Question

given that (overline{ab}paralleloverline{dc}), (mangle1 = 112), (mangle2 = 4x), and (mangle3 = 3x + 12), find (mangle2). after you enter your answer press go. (mangle2=)

Explanation:

Step1: Find the measure of $\angle BAC$

Since $\angle1 = 112^{\circ}$, and $\angle1+\angle BAC = 180^{\circ}$ (linear - pair angles), then $\angle BAC=180 - 112=68^{\circ}$.

Step2: Use the property of parallel lines

Because $\overline{AB}\parallel\overline{DC}$, then $\angle2+\angle3=\angle BAC$ (alternate - interior angles).
Substitute $m\angle2 = 4x$ and $m\angle3 = 3x + 12$ into the equation: $4x+(3x + 12)=68$.

Step3: Solve the equation for $x$

Combine like terms: $4x+3x+12 = 68$, which simplifies to $7x+12 = 68$.
Subtract 12 from both sides: $7x=68 - 12=56$.
Divide both sides by 7: $x=\frac{56}{7}=8$.

Step4: Find $m\angle2$

Since $m\angle2 = 4x$, substitute $x = 8$ into the formula. Then $m\angle2=4\times8 = 32$.

Answer:

$32$