Question

Explain how you can use a straight edge and a compass to construct an angle that is both congruent an adjacent to a given angle

Answer 1

An adjacent angle is one of two angles that lie next to each other. They both share a common boundary line but do not overlap each other.  Congruent angles are angles that are opposite each other. They  only share a center point. You construct an adjacent angle and a congruent angle using a straight line and a compass by drawing two congruent angles with the same size angle in both angles. Because both angles are constructed using the same compass measurement; both angles are considered to be both congruent and adjacent. 

Answer 2

The correct answer is:
We will use an edge of the given angle to construct another angle, with the same vertex; we will call that vertex O.
Mark an arc through the sides of the angle. Name the points where the arc intercepts the angle A and B; extend the arc past B.
Set the compass at point B. Open it to the width of AB.
With the compass still set on point B, mark an arc to intersect the first arc at a point C.
Now you have two congruent segments, AB = BC.
Since the distance OA=OB=OC, ∠AOB and ∠BOC are congruent.