Question

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

Explanation:

Step1: Find the value of \(x\)

Since \(\overline{AB}\parallel\overline{DC}\), \(\angle 2=\angle 3\) (alternate - interior angles).
So, \(4x = 3x + 12\).
Subtract \(3x\) from both sides: \(4x-3x=3x + 12-3x\), which gives \(x = 12\).

Step2: Find \(m\angle 2\)

Substitute \(x = 12\) into \(m\angle 2=4x\). Then \(m\angle 2=4\times12 = 48\).

Step3: Use the angle - sum property in \(\triangle ADC\)

We know that \(\angle 1\) and \(\angle DAC\) form a linear pair. So \(m\angle DAC=180 - m\angle 1\). Given \(m\angle 1 = 112\), then \(m\angle DAC=180 - 112=68\).
In \(\triangle ADC\), we use the angle - sum property of a triangle (\(m\angle D+m\angle DAC+m\angle 2 = 180\)).
Let \(m\angle D=y\). Then \(y+68 + 48=180\).
Simplify the left - hand side: \(y+116 = 180\).
Subtract 116 from both sides: \(y=180 - 116\).

Answer:

\(64\)