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

Answer 1

Call the vertex of the angle O

Place the compass at O and draw a circle. Extend the sides of the angle so they intersect circle O at A and B.

Place the compass at A and open it to the distance AB.

Draw a circle with center A and radius AB.

Circle A will intersect circle O at another point, C.

Then angle AOB is congruent and adjacent to angle AOC.

Answer 2

ANGLE ABC, center at B

swing compass radius R (any R) around B to make a circle of radius R around B

now put the point of your compass on C and the pencil of it on A. Get distance x between point and pencil

move point of compass to A.
label point where compass pencil crosses your circle D
now <DBA = <ABC