What additional congruence is needed to prove that ALMN and ARST are congruent by SAS postulate?

Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
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Question 12

### Question 12:
In \( \triangle LMN \) and \( \triangle RST \), \( \angle MNL \cong \angle STR \) and \( \overline{MN} \cong \overline{ST} \).

What additional congruence is needed to prove that \( \triangle LMN \) and \( \triangle RST \) are congruent by SAS postulate?

#### Options:
A. \( \angle LMN \cong \angle RST \)

B. \( \angle NLM \cong \angle TRS \)

C. \( \overline{LM} \cong \overline{RS} \)

D. \( \overline{LN} \cong \overline{RT} \)  *(selected)*

Explanation:
- **SAS Postulate**: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
  
- Given that:
  - \( \angle MNL \cong \angle STR \)
  - \( \overline{MN} \cong \overline{ST} \)
  
  To apply the SAS postulate, we need one more pair of corresponding congruent sides.

The correct additional congruence required is \( \overline{LN} \cong \overline{RT} \).

Graph/Diagram Explanation:
- The image contains a multiple-choice question with congruence conditions related to two triangles.
- Four options are given, out of which option D is highlighted as the correct answer.
Transcribed Image Text:### Question 12: In \( \triangle LMN \) and \( \triangle RST \), \( \angle MNL \cong \angle STR \) and \( \overline{MN} \cong \overline{ST} \). What additional congruence is needed to prove that \( \triangle LMN \) and \( \triangle RST \) are congruent by SAS postulate? #### Options: A. \( \angle LMN \cong \angle RST \) B. \( \angle NLM \cong \angle TRS \) C. \( \overline{LM} \cong \overline{RS} \) D. \( \overline{LN} \cong \overline{RT} \) *(selected)* Explanation: - **SAS Postulate**: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. - Given that: - \( \angle MNL \cong \angle STR \) - \( \overline{MN} \cong \overline{ST} \) To apply the SAS postulate, we need one more pair of corresponding congruent sides. The correct additional congruence required is \( \overline{LN} \cong \overline{RT} \). Graph/Diagram Explanation: - The image contains a multiple-choice question with congruence conditions related to two triangles. - Four options are given, out of which option D is highlighted as the correct answer.
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