Question

consider the triangles shown.
if ( mangle utv < mangle uts < mangle str ), which statement is true?
( \bigcirc ) ( vu < us < sr ) by the hinge theorem.
( \bigcirc ) ( vu = us = sr ) by the hinge theorem.
( \bigcirc ) ( mangle utv = mangle ust = mangle str ) by the converse of the hinge theorem.
( \bigcirc ) ( mangle utv > mangle uts > mangle str ) by the converse of the hinge theorem.

Explanation:

Step1: Identify Hinge Theorem conditions

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, the triangle with the larger included angle has the longer third side. From the diagram, \(UT = ST\) (marked congruent), and \(TV = TR\) (marked congruent). The included angles are \(\angle UTV\), \(\angle UTS\), \(\angle STR\) (note: \(\angle UTS\) is the included angle for sides \(UT, ST\) and the segment between them, but since \(UT=ST\) and \(TV=TR\), the third sides opposite these angles are \(VU\), \(US\), \(SR\) respectively).

Step2: Apply angle order to side lengths

Given \(m\angle UTV < m\angle UTS < m\angle STR\), by the Hinge Theorem, the corresponding third sides must follow the same order: \(VU < US < SR\).

Answer:

A. VU < US < SR by the hinge theorem.