Question

lesson 9.2 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. does each set of given information guarantee that quadrilateral \\(jklm\\) is a parallelogram? select yes or no for each situation.

a. \\(jn = 25\text{ cm}, jl = 50\text{ cm}, kn = 13\text{ cm}, km = 26\text{ cm}\\) yes no
b. \\(\angle mjl \cong \angle klj, jm \cong lk\\) yes no
c. \\(jm \cong jk, kl \cong lm\\) yes no
d. \\(\angle mjl \cong \angle mlj, \angle kjl \cong \angle klj\\) yes no
e. \\(\delta jkn \cong \delta lmn\\) yes no

Explanation:

Analyze diagonal bisection condition

Using the Parallelogram Diagonal Properties and Converse of Parallelogram Theorems knowledge points

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Since the diagonals bisect each other, \(JKLM\) is guaranteed to be a parallelogram.

Analyze alternate interior angles and opposite sides

Using the Converse of Parallelogram Theorems knowledge point

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One pair of opposite sides is both parallel and congruent, which guarantees a parallelogram.

Analyze adjacent side congruence

Using the Converse of Parallelogram Theorems knowledge point

$$ JM \cong JK,\quad KL \cong LM $$

Congruence of adjacent sides defines a kite, not necessarily a parallelogram. Thus, it does not guarantee a parallelogram.

Analyze angle bisector conditions

Using the Converse of Parallelogram Theorems knowledge point

$$ \angle MJL \cong \angle MLJ,\quad \angle KJL \cong \angle KLJ $$

This implies \(MJ \cong ML\) and \(KJ \cong KL\), which describes a kite, not necessarily a parallelogram.

Analyze triangle congruence condition

Using the Converse of Parallelogram Theorems knowledge point

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Answer:

No.	Problem	Answer
B	\(\angle MJL \cong \angle KLJ, JM \cong LK\)	Yes
C	\(JM \cong JK, KL \cong LM\)	No
D	\(\angle MJL \cong \angle MLJ, \angle KJL \cong \angle KLJ\)	No
E	\(\Delta JKN \cong \Delta LMN\)	Yes