select the correct answer. given: line l and line m intersect prove: diagram of intersecting lines l and m with angles 1,2,3,4 statements: 1. line l and line m intersect 2. ∠1 is supplementary to ∠2 3.? 4. ∠1 ≅ ∠3 reasons: 1. given 2. linear pair theorem 3.? 4. congruent supplements theorem which statement and reason best completes the proof? a. 3. m∠1 + m∠3 = 180° b. 3. ∠3 is supplementary to ∠2 c. 3. m∠1 + m∠2 = 180° d. 3. ∠1 is supplementary to ∠4

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# Brief Explanations:
To solve this, we analyze the proof structure. We know that ∠1 and ∠2 are supplementary (linear pair theorem), so $ m\angle1 + m\angle2 = 180^\circ $. For the next step, we need a statement that relates to vertical angles or supplementary angles. ∠2 and ∠3 are vertical angles? No, wait, ∠1 and ∠3: Wait, actually, ∠2 and ∠3 are supplementary? No, let's check the options. The correct statement for step 3 should be that ∠1 is supplementary to ∠3? Wait, no. Wait, the linear pair theorem gives $ m\angle1 + m\angle2 = 180^\circ $. Then, if we can say $ m\angle2 + m\angle3 = 180^\circ $ (linear pair again), but no. Wait, the options: Option D says "∠1 is supplementary to ∠4"? No, wait the options for the statement: Wait, the problem is about the proof of vertical angles or supplementary angles. Wait, the correct statement for step 3 (to lead to ∠1 ≅ ∠3 by congruent supplements theorem) should be that ∠1 and ∠3 are both supplementary to ∠2. Wait, no, the linear pair theorem gives $ m\angle1 + m\angle2 = 180^\circ $, and if we can say $ m\angle2 + m\angle3 = 180^\circ $ (so ∠2 and ∠3 are supplementary), then by congruent supplements theorem, ∠1 ≅ ∠3. But the options for the statement: Let's check the options. Wait, the options are:

A. $ m\angle1 + m\angle3 = 180^\circ $

B. ∠3 is supplementary to ∠2

C. $ m\angle1 + m\angle2 = 180^\circ $ (but that's step 2? No, step 2 is ∠1 supplementary to ∠2, so step 3 should be another supplementary relationship. Wait, no, the linear pair theorem is step 2: $ m\angle1 + m\angle2 = 180^\circ $ (linear pair theorem). Then step 3 should be $ m\angle2 + m\angle3 = 180^\circ $ (linear pair theorem again), but that's not an option. Wait, the options given:

Wait the options for the statement (step 3) are:

A. $ m\angle1 + m\angle3 = 180^\circ $

B. ∠3 is supplementary to ∠2

C. $ m\angle1 + m\angle2 = 180^\circ $ (but that's step 2)

D. ∠1 is supplementary to ∠4

Wait, no, maybe I misread. Wait the problem is: Given line l and line m intersect, ∠1 supplementary to ∠2 (linear pair), then step 3: which statement? Then step 4: ∠1 ≅ ∠3 (congruent supplements theorem). So the congruent supplements theorem says that if two angles are supplementary to the same angle, they are congruent. So ∠1 is supplementary to ∠2, and ∠3 is supplementary to ∠2, so ∠1 ≅ ∠3. Therefore, step 3 should be that ∠3 is supplementary to ∠2. Wait, but ∠2 and ∠3: are they supplementary? Yes, because they form a linear pair. So step 3: ∠3 is supplementary to ∠2 (statement), and the reason would be linear pair theorem. Then step 4: ∠1 ≅ ∠3 (congruent supplements theorem). So the correct statement for step 3 is "∠3 is supplementary to ∠2", which is option B? Wait no, option B says "∠3 is supplementary to ∠2"? Wait the options:

Wait the options are:

A. $ m\angle1 + m\angle3 = 180^\circ $

B. ∠3 is supplementary to ∠2

C. $ m\angle1 + m\angle2 = 180^\circ $

D. ∠1 is supplementary to ∠4

Wait, no, the original options (from the image):

A. $ 3. m\angle1 + m\angle3 = 180^\circ $

B. $ 3. \angle3 $ is supplementary to $ \angle2 $

C. $ 3. m\angle1 + m\angle2 = 180^\circ $

D. $ 3. \angle1 $ is supplementary to $ \angle4 $

Wait, no, maybe the options are:

Wait the user's image:

Options:

A. 3. $ m\angle1 + m\angle3 = 180^\circ $

B. 3. $ \angle3 $ is supplementary to $ \angle2 $

C. 3. $ m\angle1 + m\angle2 = 180^\circ $

D. 3. $ \angle1 $ is supplementary to $ \angle4 $

Wait, but step 2 is $ \angle1 $ is supplementary to $ \angle2 $ (linear pair theorem, so $ m\angle1 + m\angle2 = 180^\circ $). Then step 3 should be that $ \angle3 $ is supplementary to $ \angle2 $ (so $ m\angle2 + m\angle3 = 180^\circ $), which is option B. Then by congruent supplements theorem, since ∠1 and ∠3 are both supplementary to ∠2, they are congruent. So the correct statement for step 3 is that ∠3 is supplementary to ∠2, which is option B. Wait, but let's check again. The linear pair theorem gives $ m\angle1 + m\angle2 = 180^\circ $. Then, if we can say $ m\angle2 + m\angle3 = 180^\circ $ (so ∠3 is supplementary to ∠2), then by congruent supplements theorem, $ \angle1 \cong \angle3 $. So the correct statement for step 3 is option B: "∠3 is supplementary to ∠2".

# Answer:
B. 3. $ \angle3 $ is supplementary to $ \angle2 $

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