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: Recall the Hinge Theorem

The Hinge Theorem (also known as the SAS Inequality Theorem) states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is smaller than the included angle of the second triangle, then the third side of the first triangle is shorter than the third side of the second triangle. Its converse states that if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is shorter than the third side of the second triangle, then the included angle of the first triangle is smaller than the included angle of the second triangle.

Step2: Analyze the Triangles

Looking at the diagram, we can see that \( VT = TR \) (marked with tick marks) and \( UT = ST \) (also marked with tick marks). So, in triangles \( \triangle UTV \), \( \triangle UTS \), and \( \triangle STR \), we have two sides congruent ( \( VT = TR \), \( UT = ST \) for the relevant pairs) and the included angles are \( \angle UTV \), \( \angle UTS \), and \( \angle STR \) respectively.

Step3: Apply the Hinge Theorem

Given \( m\angle UTV < m\angle UTS < m\angle STR \), by the Hinge Theorem, the third sides opposite these angles ( \( VU \), \( US \), and \( SR \) respectively) should satisfy \( VU < US < SR \).

Let's check the other options:

• The second option says \( VU = US = SR \), but the angles are not equal, so this is wrong.
• The third and fourth options refer to the converse of the Hinge Theorem, but the given information is about the angles being in order, not the sides, so we use the Hinge Theorem (not its converse) here. So these options are incorrect.

Answer:

A. \( VU < US < SR \) by the hinge theorem.