### Chaim's Triangle Congruence Proof#### Problem Statement:Chaim attempted to prove that the triangles $ \triangle JKL \cong \triangle LMN $.#### Illustration:The illustration features two triangles:- Triangle $JKL$ with sides $ JK = 3 $, $ JL = 5 $, and angle $ \angle KLJ = 27^\circ $.- Triangle $LMN$ with sides $ LM = 3 $, $ LN = 5 $, and angle $ \angle MNL = 27^\circ $.#### Proof Table:| Statement | Reason ||-------------------------------------|----------------------------------------|| 1. $ JK = LM = 3 $ | Given || 2. $ JL = LN = 5 $ | Given || 3. $ m\angle KLJ = m\angle MNL = 27^\circ $ | Given || 4. $ \triangle JKL \cong \triangle LMN $ | Side-Side-Angle congruence |#### Question:What is the first error Chaim made in his proof?##### Possible Answers:A. Chaim used an invalid reason to justify the congruence of a pair of sides or angles.B. Chaim only established some of the necessary conditions for a congruence criterion.C. Chaim established all necessary conditions but then used an inappropriate congruence criterion.D. Chaim used a criterion that does not guarantee congruence.##### Explanation:To evaluate the correctness of Chaim's proof, it is essential to understand which congruence conditions (criteria) are valid in proving triangle congruence. In this case:- **Side-Side-Angle (SSA)** is not a valid criterion for triangle congruence.- **Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS),** and **Side-Side-Side (SSS)** are the recognized criteria for triangle congruence.Given this, Chaim's error lies in the inappropriate use of the SSA criterion in statement 4.The correct answer is:**D. Chaim used a criterion that does not guarantee congruence.**

Chaim tried to prove that AJKLE ALMN. K 27 27 J Statement Reason 1 JK = LM = 3 Given JL = LN = 5 Given MZKLJ = mZMNL = 27° Given 4. AJKLE ALMN Side-side- angle congruence What is the first error Chaim made in his proof? Choose 1 answer: Chaim used an invalid reason to justify the congruence of a pair of sides or angles. Chaim only established some of the necessary conditions for a congruence criterion. Chaim established all necessary conditions, but then used an inappropriate congruence criterion. Chaim used a criterion that does not guarantee congruence.

Chaim tried to prove that AJKLE ALMN. K 27 27 J Statement Reason 1 JK = LM = 3 Given JL = LN = 5 Given MZKLJ = mZMNL = 27° Given 4. AJKLE ALMN Side-side- angle congruence What is the first error Chaim made in his proof? Choose 1 answer: Chaim used an invalid reason to justify the congruence of a pair of sides or angles. Chaim only established some of the necessary conditions for a congruence criterion. Chaim established all necessary conditions, but then used an inappropriate congruence criterion. Chaim used a criterion that does not guarantee congruence.

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

Problem 1CT

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Question

### Chaim's Triangle Congruence Proof

#### Problem Statement:
Chaim attempted to prove that the triangles $ \triangle JKL \cong \triangle LMN $.

#### Illustration:
The illustration features two triangles:

- Triangle $JKL$ with sides $ JK = 3 $, $ JL = 5 $, and angle $ \angle KLJ = 27^\circ $.
- Triangle $LMN$ with sides $ LM = 3 $, $ LN = 5 $, and angle $ \angle MNL = 27^\circ $.

#### Proof Table:

| Statement | Reason |
|-------------------------------------|----------------------------------------|
| 1. $ JK = LM = 3 $ | Given |
| 2. $ JL = LN = 5 $ | Given |
| 3. $ m\angle KLJ = m\angle MNL = 27^\circ $ | Given |
| 4. $ \triangle JKL \cong \triangle LMN $ | Side-Side-Angle congruence |

#### Question:
What is the first error Chaim made in his proof?

##### Possible Answers:
A. Chaim used an invalid reason to justify the congruence of a pair of sides or angles.
B. Chaim only established some of the necessary conditions for a congruence criterion.
C. Chaim established all necessary conditions but then used an inappropriate congruence criterion.
D. Chaim used a criterion that does not guarantee congruence.

##### Explanation:
To evaluate the correctness of Chaim's proof, it is essential to understand which congruence conditions (criteria) are valid in proving triangle congruence. In this case:

- **Side-Side-Angle (SSA)** is not a valid criterion for triangle congruence.
- **Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS),** and **Side-Side-Side (SSS)** are the recognized criteria for triangle congruence.

Given this, Chaim's error lies in the inappropriate use of the SSA criterion in statement 4.

The correct answer is:
**D. Chaim used a criterion that does not guarantee congruence.**

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