If $$ riangle T I N \cong riangle G E L $$, 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.

Question

If $$ 	riangle T I N \cong 	riangle G E L $$, 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.

UpStudy AI · Accepted Answer

To determine the truth of each statement regarding the congruence of triangles $$ 	riangle TIN $$ and $$ 	riangle GEL $$, we need to analyze the corresponding parts of the triangles based on the properties of congruent triangles.

### Given:
- $$ 	riangle TIN \cong 	riangle GEL $$

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
- (a) Yes, $$ \overline{TN} \cong \overline{GL} $$ - (b) Yes, $$ \angle I \cong \angle G $$ - (c) No, $$ \overline{N} $$ does not correspond to $$ \overline{GE} $$

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