Question

chaim tried to prove that $\triangle jkl \cong \triangle lmn$.

image of two triangles with labeled sides and angles

statement	reason
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

what is the first error chaim made in his proof?

choose 1 answer:

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 determine the error in Chaim's proof of triangle congruence, we analyze the congruence criteria:

1. Recall Congruence Criteria: The valid triangle congruence criteria are SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and HL (Hypotenuse - Leg) for right triangles. There is no "Side - Side - Angle" (SSA) congruence criterion because SSA does not guarantee that two triangles are congruent (it can lead to the ambiguous case in triangles, especially for non - right triangles).

1. Analyze Chaim's Proof:

• Chaim identified two sides (\(JK = LM = 3\), \(JL=LN = 5\)) and a non - included angle (\(m\angle KLJ=m\angle MNL = 27^{\circ}\)) and then used the non - existent "Side - Side - Angle" criterion to prove \(\triangle JKL\cong\triangle LMN\).
• Option A is incorrect because the reasons for the congruence of sides and angles (given) are valid. Option B is incorrect because it's not about establishing some conditions, but using an invalid criterion. Option C is incorrect because the "Side - Side - Angle" is not a valid congruence criterion, so he did not establish all necessary conditions for a valid criterion. Option D is correct because the "Side - Side - Angle" criterion he used does not guarantee congruence.

Answer:

D. Chaim used a criterion that does not guarantee congruence.