GRE geometry question: Find the angle $s$ from the figure which consists of four lines among which two are parallel to each other

Question

I have to find $s$ from the above figure. The solution given is:

They say that $t$ and $3t+8$ are supplementary angles, and then proceed to find $t$ and tuse its value to solve for $s$.

Looking at the figure I am unable to see how $t$ and $3t+8$ are supplementary angles. I know that supplementary angles sum to $180$ degrees.

Can anyone explain to me how $t$ and $3t+8$ are supplementary? Thanks

EDIT: I think I figured out the supplementary part but how did they find the topmost angle $t+8$?

Answer 1

Because those two lines $a$ and $b$ are parallel therefore:

Answer 2

If you look at the original question, there is an angle of measure $t$ on the exterior of the transversal. Using vertical angles and the fact that $a$ and $b$ are parallel, you can deduce that $3t+8$ is supplementary to $t$ through the same-side interior angle theorem.

You also mention $t+8$as the top-most angle of the triangle. Since you just found that $t + (3t + 8) = 180$, solve for $t$. Now look at the other pair of same-side interior angles: $2t + s = 180$. You can then solve for $s$ and find $t+8$ in the triangle.