Question

lines b and c are parallel.
what is the measure of ∠2?
(7x+1)° (18x+4)°
options: m∠2 = 31°, m∠2 = 50°, m∠2 = 120°, m∠2 = 130°

Explanation:

Step1: Identify supplementary angles

∠1 and ∠2 are supplementary (form a linear pair), but also, since lines b and c are parallel, ∠1 and ∠5 are equal (corresponding angles), and ∠5 and ∠6 are supplementary? Wait, no, actually ∠1 and ∠2 are adjacent and form a straight line, so \( (7x + 1) + (18x + 4) = 180 \) (since they are supplementary, sum to 180°).

Step2: Solve for x

Combine like terms: \( 7x + 18x + 1 + 4 = 180 \) → \( 25x + 5 = 180 \). Subtract 5: \( 25x = 175 \). Divide by 25: \( x = 7 \).

Step3: Find measure of ∠2

Substitute x = 7 into \( 18x + 4 \): \( 18(7) + 4 = 126 + 4 = 130 \). Wait, but wait, maybe I made a mistake. Wait, ∠1 and ∠2 are supplementary, but also, ∠1 and ∠5 are equal (corresponding angles), and ∠5 and ∠6 are supplementary? No, actually, ∠1 and ∠2 are adjacent angles on a straight line, so their sum is 180. So solving \( 7x + 1 + 18x + 4 = 180 \) gives x = 7. Then ∠2 is \( 18x + 4 = 18*7 + 4 = 130 \). Wait, but let's check ∠1: \( 7*7 + 1 = 50 \). Then 50 + 130 = 180, which is correct. Also, since lines b and c are parallel, ∠2 and ∠6 should be equal (corresponding angles), and ∠5 and ∠1 are equal. So ∠2 is 130 degrees.

Answer:

\( m\angle2 = 130^\circ \) (the option: \( m\angle2 = 130^\circ \))