Question

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if \\(\overline{sv} \cong \overline{uv}\\) and \\(m\angle stv = 47^\circ\\), what is \\(m\angle utv\\)?
\\(m\angle utv = \square^\circ\\)

Explanation:

Step1: Identify Right Triangles

Triangles \( \triangle UTV \) and \( \triangle STV \) are right triangles (since \( \angle TUV \) and \( \angle TSV \) are right angles). \( \overline{SV} \cong \overline{UV} \) and \( \overline{TV} \) is common to both triangles. So, \( \triangle UTV \cong \triangle STV \) by HL (Hypotenuse-Leg) congruence.

Step2: Use Congruence to Find Angle

Since \( \triangle UTV \cong \triangle STV \), their corresponding angles \( \angle UTV \) and \( \angle STV \) are equal? Wait, no—wait, actually, the right angles are \( \angle U \) and \( \angle S \), hypotenuse \( TV \), leg \( UV \cong SV \). Wait, maybe \( \angle UTV \) and \( \angle STV \) are related. Wait, no, let's re-examine. The triangles \( \triangle UTV \) and \( \triangle STV \): right angles at \( U \) and \( S \), \( UV = SV \), \( TV = TV \). So HL congruence. Therefore, \( \angle UTV = \angle STV \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the problem says \( \overline{SV} \cong \overline{UV} \), and \( TV \) is the hypotenuse. So triangles \( TUV \) and \( TSV \) are right-angled at \( U \) and \( S \), with \( UV = SV \) and \( TV \) common. So by HL, \( \triangle TUV \cong \triangle TSV \). Therefore, \( \angle UTV = \angle STV \)? Wait, but \( m\angle STV = 47^\circ \), so \( m\angle UTV = 47^\circ \)? Wait, no, that seems off. Wait, maybe the triangles are congruent, so corresponding angles: \( \angle UTV \) corresponds to \( \angle STV \)? Wait, maybe I messed up the correspondence. Let's label the triangles: \( \triangle TUV \) has right angle at \( U \), legs \( TU, UV \); \( \triangle TSV \) has right angle at \( S \), legs \( TS, SV \). Since \( UV = SV \) and \( TV \) is hypotenuse, HL congruence implies \( \triangle TUV \cong \triangle TSV \). Therefore, \( \angle UTV = \angle STV \). Wait, but then \( m\angle UTV = 47^\circ \)? Wait, maybe that's correct. Wait, let's check again. The right angles are at \( U \) and \( S \), \( UV = SV \), \( TV \) is common. So HL: hypotenuse \( TV \), leg \( UV = SV \). So triangles are congruent. Therefore, corresponding angles: \( \angle UTV \) and \( \angle STV \) are equal. Wait, but the problem says \( m\angle STV = 47^\circ \), so \( m\angle UTV = 47^\circ \)? Wait, maybe I made a mistake. Wait, no—wait, maybe the angles at \( T \) are equal. So yes, \( m\angle UTV = 47^\circ \).

Answer:

\( 47 \)