Exercise 3 Page 77

As we can see, this construction requires drawing only one arc with a compass! Let's check the other ones to see if it is the only one that satisfies this condition.

Constructing congruent angles

The following steps will help us construct congruent angles.

1. We draw a ray with endpoint S.
2. With the compass point on the vertex of the given ∠ A, we draw an arc that intersects the sides of ∠ A, and label the points of intersection B and C.
3. With the same compass setting, we put the compass point on point S, draw an arc, and label its point of intersection with the ray as R.
4. We open the compass to the length BC. Then, keeping the same compass setting, we put the compass point on point R and draw an arc to locate point T.
5. Finally, we draw the ray ST.

Ray	Arc on ∠ A	Arc on ray	Arc from point R	ST

We can tell that constructing congruent angles requires drawing three arcs with a compass.

Constructing the perpendicular bisector

The following steps will help us construct a perpendicular bisector.

1. We put the compass point on endpoint A in the given segment and draw a long arc. The opening has to be greater than 12AB.
2. With the same compass setting, we put the compass point on point B and draw another long arc. Also, we label the points where the two arcs intersect as X and Y.
3. We draw the line XY and label its point of intersection with the segment AB as M. The line XY is perpendicular to AB at midpoint M, so it is the perpendicular bisector of AB.

Segment	Arc	Arcs	Bisector

As we can see, this construction requires drawing two arcs with a compass.

Constructing the angle bisector

The following steps will help us construct an angle bisector.

1. We put the compass point on vertex of the given angle ∠ A, draw an arc that intersects the sides of ∠ A, and label the points of intersection B and C.
2. We put the compass point on point C and draw an arc. With the same compass setting, we draw an arc using point B and label the point of intersection D.
3. We draw a ray AD. This is the bisector of ∠ A.

Angle	Arc	Arcs	Bisector

We can tell that constructing the angle bisector requires drawing three arcs with a compass.

Conclusion

As we can see, constructing congruent segments is the only construction that requires drawing only one arc with a compass. This corresponds to option A.