Question

in the triangles, \\(\overline{bc} \cong \overline{rs}\\) and \\(\overline{ac} \cong \overline{ts}\\). if rt is greater than ba, which correctly compares angles c and s? \\(\bigcirc\\) (m\angle c = m\angle s) \\(\bigcirc\\) (m\angle c < m\angle s) \\(\bigcirc\\) (m\angle c > m\angle s) \\(\bigcirc\\) (m\angle c \ge m\angle s)

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 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. Conversely, 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.

In $\triangle ABC$ and $\triangle RTS$:

• $\overline{BC} \cong \overline{RS}$ (given)
• $\overline{AC} \cong \overline{TS}$ (given)
• The included angles are $\angle C$ (between $\overline{BC}$ and $\overline{AC}$) and $\angle S$ (between $\overline{RS}$ and $\overline{TS}$)
• The third sides are $\overline{BA}$ (opposite $\angle C$) and $\overline{RT}$ (opposite $\angle S$)

Step2: Apply the Hinge Theorem

We know that $RT > BA$. By the converse of the Hinge Theorem, since two sides of $\triangle RTS$ ($\overline{RS}$ and $\overline{TS}$) are congruent to two sides of $\triangle ABC$ ($\overline{BC}$ and $\overline{AC}$) respectively, and the third side $RT$ of $\triangle RTS$ is longer than the third side $BA$ of $\triangle ABC$, the included angle $\angle S$ (in $\triangle RTS$) must be larger than the included angle $\angle C$ (in $\triangle ABC$). So, $m\angle C < m\angle S$.

Answer:

$\boldsymbol{m\angle C < m\angle S}$ (corresponding option: the one with $m\angle C < m\angle S$)