Question

1. in the accompanying diagram, $overline{ab}$ is parallel to $overline{cd}$, and $overline{ef}$ is a transversal. if $mangle bef=(2x + 60)^{circ}$ and $mangle dfe=(3x + 20)^{circ}$, what is $mangle bef$?

Explanation:

Step1: Use property of alternate - interior angles

Since $\overline{AB}\parallel\overline{CD}$ and $\overline{EF}$ is a transversal, $\angle BEF$ and $\angle DFE$ are alternate - interior angles, so $\angle BEF=\angle DFE$.
$2x + 60=3x+20$

Step2: Solve the equation for $x$

Subtract $2x$ from both sides: $60=x + 20$.
Then subtract 20 from both sides: $x=40$.

Step3: Find the measure of $\angle BEF$

Substitute $x = 40$ into the expression for $\angle BEF$.
$m\angle BEF=2x+60=2\times40 + 60=80 + 60=140^{\circ}$

Answer:

$140^{\circ}$