Explain how you can use a
straightedge and a compass to
construct an angle that is both
congruent and adjacent to a
given angle.
To construct an angle that is both congruent and adjacent to a given angle, you can follow these steps using a straightedge and compass:
1. Place the straightedge on the given angle, aligning it with one of the rays of the angle.
2. Draw a line segment of any length along the straightedge, starting from the vertex of the given angle.
3. Place the point of the compass on the vertex of the given angle and adjust the width to a suitable distance.
4. Without changing the width of the compass, place the compass point on one end of the line segment you drew in step 2 and draw an arc.
5. Again, without changing the width of the compass, place the compass point on the other end of the line segment and draw another arc that intersects the first arc you drew.
6. Use the straightedge to draw a line segment connecting the vertex of the given angle to the point where the two arcs intersect. This line segment will be the congruent and adjacent angle to the given angle.
By constructing the two arcs of the same radius from the endpoints of the line segment, you are essentially copying the width of the compass to create an angle congruent to the given angle. The line segment connecting the vertex of the given angle to the intersection of the two arcs will create an adjacent angle as it shares a side with the given angle.