Question

the diagram shows the first few steps of a construction using a compass and straightedge. the three arcs were constructed using a compass set to the same width. the following statements were made: 1. connecting points u and t completes the construction of a bisector of ∠ytz. ii. connecting points u and y completes the construction of a line parallel to line j through point y. iii. connecting points u and z completes the construction of a line perpendicular to line j through point z. which of the statements made are true?

Explanation:

Step1: Recall angle - bisector construction

For constructing an angle - bisector of an angle, we use a compass to create equal - radius arcs from the vertex of the angle and then connect the intersection of the arcs within the angle to the vertex. Here, for $\angle YTZ$, the construction does not follow the correct steps for an angle - bisector. So, statement I is false.

Step2: Recall parallel - line construction

To construct a line parallel to a given line $j$ through a point $Y$, we use the property of equal corresponding angles. By creating equal - radius arcs from points on the given line and the external point, and then using the intersection of the arcs to draw a line. Connecting $U$ and $Y$ creates a line with equal corresponding angles with respect to line $j$ and the transversal passing through $Y$ and a point on $j$. So, statement II is true.

Step3: Recall perpendicular - line construction

For constructing a line perpendicular to a given line $j$ through a point $Z$, the construction steps are not followed correctly here. So, statement III is false.

Answer:

II. Connecting points $U$ and $Y$ completes the construction of a line parallel to line $j$ through point $Y$.