Question

problem 6
line de is parallel to line fg.
is it possible to determine all five missing angle measures with the given information?
show or explain your thinking.

Explanation:

Step1: Use linear - pair property

The angle adjacent to the $108^{\circ}$ angle on line $FG$ forms a linear - pair. Let this angle be $\angle FBA$. Since the sum of angles in a linear - pair is $180^{\circ}$, $\angle FBA=180 - 108=72^{\circ}$.

Step2: Use corresponding - angles property

Since $DE\parallel FG$, the angle corresponding to the $30^{\circ}$ angle with respect to the transversal $AB$ is $\angle FBA$.

Step3: Use triangle - angle - sum property

In $\triangle ABC$, we know one angle is $30^{\circ}$ and $\angle ABC = 72^{\circ}$. Let the third angle $\angle ACB$ be $x$. By the triangle - angle - sum property ($\angle BAC+\angle ABC+\angle ACB = 180^{\circ}$), we have $x=180-(30 + 72)=78^{\circ}$.
The angle adjacent to $\angle ACB$ on line $DE$ forms a linear - pair with it, so it is $180 - 78=102^{\circ}$. And the angle corresponding to this $102^{\circ}$ angle with respect to the transversal $BC$ and parallel lines $DE$ and $FG$ can also be found.

Answer:

Yes