Question

given that ( overline{ab} parallel overline{dc} ), ( mangle1 = 112 ), ( mangle2 = 4x ), and ( mangle3 = 3x + 12 ), find ( mangle3 ).

Explanation:

Step1: Find the sum of angles on a straight line

Since \(\angle1\), \(\angle2\) and \(\angle3\) are on a straight line, \(\angle1+\angle2+\angle3 = 180^{\circ}\) (angles on a straight line sum to \(180^{\circ}\)).
Given \(m\angle1 = 112\), \(m\angle2=4x\), \(m\angle3 = 3x + 12\), we substitute these values into the equation:
\(112+4x+(3x + 12)=180\)

Step2: Simplify the equation

Combine like - terms:
\(112 + 12+4x+3x=180\)
\(124 + 7x=180\)
Subtract \(124\) from both sides:
\(7x=180 - 124\)
\(7x=56\)
Divide both sides by \(7\):
\(x=\frac{56}{7}=8\)

Step3: Find \(m\angle3\)

Substitute \(x = 8\) into the formula for \(m\angle3\):
\(m\angle3=3x + 12\)
\(m\angle3=3\times8+12\)
\(m\angle3=24 + 12\)

Answer:

\(36\)