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.

Linear pairs are adjacent angles that are supplementary.
Their measures add to 180 deg.
Vertical angles are congruent.

Let's say two lines intersect, and you know the measure of one of the 4 angles that were formed. One angle measures k degrees. You can find all the other measures.
Since one angle measures k degrees, the angle vertical to it also measures k degrees.
Each of the other two angles forms a linear pair with the angle that measures k. That means each angle is supplementary to the angle measuring k degrees and has a measure of (180 - k) degrees.