Imagine two lines intersect. how can the properties of linear pairs and vertical angles to determine the angle measures created by the intersecting lines? explain.

Vertical angles are opposite angles that share only a vertex.
When two lines intersect, we have two sets of vertical angles.

Linear pairs are two angles that form a straight line; when two lines intersect, we have two linear pairs. 

If we know one of the angles formed when two lines intersect, we know that the angle opposite it, the vertical angle, has the same measure.
We also know that the angle beside it, that forms a linear pair, will be supplementary with it (add to 180); this means we can find its measure by subtracting from 180.
The remaining missing angle will be the same as this supplementary angle, since they are vertical angles