Question

some steps to construct an angle mnt congruent to angle pqr are listed below
step 1: use a compass to draw an arc from point q that intersects the side pq at point a and the side qr at point b
step 2: draw a segment nt, and use the same width of the compass to draw an arc from point n that intersects the segment nt at a point x
step 3:
step 4: join points n and y using a straightedge
which statement describes step 3 correctly?
adjust the width of the compass to ab, and draw an arc from point x such that it intersects the arc drawn from n in a point y
adjust the width of the compass to aq, and draw an arc from point x such that it intersects the arc drawn from n in a point y
adjust the width of the compass to bq, and draw an arc from point x such that it intersects the arc drawn from n in a point y
adjust the width of the compass to nx, and draw an arc from point x such that it intersects the arc drawn from n in a point y

Explanation:

Step1: Analyze the construction of congruent angles

When constructing an angle congruent to another angle, after creating the initial arcs on the original angle (\(\angle PQR\)) and the new segment (\(NT\)), the next step is to transfer the length of the arc - segment on the original angle (in this case \(AB\)) to the new construction.

Step2: Consider the role of each segment

The length \(AB\) (the distance between the two intersection points of the arc with the sides of \(\angle PQR\)) is used to ensure the congruency of the angles. By setting the compass width to \(AB\) and drawing an arc from \(X\) (the intersection point of the arc with \(NT\)), we can find the point \(Y\) which helps in forming the congruent angle \(\angle MNT\) to \(\angle PQR\).

Answer:

Adjust the width of the compass to \(AB\), and draw an arc from point \(X\) such that it intersects the arc drawn from \(N\) in a point \(Y\)