If \( \triangle T I N \cong \triangle G E L \), determine which of parts (a) through (c), if any, are necessarily true. Explain why. a. Is \( \overline{\mathrm{TN}} \cong \overline{\mathrm{GL}} \) necessarily true? A. Yes. Since the triangles are congruent and \( \overline{\mathrm{TN}} \) corresponds to \( \overline{\mathrm{GL}}, \overline{\mathrm{TN}} \cong \overline{\mathrm{GL}} \). B. No. \( \overline{\mathrm{TN}} \) and \( \overline{\mathrm{GL}} \) are not corresponding line segments in the two triangles. b. Is \( \angle I \cong \angle G \) necessarily true? A. Yes. Since the triangles are congruent and \( \angle \mathrm{I} \) corresponds to \( \angle \mathrm{G}, \angle \mathrm{I} \cong \angle \mathrm{G} \). B. No. \( \angle 1 \) and \( \angle G \) are not corresponding angles in the triangles. c. Is \( \overline{\mathrm{N}} \cong \overline{\mathrm{GE}} \) necessarily true? A. Yes. Since the triangles are congruent and \( \overline{\mathbb{N}} \) corresponds to \( \overline{\mathrm{GE}}, \overline{\mathrm{N}} \cong \overline{\mathrm{GE}} \). B. No. \( \overline{\mathbb{N}} \) and \( \overline{\mathrm{GE}} \) are not corresponding line segments in the two triangles.

To determine the truth of each statement regarding the congruence of triangles \( \triangle TIN \) and \( \triangle GEL \), we need to analyze the corresponding parts of the triangles based on the properties of congruent triangles. ### Given: - \( \triangle TIN \cong \triangle GEL \) This means that all corresponding sides and angles of the triangles are equal. ### Part (a): Is \( \overline{TN} \cong \overline{GL} \) necessarily true? - **A. Yes. Since the triangles are congruent and \( \overline{TN} \) corresponds to \( \overline{GL} \), \( \overline{TN} \cong \overline{GL} \).** - **B. No. \( \overline{TN} \) and \( \overline{GL} \) are not corresponding line segments in the two triangles.** **Explanation:** Since the triangles are congruent, the sides correspond to each other. In this case, if we assume the correspondence is as follows: - \( T \) corresponds to \( G \) - \( I \) corresponds to \( E \) - \( N \) corresponds to \( L \) Then, \( \overline{TN} \) corresponds to \( \overline{GL} \). Therefore, the correct answer is **A**. ### Part (b): Is \( \angle I \cong \angle G \) necessarily true? - **A. Yes. Since the triangles are congruent and \( \angle I \) corresponds to \( \angle G \), \( \angle I \cong \angle G \).** - **B. No. \( \angle I \) and \( \angle G \) are not corresponding angles in the triangles.** **Explanation:** Using the same correspondence as before, \( \angle I \) corresponds to \( \angle G \). Therefore, since the triangles are congruent, \( \angle I \cong \angle G \) is true. The correct answer is **A**. ### Part (c): Is \( \overline{N} \cong \overline{GE} \) necessarily true? - **A. Yes. Since the triangles are congruent and \( \overline{N} \) corresponds to \( \overline{GE} \), \( \overline{N} \cong \overline{GE} \).** - **B. No. \( \overline{N} \) and \( \overline{GE} \) are not corresponding line segments in the two triangles.** **Explanation:** In the assumed correspondence, \( N \) corresponds to \( L \) and \( E \) corresponds to \( I \). Therefore, \( \overline{N} \) does not correspond to \( \overline{GE} \). Thus, the correct answer is **B**. ### Summary of Answers: - (a) A - (b) A - (c) B