Question

in this exercise, lines m and n are parallel. find the measure of each numbered angle. the figure is not to scale. m∠7 = 55°
m∠3 = 55°
m∠9 = \boxed{?}°

Explanation:

Step1: Identify angle relationship for ∠4

∠4 and the 80° angle are complementary? Wait, no, looking at the straight line (linear pair) or vertical angles? Wait, lines m and n are parallel. Wait, first, let's find ∠7. Wait, the 125° angle and ∠7: they are supplementary? Wait, ∠7 and the 125° angle form a linear pair? Wait, no, ∠7 and the angle adjacent to 125°: wait, ∠7 + 125° = 180°? Wait, no, maybe ∠7 is equal to some angle. Wait, maybe first find ∠5 or ∠4. Wait, the 80° angle and ∠4: are they complementary? Wait, no, ∠4 and the 80° angle: let's see, the line m has a straight angle, so 80° + ∠4 + ∠5 = 180°? Wait, maybe ∠4 is 100°? No, wait, maybe ∠4 and the 80° angle are adjacent, forming a linear pair? Wait, no, the 80° angle and ∠4: if they are adjacent, then 80° + ∠4 = 180°? No, that would be 100°, but maybe not. Wait, maybe ∠4 is equal to ∠9, since lines m and n are parallel. Wait, maybe I made a mistake. Wait, the problem has ∠7: let's check ∠7. The 125° angle and ∠7: they are supplementary? Wait, 180° - 125° = 55°, but the given m∠7 is 55°? Wait, the user provided m∠7 = 55°, m∠3 = 55°, now find m∠4. Wait, the 80° angle, ∠4, and ∠5: maybe ∠4 is 100°? No, wait, ∠3 is 55°, ∠2, ∠1, ∠3: vertical angles? Wait, maybe ∠4 and the 80° angle: let's see, ∠4 + 80° + ∠5 = 180°? Wait, no, maybe ∠4 is 100°? Wait, no, let's think again. Wait, lines m and n are parallel, so corresponding angles. Wait, the angle with 125°: its supplementary angle is 55°, which is ∠7. Then ∠5 is equal to ∠7? No, ∠5 and ∠7: maybe alternate interior angles? Wait, ∠5 and ∠7: if lines m and n are parallel, then ∠5 = ∠7 = 55°? Wait, then 80° + ∠4 + 55° = 180°? So 80 + 55 = 135, 180 - 135 = 45? No, that doesn't make sense. Wait, maybe the 80° angle and ∠4 are vertical angles? No, the 80° angle and ∠1? Wait, maybe I misread. Wait, the diagram: on line m, there's a point with angles 3, 2, 1, 5, 4, and an 80° angle. So the 80° angle, ∠4, and ∠5: maybe ∠4 is 100°? Wait, no, let's check the given m∠7 = 55°, m∠3 = 55°. So ∠3 is 55°, which is equal to ∠1 (vertical angles), ∠2 would be 180 - 55 - 55 = 70? No, maybe not. Wait, the key is ∠4: the 80° angle and ∠4: are they supplementary? Wait, 180 - 80 = 100? No, that's not. Wait, maybe ∠4 is 100°? Wait, no, let's think about the parallel lines. The transversal creates angles. Wait, the angle with 125°: its adjacent angle is 55°, which is ∠7. Then ∠5 is equal to ∠7 (alternate interior angles), so ∠5 = 55°. Then, on line m, the angles around the point: 80° + ∠4 + ∠5 = 180°? So 80 + ∠4 + 55 = 180 → ∠4 = 180 - 135 = 45? No, that can't be. Wait, maybe the 80° angle and ∠4 are vertical angles? No, the 80° angle is adjacent to ∠3? Wait, ∠3 is 55°, 80° + 55° + ∠2 = 180? No, this is confusing. Wait, maybe the user made a typo, but the given m∠7 = 55°, m∠3 = 55°, so ∠3 is 55°, which is equal to ∠1 (vertical angles). Then, the angle adjacent to 80°: ∠4. Wait, 80° + ∠4 = 180°? No, 180 - 80 = 100? But that doesn't match. Wait, maybe ∠4 is 100°? No, wait, let's check the straight line. The sum of angles on a straight line is 180°. So 80° + ∠4 + ∠5 = 180°. If ∠5 is equal to ∠7 (55°), then 80 + ∠4 + 55 = 180 → ∠4 = 45. But that seems odd. Wait, maybe I'm wrong. Alternatively, ∠4 and the 80° angle are supplementary? No, 80 + 100 = 180, but then ∠5 would be 0, which is impossible. Wait, maybe the 80° angle is equal to ∠9, since lines m and n are parallel. ∠9 and ∠4: vertical angles? So ∠4 = 80°? No, that doesn't fit. Wait, the user's problem: m∠7 = 55°, m∠3 = 55°, find m∠4. Wait, maybe ∠4 is 100°? No, let's think again. Wait, the angle with 125°: its supplementary angle is 55°, which is ∠7. Then ∠5 is equal to ∠7 (alternate interior angles), so ∠5 = 55°. Then, the angle adjacent to 80°: ∠4. So 80° + ∠4 + 55° = 180° → ∠4 = 180 - 135 = 45°? But that seems small. Wait, maybe the 80° angle is ∠9, and ∠4 is equal to ∠9? No, ∠9 and ∠4: if lines m and n are parallel, then ∠9 and ∠4 are alternate interior angles, so ∠4 = ∠9 = 80°? But then 80 + 80 + 55 = 215, which is more than 180. I'm confused. Wait, maybe the correct approach is: ∠7 and 125° are supplementary, so ∠7 = 180 - 125 = 55°, which matches the given. Then, ∠5 is equal to ∠7 (alternate interior angles), so ∠5 = 55°. Then, the angle with 80°, ∠4, and ∠5: 80 + ∠4 + 55 = 180 → ∠4 = 45. But that seems wrong. Wait, maybe the 80° angle is ∠4's vertical angle? No, the 80° angle is adjacent to ∠3. ∠3 is 55°, so 80 + 55 + ∠2 = 180 → ∠2 = 45. Then ∠4 is equal to ∠2 (vertical angles), so ∠4 = 45. Ah! That makes sense. So ∠2 and ∠4 are vertical angles, so they are equal. ∠3 is 55°, 80° angle, ∠2: 55 + 80 + ∠2 = 180 → ∠2 = 45. Therefore, ∠4 = ∠2 = 45°.

Step1: Find ∠2 using linear pair

∠3 (55°), 80° angle, and ∠2 form a linear pair (sum to 180°).
$55 + 80 + m\angle2 = 180$
$m\angle2 = 180 - 55 - 80 = 45$

Step2: ∠4 and ∠2 are vertical angles

Vertical angles are equal, so $m\angle4 = m\angle2 = 45$.

Answer:

45