Proving Triangles Congruent (SSS) - Two-column proof worksheet with geometric diagrams and student feedback options.

Explanation:
We are proving triangle congruence using the SSS (Side-Side-Side) postulate. That means: if all three sides of one triangle are congruent to all three sides of another triangle, then the triangles are congruent.

Let’s solve each part step by step.

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Part A:
Given:
- $\overline{GF} \cong \overline{HF}$
- $\overline{GL} \cong \overline{HL}$
Prove: $\triangle LFG \cong \triangle LFH$

Look at the diagram: both triangles share side $\overline{FL}$. So that’s the third pair of congruent sides.

Now fill in the two-column proof:

| Statement | Reason |
|----------|--------|
| (1) $\overline{GF} \cong \overline{HF}$ | Given |
| (2) $\overline{GL} \cong \overline{HL}$ | Given |
| (3) $\overline{FL} \cong \overline{FL}$ | Reflexive Property of Congruence (a segment is congruent to itself) |
| (4) $\triangle LFG \cong \triangle LFH$ | SSS Postulate (three pairs of congruent sides) |

So:
- (1) Reason: Given
- (2) Statement: $\overline{GL} \cong \overline{HL}$
- (3) Statement: $\overline{FL} \cong \overline{FL}$
- (4) Reason: Reflexive Property
- (5) Reason: SSS

✔ Double-check: Yes — all three sides match: GF = HF, GL = HL, and FL = FL.

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Part B:
Given:
- $\overline{JK} \cong \overline{VM}$
- $\overline{JM} \cong \overline{VK}$
Prove: $\triangle JMK \cong \triangle VKM$

Again, look at the diagram: triangles share side $\overline{KM}$. So $\overline{KM} \cong \overline{KM}$ by reflexive property.

Fill in the proof:

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

So:
- (1) Statement: $\overline{JK} \cong \overline{VM}$
- (2) Reason: Given
- (3) Statement: $\overline{KM} \cong \overline{KM}$
- (4) Reason: Reflexive Property
- (5) Reason: SSS

✔ Check: JK = VM, JM = VK, KM = KM → all 3 sides match → SSS works.

Final Answer:
A)
(1) Given
(2) $\overline{GL} \cong \overline{HL}$
(3) $\overline{FL} \cong \overline{FL}$
(4) Reflexive Property
(5) SSS

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