2 form a linear pair; ∠3 and ∠4 form a linear pair" is given information. So the reason is "Def. of linear pair".

Step2: Fill in Statement 2

Since the reason is "Supplement Thm." (the Supplement Theorem which states that if two angles form a linear pair, then they are supplementary), the statement should be "∠1 and ∠2 are supp.; ∠3 and ∠4 are supp."

Step3: Fill in Reason 3

If ∠1 and ∠2 are supplementary (∠1 + ∠2=180°), ∠3 and ∠4 are supplementary (∠3 + ∠4 = 180°), and ∠1≅∠3 (given as statement 3), then by the Congruent Supplements Theorem (if two angles are supplementary to congruent angles, then they are congruent), we can get the next relationship. But first, for the reason of ∠1≅∠3, if we assume it is given (but in the problem - solving context with the given options, if we consider the flow, if we assume some prior - knowledge given - like maybe from a diagram or previous part not shown, but with the given options, if we consider the structure. Wait, no, actually, if we follow the two - column proof structure: Since ∠1 and ∠2 are supplementary, ∠3 and ∠4 are supplementary (from step 2) and if we assume ∠1≅∠3 (maybe given in a broader context, but with the options, if we consider the flow of using the Congruent Supplements Theorem later. But for the reason of ∠1≅∠3, if we assume it is given (but in the problem, the options have a "Given" box. Wait, no, the first statement's reason is "Def. of linear pair", the second statement's reason is "Supplement Thm.", for the third statement ∠1≅∠3, if we assume it is given (but in the problem - solving with the given options, maybe there was a mis - order. Wait, no, actually, using the Congruent Supplements Theorem: If ∠1 + ∠2=180°, ∠3+∠4 = 180°, and ∠1≅∠3, then ∠2≅∠4. But for the proof:

1. The first reason is "Def. of linear pair" (because that's the definition of a linear pair: two adjacent angles whose non - common sides are opposite rays and they are supplementary).
2. The second statement is "∠1 and ∠2 are supp.; ∠3 and ∠4 are supp." (by the Supplement Theorem: linear pair → supplementary).
3. If ∠1≅∠3 (maybe given in a broader geometric figure context, but with the options, if we consider the flow, after having ∠1 + ∠2=180°, ∠3+∠4 = 180°, and ∠1≅∠3, then by the Congruent Supplements Theorem (if two angles are supplementary to congruent angles, then they are congruent), ∠2≅∠4.
4. So statement 4 is "∠2≅∠4" and the reason is "Congruent Supplements Theorem" (but in the options, we use the given options. Since we have "∠2≅∠4" as an option.

Answer:

1. Def. of linear pair
2. ∠1 and ∠2 are supp.; ∠3 and ∠4 are supp.
3. (Assume given in a broader context, but with the flow of using congruent supplements, if we consider the structure of two - column proof and the given options)
4. ∠2≅∠4