Question

lines b and c are parallel. what is the measure of ∠2? diagram: line ( a ) (transversal) intersects parallel lines ( b ) and ( c ). on line ( b ), adjacent angles ( angle 1 ) (labeled ( (7x+1)^circ )) and ( angle 2 ) (labeled ( (18x+4)^circ )) are shown. on line ( c ), angles ( angle 5, angle 6, angle 7, angle 8 ) are shown. options: ( circ mangle 2 = 31^circ ), ( circ mangle 2 = 50^circ ), ( circ mangle 2 = 120^circ ), ( circ mangle 2 = 130^circ )

Explanation:

Step1: Identify supplementary angles

∠1 and ∠2 are supplementary (linear pair), but also, since lines b and c are parallel, ∠1 and ∠6 are equal (corresponding angles), and ∠2 and ∠6 are same - side interior angles? Wait, no, actually ∠1 and ∠2 are adjacent and form a linear pair, so \(m\angle1 + m\angle2=180^{\circ}\). But also, ∠1 and ∠5 are equal (alternate interior angles), and ∠5 and ∠6 are supplementary? Wait, no, let's correct. ∠1 and ∠2 are adjacent angles on a straight line, so they are supplementary: \((7x + 1)+(18x + 4)=180\).

Step2: Solve for x

Combine like terms: \(7x+18x + 1 + 4=180\)
\(25x+5 = 180\)
Subtract 5 from both sides: \(25x=180 - 5=175\)
Divide both sides by 25: \(x=\frac{175}{25}=7\)

Step3: Calculate \(m\angle2\)

Substitute \(x = 7\) into the expression for ∠2: \(18x+4\)
\(18\times7+4=126 + 4=130^{\circ}\)

Answer:

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