4. use the figure below to answer the question. lines a and b are parallel. line c is a transversal. which statement is true? a. ( mangle 1
eq mangle 4 ), because they are opposite angles b. ( mangle 1 = mangle 6 ), because they are supplementary angles c. ( mangle 1 = mangle 8 ), because they are alternate angles d. ( mangle 1 = mangle 4 ), because they are vertical angles

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# Brief Explanations:
- Option A: Angles 1 and 4 are not "opposite angles" in the standard sense. Vertical angles are opposite and equal, but ∠1 and ∠4 are supplementary (form a linear pair), so $ m\angle1 + m\angle4 = 180^\circ $, not equal. Eliminate A.
- Option B: ∠1 and ∠6 are not supplementary. Since lines $ a \parallel b $ and $ c $ is a transversal, ∠1 and ∠5 are corresponding angles (equal), and ∠5 and ∠6 are supplementary. So ∠1 and ∠6 are supplementary? Wait, no—wait, let's re - check. ∠1 and ∠3 are vertical angles, ∠3 and ∠7 are corresponding (since $ a\parallel b $), ∠7 and ∠6 are supplementary. So ∠1 and ∠6: $ m\angle1 = m\angle3 $, $ m\angle3 + m\angle7 = 180^\circ $ (linear pair), $ m\angle7 = m\angle6 $ (vertical angles)? No, ∠7 and ∠6 are adjacent and form a linear pair? Wait, the diagram: line $ c $ is a transversal, intersecting $ a $ at one point (forming ∠1, ∠2, ∠3, ∠4) and $ b $ at another (forming ∠5, ∠6, ∠7, ∠8). So ∠1 and ∠5 are corresponding (equal, since $ a\parallel b $), ∠5 and ∠6 are supplementary (linear pair). So $ m\angle1 + m\angle6 = 180^\circ $, so they are supplementary, but the option says "because they are supplementary angles"—but are they equal? No, supplementary angles sum to 180, not necessarily equal. So B is wrong.
- Option C: ∠1 and ∠8: Let's see. ∠1 and ∠3 are vertical, ∠3 and ∠7 are corresponding ( $ a\parallel b $ ), ∠7 and ∠8 are vertical. So $ m\angle1 = m\angle3 = m\angle7 = m\angle8 $? Wait, no—∠7 and ∠8 are supplementary (linear pair). Wait, no, ∠5 and ∠6 are supplementary, ∠5 and ∠7 are vertical, ∠6 and ∠8 are vertical. So ∠1 and ∠8: ∠1 and ∠5 are corresponding (equal), ∠5 and ∠8 are alternate exterior angles? Wait, lines $ a $ and $ b $ are parallel, transversal $ c $. ∠1 is at the top - left of the first intersection, ∠8 is at the bottom - right of the second intersection. Wait, maybe I made a mistake. Let's recall angle types: vertical angles (opposite each other when two lines intersect, equal), corresponding angles (equal when lines are parallel), alternate interior/exterior (equal when lines are parallel), supplementary (sum to 180). ∠1 and ∠4: linear pair (supplementary). ∠1 and ∠2: linear pair. ∠1 and ∠3: vertical (equal). Now, ∠1 and ∠8: Let's see, ∠1 and ∠5 are corresponding (equal, $ a\parallel b $ ), ∠5 and ∠8 are vertical? No, ∠5 and ∠7 are vertical, ∠6 and ∠8 are vertical. Wait, maybe the option C says "m∠1 = m∠8, because they are alternate angles"—wait, alternate exterior angles: ∠1 and ∠8—if we consider the two parallel lines $ a $ and $ b $, and transversal $ c $, ∠1 is an exterior angle on the left of line $ a $, ∠8 is an exterior angle on the right of line $ b $? No, maybe I mislabel. Wait, the correct reasoning for option D: ∠1 and ∠4—no, ∠1 and ∠3 are vertical? Wait, no, the diagram: when two lines intersect, vertical angles are opposite. So at the first intersection (line $ a $ and $ c $), ∠1 and ∠4 are adjacent? Wait, no, the labels: ∠1, ∠2, ∠3, ∠4 around the intersection of $ a $ and $ c $. So ∠1 and ∠3 are vertical (opposite), ∠2 and ∠4 are vertical. Wait, maybe the option D is $ m\angle1 = m\angle4 $ because they are vertical angles? No, that's wrong. Wait, I think I misread the options. Let's re - read:

Option D: $ m\angle1 = m\angle4 $ because they are vertical angles? No, vertical angles are equal, but ∠1 and ∠4—wait, maybe the labels are ∠1, ∠2 (top), ∠3, ∠4 (bottom) at the first intersection. So ∠1 and ∠4: ∠1 + ∠2 = 180, ∠2 + ∠4 = 180, so ∠1 = ∠4 (vertical angles? Wait, no, vertical angles are opposite. If ∠1 is top - left, ∠4 is bottom - right, then they are vertical angles. So $ m\angle1 = m\angle4 $ because they are vertical angles. Wait, but earlier I thought ∠1 and ∠3 are vertical. Maybe the diagram has ∠1, ∠2 (adjacent, linear pair), ∠3, ∠4 (adjacent, linear pair), with ∠1 opposite ∠4 and ∠2 opposite ∠3. So vertical angles: ∠1 and ∠4, ∠2 and ∠3. So then $ m\angle1 = m\angle4 $ (vertical angles are equal). Then option D: "m∠1 = m∠4, because they are vertical angles"—that's correct.

Wait, let's re - analyze each option:

- Option A: "m∠1 = m∠4, because they are opposite angles"—the term "opposite angles" is not standard; vertical angles are the correct term, but if ∠1 and ∠4 are vertical, then why is A wrong? Wait, maybe the diagram has ∠1 and ∠3 as vertical. I think I messed up the angle labels. Let's start over. When two lines intersect, the vertical angles are the pairs opposite each other. So if we have two intersecting lines, forming four angles: ∠1 (top - left), ∠2 (top - right), ∠3 (bottom - left), ∠4 (bottom - right). Then ∠1 and ∠4 are not vertical—∠1 and ∠3 are vertical, ∠2 and ∠4 are vertical. So in that case, ∠1 and ∠3 are vertical (equal), ∠2 and ∠4 are vertical (equal). Then:

- Option A: "m∠1 = m∠4"—∠1 and ∠4: ∠1 + ∠2 = 180, ∠2 + ∠4 = 180, so ∠1 = ∠4 (supplementary to the same angle), but they are not vertical angles. So the reason "because they are opposite angles" is wrong.

- Option B: "m∠1 = m∠6"—∠1 and ∠6: lines $ a\parallel b $, transversal $ c $. ∠1 and ∠5 are corresponding (equal), ∠5 and ∠6 are supplementary (linear pair), so ∠1 + ∠6 = 180, not equal. So B is wrong.

- Option C: "m∠1 = m∠8"—∠1 and ∠8: ∠1 and ∠3 are vertical, ∠3 and ∠7 are corresponding ( $ a\parallel b $ ), ∠7 and ∠8 are supplementary (linear pair), so ∠3 + ∠8 = 180, so ∠1 + ∠8 = 180, not equal. So C is wrong.

- Option D: "m∠1 = m∠4"—∠1 and ∠4: as ∠1 + ∠2 = 180 and ∠2 + ∠4 = 180, by the transitive property, ∠1 = ∠4. And they are vertical angles (if ∠1 and ∠4 are opposite each other when the two lines intersect). So the reason "because they are vertical angles" is correct, and $ m\angle1 = m\angle4 $. So D is correct.

# Answer:
D. $ m\angle1 = m\angle4 $, because they are vertical angles

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