Two angles that are on the exterior of a pair of parallel lines are supplementary. This means that their measures add up to 180°. So, given two linear angles whose measures are given by expressions with variables, add these expressions and set their sum equal to 0. Solve using algebra techniques.
Same side interior angles can be recognized by being between two parallel lines and on the same side of the transversal. We know that same side interior angles are supplementary, so their measures add up to 180°. Again, if these same side interior angles are given in variables, add the expressions together, set the sum equal to 180°, and use algebra to solve for the variable.
If we look at this example right here, we’re being asked to solve for two different variables, x and y. Let’s start with the Xs what do we know?
We have 2 parallel lines and we have a transversal. 2x and 3x are on the same side of that transversal. They’re also on the exterior of the two parallel lines which means if I add them up 2x plus 3x they’re going to have to be supplementary. If I solve this for x I’m going to combine like terms so I have 5x equals 180. I’m going to divide by 5 and everyone knows that that’s 36, so x equals 36.
Let’s look at the Ys. We have 2 parallel lines and a transversal. 6y and 9y are on the same side of that transversal and they’re in between the parallel lines or the interior, so these two are same side interior angles which means they’re also supplementary, so we’re going to say 6y plus 9y equals 180. Combine like terms 15y equals 180 and if we divide by 15, 15 goes into 180 12 times, so we’ve solved this problem by saying that same side exterior angles, same side interior angles are always supplementary.