Question

consider the two triangles. if rt is greater than ba, which statement is true? by the converse of the hinge theorem, m∠c = m∠s by the hinge theorem,ts > ac. by the converse of the hinge theorem, m∠s > m∠c by the hinge theorem, ba = rt

Explanation:

Step1: Recall hinge - theorem and its converse

The hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third - side of the first triangle is longer than the third - side of the second triangle. The converse of the hinge theorem 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 longer than the third - side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

Step2: Analyze the given triangles

In \(\triangle ABC\) and \(\triangle RTS\), assume \(AB = RT\) and \(BC=TS\). Given that \(RT>BA\) (which is incorrect as we assume \(AB = RT\), but if we assume the correct comparison based on the structure of the hinge - theorem application), if \(RT>BA\) (should be considered in the context of the congruent - side pairs), and we know that the sides \(BC = TS\) and \(AB\) corresponds to \(RT\), \(AC\) and \(RS\) are the third sides. By the converse of the hinge theorem, if \(RT>BA\) (correcting the comparison logic and assuming the right side - pair congruence), and \(BC = TS\), then the angle opposite to \(RT\) in \(\triangle RTS\) which is \(\angle S\) and the angle opposite to \(BA\) in \(\triangle ABC\) which is \(\angle C\), we have \(m\angle S>m\angle C\).

Answer:

By the converse of the hinge theorem, \(m\angle S > m\angle C\)