Question

the diagram shows two parallel lines and a transversal. if the measure of $\angle 6$ is $140^\circ$ , what is the measure of $\angle 1$? \
\
a. $50^\circ$ \
b. $60^\circ$ \
c. $40^\circ$ \
d. $140^\circ$

Explanation:

Step1: Identify relationship between ∠6 and ∠7

∠6 and ∠7 are same - side interior angles? No, wait, ∠6 and ∠3 are adjacent supplementary? Wait, no, the two lines are parallel. ∠6 and ∠7: since the two lines are parallel, ∠6 + ∠7 = 180°? Wait, no, ∠6 and ∠3 are adjacent, ∠6 and ∠5 are vertical? Wait, first, ∠6 and ∠3: they are adjacent angles forming a linear pair? Wait, no, the transversal cuts the parallel lines. Let's see, ∠6 and ∠7: since the two lines are parallel, ∠6 and ∠7 are same - side interior angles? Wait, no, ∠6 and ∠2: corresponding angles? Wait, maybe first find ∠3. ∠6 and ∠3 are adjacent supplementary? Wait, ∠6 and ∠3: they form a linear pair? Wait, ∠6 + ∠3 = 180°? No, ∠6 and ∠4: vertical angles? Wait, maybe better to find ∠7 first. ∠6 and ∠7: since the two lines are parallel, ∠6 + ∠7 = 180°? Wait, ∠6 is 140°, so ∠7 = 180° - 140° = 40°? Wait, no, ∠7 and ∠1: vertical angles? Wait, ∠7 and ∠1: are they vertical? Wait, ∠7 and ∠2: adjacent, ∠2 and ∠1: vertical? Wait, let's label the angles. The two parallel lines, transversal. ∠6 and ∠2: corresponding angles? Wait, no, ∠6 and ∠7: alternate interior angles? Wait, no, ∠6 and ∠3: alternate interior? Wait, maybe I made a mistake. Let's start over.

∠6 and ∠3: they are adjacent angles on a straight line, so ∠6 + ∠3 = 180°? Wait, no, ∠6 and ∠4: vertical angles, ∠6 and ∠5: vertical? Wait, the first parallel line: angles 5,4,3,6. The second:7,2,1,8. ∠6 and ∠7: same - side interior angles? Since the lines are parallel, same - side interior angles are supplementary. So ∠6 + ∠7 = 180°. So ∠7 = 180° - 140° = 40°. Then ∠7 and ∠1: are they vertical angles? Wait, ∠7 and ∠1: ∠7 and ∠8 are adjacent supplementary, ∠8 and ∠1 are vertical? Wait, no, ∠7 and ∠1: let's see, ∠7 and ∠2: adjacent, ∠2 and ∠1: vertical. Wait, ∠7 and ∠1: are they equal? Wait, ∠7 and ∠1: ∠7 is 40°, then ∠1 is also 40°? Wait, but let's check the options. Option C is 40°. Wait, maybe my steps:

Step1: Find ∠7

Since the two lines are parallel, ∠6 and ∠7 are same - side interior angles, so they are supplementary. So \( \angle 6+\angle 7 = 180^{\circ} \). Given \( \angle 6 = 140^{\circ} \), then \( \angle 7=180^{\circ}- 140^{\circ}=40^{\circ} \).

Step2: Find ∠1

∠7 and ∠1 are vertical angles, so they are equal. So \( \angle 1=\angle 7 = 40^{\circ} \).

Answer:

C. 40°