consider the two triangles. if rt is greater than ba, which statement is true? by the converse of the hinge theorem, ( mangle c=mangle s ). by the hinge theorem, ( tsac ). by the converse of the hinge theorem, ( mangle smangle c ). by the hinge theorem, ( ba = rt ).

Question

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

Accepted Answer

# Brief Explanations:
1. First, recall the Hinge Theorem (SAS Inequality Theorem) and its converse:
   - Hinge Theorem: 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.
   - Converse of Hinge Theorem: 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.
2. Analyze the triangles:
   - From the diagram, $AC = TS$ (marked congruent) and $BC = RS$ (marked congruent). So two sides of $\triangle ABC$ are congruent to two sides of $\triangle RTS$ ($AC\cong TS$, $BC\cong RS$).
   - We are given $RT>BA$. So the third side of $\triangle RTS$ ($RT$) is longer than the third side of $\triangle ABC$ ($BA$).
   - By the converse of the Hinge Theorem, the included angle of the triangle with the longer third side should be larger. The included angle for $RT$ in $\triangle RTS$ is $\angle S$, and the included angle for $BA$ in $\triangle ABC$ is $\angle C$. So $m\angle S>m\angle C$.
   - Let's check other options:
     - Option 1: Says $m\angle C = m\angle S$, which is wrong as we just deduced $m\angle S>m\angle C$.
     - Option 2: Hinge Theorem (not converse) would relate included angle to third side. Here we have third side relation, so converse applies, and this option misapplies the theorem and the conclusion $TS>AC$ is wrong as $AC = TS$ (marked congruent).
     - Option 4: Says $BA = RT$, but we are given $RT>BA$, so this is wrong.

# Answer: By the converse of the hinge theorem, $m\angle S>m\angle C$ (the third option: "By the converse of the hinge theorem, $m\angle S>m\angle C$")

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