Question

central angles
applying angles-chords-arcs congruency theorems
the minor arc measures of circle z are shown in the figure. \t\t\tuse the drop-down menus to complete each statement.
\t\t\tangle uzt is congruent to angle
\t\t\tangle vzw is congruent to angle
\t\t\toptions: tzy, xzw, vzw

Explanation:

Step1: Recall Central Angle Theorem

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 for ∠UZT

Arc \( UT \) measures \( 54^\circ \). Now check arcs for other angles:

• Arc \( TY \) is \( 54^\circ \), so the central angle intercepting arc \( TY \) is \( \angle TZY \). Wait, no, wait: \( \angle UZT \) intercepts arc \( UT \) (54°), and \( \angle TZY \) intercepts arc \( TY \) (54°)? Wait, no, looking at the circle: points T, Y, X, W, U, V (wait, V is not labeled? Wait, maybe typo, but the arcs: arc \( UT = 54^\circ \), arc \( TY = 54^\circ \)? Wait, the diagram shows arc \( T \) to \( Y \) is \( 54^\circ \), arc \( U \) to \( T \) is \( 54^\circ \). So central angles \( \angle UZT \) (intercepts \( UT \)) and \( \angle TZY \) (intercepts \( TY \))? Wait, no, maybe I misread. Wait, the first statement: Angle \( UZT \) is congruent to which angle? Let's check the arcs. Arc \( UT = 54^\circ \), arc \( TY = 54^\circ \), so their central angles (angles with vertex at Z) should be congruent. So \( \angle UZT \) (intercepts \( UT \)) and \( \angle TZY \) (intercepts \( TY \))? Wait, but the dropdown has \( TZY \), \( XZW \), \( VZW \). Wait, maybe another arc. Wait, maybe I made a mistake. Wait, let's re-express: Central angle \( \angle UZT \): points U, Z, T. So the arc between U and T is \( 54^\circ \). Now, which other central angle has an arc of \( 54^\circ \)? Arc \( T \) to \( Y \) is \( 54^\circ \), so central angle \( \angle TZY \) (points T, Z, Y) intercepts arc \( TY \) (54°), so \( \angle UZT \cong \angle TZY \) because their intercepted arcs are congruent (both 54°).

For the second part, Angle \( VZW \): Let's find its intercepted arc. Wait, maybe arc \( VW \) is \( 71^\circ \)? Wait, arc \( YX \) is \( 71^\circ \), arc \( XW \) is \( 32^\circ \), arc \( W \) to... Wait, maybe arc \( V \) to \( W \) is \( 71^\circ \), and arc \( Y \) to \( X \) is \( 71^\circ \), so central angle \( \angle VZW \) (intercepts \( VW \)) and \( \angle YZX \) (intercepts \( YX \))? But the dropdown has \( TZY \), \( XZW \), \( VZW \). Wait, maybe I mislabeled. Wait, the second statement: Angle \( VZW \) is congruent to angle... Let's check the arcs. If arc \( V \) to \( W \) is \( 71^\circ \), and arc \( Y \) to \( X \) is \( 71^\circ \), then central angle \( \angle VZW \) (intercepts \( VW \)) and \( \angle YZX \) (intercepts \( YX \))? But the options are \( TZY \), \( XZW \), \( VZW \). Wait, maybe \( \angle XZW \): arc \( XW \) is \( 32^\circ \), no. Wait, maybe the first answer is \( TZY \) for \( \angle UZT \cong \angle TZY \) (since their arcs are 54° each).

Answer:

For "Angle UZT is congruent to angle", the answer is TZY (from the dropdown: A. TZY, B. XZW, C. VZW? Wait, the dropdown shows TZY, XZW, VZW. So the first blank: TZY. For the second, maybe another, but since the first is asked, the answer for the first is TZY.