given: l || m assume the transversal is perpendicular to m. find m ∠1. diagram of lines l, m, transversal, 30° angle, angle 1 m ∠1 = ○ 45 ○ 30 ○ 60

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# Explanation:
## Step1: Analyze transversal and parallel lines
Since $ l \parallel m $ and the transversal is perpendicular to $ m $, it is also perpendicular to $ l $ (perpendicular transversal theorem), so the angle between the transversal and $ m $ (or $ l $) is $ 90^\circ $.
## Step2: Use angle relationship
The $ 30^\circ $ angle and $ \angle 1 $ are complementary (sum to $ 90^\circ $) because the transversal is perpendicular to $ l $. So $ m\angle 1 + 30^\circ = 90^\circ $.
## Step3: Solve for $ m\angle 1 $
Subtract $ 30^\circ $ from both sides: $ m\angle 1 = 90^\circ - 30^\circ = 60^\circ $? Wait, no, wait. Wait, actually, since $ l \parallel m $, the alternate interior angles: the $ 30^\circ $ angle and $ \angle 1 $ – wait, no, the transversal is perpendicular to $ m $, so the angle between transversal and $ m $ is $ 90^\circ $. The $ 30^\circ $ angle and the angle adjacent to $ \angle 1 $ (on $ m $) are equal (alternate interior). Wait, maybe better: the transversal is perpendicular to $ m $, so it's $ 90^\circ $ with $ m $. Then, since $ l \parallel m $, the transversal makes $ 90^\circ $ with $ l $ too. The $ 30^\circ $ angle and $ \angle 1 $: let's see, the right angle (90°) is split into $ 30^\circ $ and $ \angle 1 $? Wait, no, the $ 30^\circ $ angle is on $ m $, and since $ l \parallel m $, the corresponding angle to $ 30^\circ $ on $ l $ would be equal, but then $ \angle 1 $ and that corresponding angle are complementary (since transversal is perpendicular to $ l $, so 90° total). Wait, no, maybe I messed up. Wait, the transversal is perpendicular to $ m $, so the angle between transversal and $ m $ is $ 90^\circ $. The $ 30^\circ $ angle is formed by the slant line and $ m $, so the angle between slant line and transversal on $ m $ is $ 90^\circ - 30^\circ = 60^\circ $? No, wait, the slant line (the transversal? No, the slant line is another line, not the perpendicular one. Wait, the diagram: there's a vertical line (perpendicular to $ m $ and $ l $, since $ l \parallel m $), a slant line intersecting both $ l $ and $ m $, forming a $ 30^\circ $ angle with $ m $ and the vertical line. Then, on $ l $, the angle $ \angle 1 $ is between the vertical line and the slant line. Since $ l \parallel m $, the alternate interior angles: the $ 30^\circ $ angle on $ m $ and the angle adjacent to $ \angle 1 $ (on $ l $) are equal? Wait, no, the vertical line is perpendicular to $ m $, so it's $ 90^\circ $ to $ m $. The slant line makes $ 30^\circ $ with $ m $, so the angle between slant line and vertical line on $ m $ is $ 90^\circ - 30^\circ = 60^\circ $? No, wait, the $ 30^\circ $ is between the slant line and $ m $, so the angle between slant line and vertical line (perpendicular to $ m $) is $ 90^\circ - 30^\circ = 60^\circ $? But since $ l \parallel m $, the angle $ \angle 1 $ should be equal to that angle? Wait, no, maybe the $ 30^\circ $ and $ \angle 1 $ are complementary? Wait, no, let's start over.

Given $ l \parallel m $, transversal (vertical line) is perpendicular to $ m $, so it's perpendicular to $ l $ (because if a transversal is perpendicular to one of two parallel lines, it's perpendicular to the other). So the vertical line and $ l $ form a right angle (90°). The slant line intersects $ l $ and $ m $, forming a $ 30^\circ $ angle with $ m $ and the vertical line. Since $ l \parallel m $, the alternate interior angle to the $ 30^\circ $ angle (on $ m $) is equal to $ 30^\circ $ on $ l $. Then, $ \angle 1 $ and that $ 30^\circ $ angle are complementary (since vertical line is perpendicular to $ l $, so their sum is 90°). Wait, no: vertical line is perpendicular to $ l $, so the angle between vertical line and $ l $ is 90°. The slant line splits that 90° into $ \angle 1 $ and the alternate interior angle (30°). So $ \angle 1 + 30° = 90° $, so $ \angle 1 = 60° $? But wait, the options are 45, 30, 60. Wait, no, maybe I got the angle wrong. Wait, maybe the $ 30^\circ $ is equal to $ \angle 1 $ because of alternate interior angles? Wait, no, the vertical line is perpendicular, so 90°. Let's look at the diagram again (as described): there's a vertical line (perpendicular to $ m $), a horizontal line $ m $, a horizontal line $ l $ above $ m $ (parallel). A slant line crosses both, making a 30° angle with $ m $ and the vertical line (so between slant line, $ m $, and vertical line is 30°). Then, on $ l $, the angle $ \angle 1 $ is between slant line, $ l $, and vertical line. Since $ l \parallel m $, the alternate interior angle: the angle between slant line, $ m $, and vertical line (30°) is equal to the angle between slant line, $ l $, and the vertical line? No, that would be if the slant line is the transversal, but the vertical line is the perpendicular transversal. Wait, maybe the $ 30^\circ $ and $ \angle 1 $ are equal because of corresponding angles? Wait, no, the vertical line is perpendicular, so 90°. Wait, maybe the answer is 60? Wait, no, let's calculate: if the vertical line is perpendicular to $ m $, then the angle between vertical line and $ m $ is 90°. The slant line makes 30° with $ m $, so the angle between slant line and vertical line on $ m $ is 90° - 30° = 60°. But since $ l \parallel m $, the angle $ \angle 1 $ (on $ l $) is equal to that 60°? No, that can't be. Wait, maybe I mixed up the angles. Wait, the problem says "the transversal is perpendicular to $ m $". So the transversal is the vertical line. Then, the slant line is another line, intersecting $ l $ and $ m $, forming a 30° angle with $ m $ and the transversal (vertical line). So on $ m $, between slant line, transversal, and $ m $ is 30°, so the angle between slant line and transversal is 90° - 30° = 60°? No, transversal is perpendicular to $ m $, so angle between transversal and $ m $ is 90°, so angle between slant line and transversal is 90° - 30° = 60°? Then, since $ l \parallel m $, the angle $ \angle 1 $ (on $ l $) is equal to that angle (60°) because of corresponding angles? Wait, but the options include 60. Wait, but let's check again. Wait, maybe the 30° is equal to $ \angle 1 $ because of alternate interior angles. Wait, no, the transversal is vertical, perpendicular to $ m $, so it's perpendicular to $ l $. The slant line intersects $ l $ and $ m $, so the alternate interior angle to the 30° angle (on $ m $) is equal to 30° on $ l $. Then, $ \angle 1 $ and that 30° angle are complementary (since transversal is perpendicular to $ l $, so their sum is 90°), so $ \angle 1 = 90° - 30° = 60° $. Yes, that makes sense. So $ m\angle 1 = 60^\circ $.

Wait, but the options are 45, 30, 60. So 60 is an option. So that's the answer.

# Answer:
$ 60^\circ $ (corresponding to the option 60)

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze transversal and parallel lines

Step2: Use angle relationship

Step3: Solve for \( m\angle 1 \)

Answer: