Question

central angles
applying angles-chords-arcs congruency theorems
the minor arc measures of circle z are shown in the figure.
use the drop-down menus to complete each statement.
angle uzt is congruent to angle tzy
angle vzw is congruent to angle
options: uzv, tzy, yzx

Explanation:

Step1: Recall Central Angle Theorem

Central angles are congruent if their intercepted arcs are congruent. So we need to find the arc intercepted by \(\angle VZW\) and check which angle has the same intercepted arc.

Step2: Find Arc for \(\angle VZW\)

The arc intercepted by \(\angle VZW\) is arc \(VW\). Let's check the measure of arc \(VW\)? Wait, no, wait. Wait, \(\angle VZW\) is a central angle, so its measure is equal to the measure of arc \(VW\)? Wait, no, let's look at the arcs. Wait, arc \(VY\)? No, wait, let's list the arcs: arc \(YT = 54^\circ\), arc \(YX = 71^\circ\), arc \(XW = 32^\circ\), arc \(WV = 71^\circ\), arc \(VU = 78^\circ\), arc \(UT = 54^\circ\). Wait, \(\angle VZW\): points \(V\), \(Z\), \(W\). So the arc between \(V\) and \(W\) is \(71^\circ\) (since arc \(WV\) is \(71^\circ\)). Now, let's check the other angles: \(\angle UZV\): arc \(VU\) is \(78^\circ\)? No, wait, \(\angle YZX\): arc \(YX\) is \(71^\circ\)? Wait, no, \(\angle YZX\): points \(Y\), \(Z\), \(X\). Arc \(YX\) is \(71^\circ\). Wait, \(\angle VZW\) intercepts arc \(VW\) which is \(71^\circ\), and \(\angle YZX\) intercepts arc \(YX\) which is \(71^\circ\)? Wait, no, maybe I made a mistake. Wait, let's re - examine the circle. The arcs: \(YT = 54^\circ\), \(YX = 71^\circ\), \(XW = 32^\circ\), \(WV = 71^\circ\), \(VU = 78^\circ\), \(UT = 54^\circ\). So \(\angle VZW\): vertices \(V\), \(Z\), \(W\), so the arc between \(V\) and \(W\) is \(WV = 71^\circ\). \(\angle YZX\): vertices \(Y\), \(Z\), \(X\), arc between \(Y\) and \(X\) is \(YX = 71^\circ\). So these two central angles have the same arc measure, so they are congruent. \(\angle UZV\) intercepts arc \(VU = 78^\circ\), \(\angle TZY\) intercepts arc \(YT = 54^\circ\). So \(\angle VZW\) is congruent to \(\angle YZX\) because their intercepted arcs (\(VW\) and \(YX\)) are both \(71^\circ\), so the central angles are congruent.

Answer:

\(\angle YZX\) (so the correct option is YZX)