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 congruent and adjacent to a given angle, follow these steps using a straightedge and compass:
1. Begin by drawing a line segment with an endpoint that serves as the vertex of the given angle.
2. Place the compass on the vertex of the given angle and draw an arc that intersects both sides of the angle.
3. Without changing the compass width, place the compass on the point where one side of the angle intersects the arc (let's call this point P) and draw an arc that cuts across the given angle.
4. Using the straightedge, draw a line passing through the vertex of the given angle and the point where the second side of the angle intersects the last drawn arc (let's call this point Q). This line should extend beyond the vertex.
5. Keeping the compass width unchanged, place the compass on point Q and draw an arc that intersects the line from step 4.
6. Now, without changing the compass width, place the compass on the vertex of the given angle and draw an arc that intersects the last drawn arc from both sides. Let the two intersection points with the last drawn arc be marked as R and S.
7. Use a straightedge to connect point R with point Q.
8. Finally, the constructed line segment RQ forms the desired angle that is congruent and adjacent to the given angle.
To construct an angle that is both congruent and adjacent to a given angle using a straightedge and a compass, follow these steps:
1. Begin by drawing the given angle, let's call it angle ABC.
2. Take a compass and place the point on point B of angle ABC. Open the compass to any convenient width.
3. Without changing the compass width, draw an arc that intersects both sides of angle ABC. Label the points where the arc intersects the sides as D and E.
4. With the same compass width, place the compass point on point D and draw an arc that intersects the side of angle ABC. Label the point of intersection as F.
5. Now, place the compass point on point E and draw another arc that intersects the previously drawn arc at point G.
6. Using a straightedge, draw a line segment from point D to point G, and label the point of intersection with the side of angle ABC as H.
7. Finally, draw an angle with the same measure as angle ABC using a straightedge by drawing a line segment from point B to point H.
The constructed angle BH will be both congruent and adjacent to angle ABC.