Question

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

Explanation:

Step1: Use corresponding - angles property

Since lines \(b\) and \(c\) are parallel, \(\angle1=(7x + 1)^{\circ}\) and \(\angle5=(7x + 1)^{\circ}\) (corresponding - angles). Also, \(\angle2\) and \(\angle6\) are corresponding - angles, and \(\angle1\) and \(\angle2\) are a linear pair. \(\angle1\) and \(\angle6\) are alternate - exterior angles, so \((7x + 1)^{\circ}=(18x+4)^{\circ}\) (alternate - exterior angles are equal for parallel lines).

Step2: Solve the equation for \(x\)

\[

$$\begin{align*} 7x+1&=18x + 4\\ 7x-18x&=4 - 1\\ - 11x&=3\\ x&=-\frac{3}{11} \end{align*}$$

\]
This is incorrect. We should use the fact that \(\angle1\) and \(\angle2\) are a linear pair, so \((7x + 1)+(18x+4)=180\) (since they are supplementary).
\[

$$\begin{align*} 7x+1+18x + 4&=180\\ 25x+5&=180\\ 25x&=180 - 5\\ 25x&=175\\ x&=7 \end{align*}$$

\]

Step3: Find the measure of \(\angle2\)

Substitute \(x = 7\) into the expression for \(\angle2=(18x + 4)^{\circ}\).
\[

$$\begin{align*} \angle2&=(18\times7+4)^{\circ}\\ &=(126 + 4)^{\circ}\\ &=130^{\circ} \end{align*}$$

\]

Answer:

\(m\angle2 = 130^{\circ}\)