Question

using the sas congruence theorem
given: $overline{jk} \parallel \overline{lm}$, $overline{jk} \cong \overline{lm}$,
l is the midpoint of $overline{jn}$.
prove: $\triangle jlk \cong \triangle lnm$
(diagram of triangles jlk and lnm with j, l, n colinear, jk || lm, jk ≅ lm, l midpoint of jn)
angles segments triangles statements reasons
sas given
def. of midpoint corresponding angles theorem
statements | reasons
✓ 1. $overline{jk} \parallel \overline{lm}$ | 1. given
✓ 2. $overline{jk} \cong \overline{lm}$ | 2. given

1. l is the midpoint of $overline{jn}$ | 3. given

✓ 4. $overline{jl} \cong \overline{ln}$ | 4. def. of midpoint
✓ 5. $\angle ljk \cong \angle nlm$ | 5. corresponding angles theorem
(empty box for next statement and reason)
correct! assemble the next statement.

Explanation:

Step1: Identify congruent parts

We have $\overline{JK} \cong \overline{LM}$ (given), $\overline{JL} \cong \overline{LN}$ (def. of midpoint), and $\angle LJK \cong \angle NLM$ (corresponding angles theorem).

Step2: Apply SAS congruence

By the SAS (Side - Angle - Side) Congruence Theorem, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. For $\triangle JLK$ and $\triangle LNM$, we have $\overline{JK} \cong \overline{LM}$, $\overline{JL} \cong \overline{LN}$, and $\angle LJK \cong \angle NLM$ (the included angle between the two sides). So the next statement should be $\triangle JLK \cong \triangle LNM$ and the reason is SAS.

Answer:

Statement: $\triangle JLK \cong \triangle LNM$; Reason: SAS