Questions: In the diagram below, KL=12 and LM=8. Which additional facts would guarantee that JKLM is a parallelogram? Check all that apply. A. JK=8, and JK is parallel to LM. B. JK=8, JM=12 C. JK=12, and JM is parallel to KL. D. JK=8, JM=8

In the diagram below, KL=12 and LM=8. Which additional facts would guarantee that JKLM is a parallelogram? Check all that apply. A. JK=8, and JK is parallel to LM. B. JK=8, JM=12 C. JK=12, and JM is parallel to KL. D. JK=8, JM=8
Transcript text: In the diagram below, $K L=12$ and $L M=8$. Which additional facts would guarantee that $J K L M$ is a parallelogram? Check all that apply. A. $J K=8$, and $\overline{J K}$ is parallel to $\overline{L M}$. B. $J K=8, J M=12$ C. $J K=12$, and $\overline{J M}$ is parallel to $\overline{K L}$. D. $J K=8, J M=8$
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Solution

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Solution Steps

Step 1: Analyze the given information

We are given that KL = 12 and LM = 8. We need to find additional facts that would guarantee that quadrilateral JKLM is a parallelogram.

Step 2: Consider the properties of parallelograms

A quadrilateral is a parallelogram if:

  1. Both pairs of opposite sides are congruent.
  2. Both pairs of opposite sides are parallel.
  3. One pair of opposite sides is both congruent and parallel.
  4. Both pairs of opposite angles are congruent.
  5. Consecutive angles are supplementary.
  6. The diagonals bisect each other.
Step 3: Check the options

  • Option A: JK = 8 and JK is parallel to LM. This is the definition of a parallelogram (one pair of opposite sides is congruent and parallel).

  • Option B: JK = 8 and JM = 12. This guarantees one pair of congruent opposite sides (JK = LM = 8) and another pair of congruent opposite sides (JM = KL = 12). Therefore, JKLM is a parallelogram.

  • Option C: JK = 12 and JM is parallel to KL. This is the definition of a parallelogram (one pair of opposite sides is congruent and parallel).

  • Option D: JK = 8 and JM = 8. This only guarantees that adjacent sides LM and JM are congruent and JK and LM are congruent, which is insufficient information to determine if JKLM is a parallelogram. For example, a kite can have adjacent sides congruent.

Final Answer: The correct options are A, B, and C.

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