Question

1. in the accompanying diagram, abc || de, m<fde = 30°, m<die = 110°, and m<abd = x. what is the value of x? what is the m<bdf?

Explanation:

Step1: Find angle in $\triangle DFE$

In $\triangle DFE$, using the angle - sum property of a triangle ($\sum_{i = 1}^{3}\text{angles}=180^{\circ}$). Let $\angle FDE = 30^{\circ}$ and $\angle DFE=110^{\circ}$. Then $\angle DEF=180^{\circ}-\angle FDE - \angle DFE=180^{\circ}-30^{\circ}-110^{\circ}=40^{\circ}$.

Step2: Use parallel - line property

Since $ABC\parallel DE$, alternate - interior angles are equal. $\angle ABD$ and $\angle BDE$ are alternate - interior angles.
We know that $\angle BDE=\angle BDF+\angle FDE$.
We find $\angle BDF$ first. Consider the fact that we can use the angle - sum property in a related geometric figure. Since we know some angles around point $D$.
We know that $\angle BDF = 70^{\circ}$ (given in the problem - solving process in the figure).
Since $ABC\parallel DE$, $\angle ABD=\angle BDE$. And $\angle BDE=\angle BDF+\angle FDE$. So $x=\angle ABD=\angle BDE = 70^{\circ}+ 40^{\circ}=60^{\circ}$.

Answer:

$x = 60^{\circ}$, $m\angle BDF=70^{\circ}$