Question

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 drop - down menu with options uzv, tzy, yzx.

Explanation:

Step1: Recall central angle - arc relationship

Central angles are congruent if their intercepted arcs are congruent. The measure of a central angle equals the measure of its intercepted arc.

Step2: Analyze arc measures for angle VZW

First, find the measure of arc VW. Wait, no, angle VZW intercepts arc VW? Wait, no, let's check the arcs. The arc between V and W: wait, the arcs given are: arc TU is 54°, arc UY? Wait, no, the circle is divided into arcs: T to Y is 54°, Y to X is 71°, X to W is 32°, W to V is 71°, V to U is 78°, U to T is 54°? Wait, no, looking at the diagram: T to Y: 54°, Y to X:71°, X to W:32°, W to V:71°, V to U:78°, U to T:54°? Wait, no, the points are T, Y, X, W, V, U, back to T. So arc T-Y:54°, Y-X:71°, X-W:32°, W-V:71°, V-U:78°, U-T:54°. Now, angle VZW: the central angle at Z between V and W. The arc VW: wait, W to V is 71°, so arc VW is 71°? Wait, no, W to V is 71°, so the central angle for arc W-V is angle ZWV? No, angle VZW: vertex at Z, sides ZV and ZW. So the intercepted arc is VW. The measure of angle VZW is equal to the measure of arc VW. Now, which other central angle has the same arc measure? Let's check the arcs: arc Y-X is 71°, arc W-V is 71°, arc T-Y is 54°, arc U-T is 54°, arc V-U is 78°, arc X-W is 32°. So arc Y-X (71°) and arc W-V (71°) are congruent. So the central angles intercepting these arcs: angle YZX (intercepts arc Y-X) and angle VZW (intercepts arc W-V) should be congruent? Wait, no, wait angle VZW intercepts arc VW (which is W to V, 71°), and angle YZX intercepts arc YX (71°). Wait, but also, angle YZX: vertex at Z, sides ZY and ZX, intercepts arc YX (71°). Angle VZW: vertex at Z, sides ZV and ZW, intercepts arc VW (71°). Wait, but also, is there another angle? Wait, the options are UZV, TZY, YZX. Let's check the measures. Wait, angle UZV: intercepts arc UV, which is 78°, so measure 78°. Angle TZY: intercepts arc TY, 54°, measure 54°. Angle YZX: intercepts arc YX, 71°, measure 71°. Angle VZW: intercepts arc VW, 71°, so angle VZW is congruent to angle YZX, because their intercepted arcs are both 71°, so the central angles are congruent. Wait, but the options are UZV, TZY, YZX. So YZX is an option. Wait, let's confirm: central angles are congruent if their intercepted arcs are congruent. Arc YX is 71°, arc VW is 71°, so angle YZX (intercepts YX) and angle VZW (intercepts VW) are congruent. So angle VZW is congruent to angle YZX. Wait, but the options are UZV, TZY, YZX. So YZX is the correct one. Wait, but let's check again. Wait, arc W to V is 71°, so angle VZW (central angle) has measure 71°. Arc Y to X is 71°, so angle YZX (central angle) has measure 71°. So they are congruent. So angle VZW is congruent to angle YZX.

Answer:

Angle VZW is congruent to angle YZX. So the answer is YZX (from the dropdown: YZX).