Question

in the diagram, ( a parallel b ). what is ( mangle1 )? a. ( 122^{circ} ) b. ( 118^{circ} ) c. ( 72^{circ} ) d. ( 62^{circ} )

Explanation:

Step1: Use the property of same - side interior angles

Since \(a\parallel b\), the sum of \((7x - 1)^{\circ}\) and \((3x + 11)^{\circ}\) is \(180^{\circ}\) (same - side interior angles are supplementary).
So, \((7x-1)+(3x + 11)=180\).
Simplify the left - hand side: \(7x-1+3x + 11=180\), \(10x+10 = 180\).
Subtract \(10\) from both sides: \(10x=180 - 10=170\).
Divide both sides by \(10\): \(x = 17\).

Step2: Find the measure of the angle adjacent to \(\angle1\)

Substitute \(x = 17\) into \((3x + 11)^{\circ}\): \(3\times17+11=51 + 11=62^{\circ}\).

Step3: Use the property of adjacent angles (linear pair)

Since \(\angle1\) and the angle \((3x + 11)^{\circ}\) (which is \(62^{\circ}\)) form a linear pair (they are adjacent and supplementary), \(m\angle1=180-(3x + 11)\).
Substitute \(x = 17\): \(m\angle1=180-62 = 118^{\circ}\).

Answer:

B. \(118^{\circ}\)