This geometry worksheet guides students through two-column proofs to demonstrate triangle congruence using the Side-Side-Side (SSS) postulate.

Here are the completed two-column proofs for both problems.

Part A) Proving $\Delta LFG \cong \Delta LFH$

We need to fill in the missing statements and reasons to prove the triangles are congruent using the Side-Side-Side (SSS) postulate.

| Statement | Reason |
| :--- | :--- |
| $\overline{GF} \cong \overline{HF}$ | (1) Given |
| (2) $\overline{GL} \cong \overline{HL}$ | Given |
| (3) $\overline{FL} \cong \overline{FL}$ | (4) Reflexive Property of Congruence |
| $\Delta LFG \cong \Delta LFH$ | (5) SSS Congruence Postulate |

Explanation of steps:
1. The first statement is already provided in the "Given" section, so the reason is simply Given.
2. The second "Given" fact from the problem description is that side $GL$ is congruent to side $HL$. So, we write $\overline{GL} \cong \overline{HL}$ as the statement.
3. Looking at the diagram, both triangles share the side in the middle, which is side $FL$. Any segment is congruent to itself. So, the statement is $\overline{FL} \cong \overline{FL}$.
4. The reason a segment is congruent to itself is the Reflexive Property of Congruence.
5. Since we have proven all three pairs of corresponding sides are congruent (Side-Side-Side), the reason for the final conclusion is the SSS Congruence Postulate.

***

Part B) Proving $\Delta JMK \cong \Delta VKM$

We follow the same logic for this quadrilateral divided by a diagonal.

| Statement | Reason |
| :--- | :--- |
| (1) $\overline{JK} \cong \overline{VM}$ | Given |
| $\overline{JM} \cong \overline{VK}$ | (2) Given |
| (3) $\overline{KM} \cong \overline{KM}$ | (4) Reflexive Property of Congruence |
| $\Delta JMK \cong \Delta VKM$ | (5) SSS Congruence Postulate |

Explanation of steps:
1. The first given fact is that side $JK$ is congruent to side $VM$. So, the statement is $\overline{JK} \cong \overline{VM}$.
2. The second statement is provided ($\overline{JM} \cong \overline{VK}$), and since it comes from the "Given" list, the reason is Given.
3. Both triangles share the diagonal side $KM$. Therefore, side $KM$ is congruent to itself. The statement is $\overline{KM} \cong \overline{KM}$.
4. The reason for a shared side being congruent to itself is the Reflexive Property of Congruence.
5. With all three sides proven congruent, the final reason is the SSS Congruence Postulate.

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Final Answer:

A)
(1) Given
(2) $\overline{GL} \cong \overline{HL}$
(3) $\overline{FL}$
(4) Reflexive Property of Congruence
(5) SSS Congruence Postulate

B)
(1) $\overline{JK} \cong \overline{VM}$
(2) Given
(3) $\overline{KM}$
(4) Reflexive Property of Congruence
(5) SSS Congruence Postulate