Geometry worksheet on proving triangle congruence with three problems involving diagrams and proof tables.

I'll work through each of these triangle congruence proofs step by step.

## Problem 1

Given: BC ≅ DC; AC ≅ EC
Prove: ΔBCA ΔDCE

Let me analyze what we have:
- Two pairs of sides are given as congruent
- Looking at the diagram, angles BCA and DCE are vertical angles (they share vertex C)
- Vertical angles are always congruent

So we have:
- Side: BC ≅ DC (given)
- Angle: ∠BCA ≅ DCE (vertical angles)
- Side: AC ≅ EC (given)

This is SAS (Side-Angle-Side) postulate - two sides and the included angle.

Completed Proof:

| Statements | Reasons |
|------------|---------|
| 1. BC ≅ DC; AC ≅ EC | 1. Given |
| 2. ∠BCA ≅ ∠DCE | 2. Vertical ∠s Theorem |
| 3. ΔBCA ≅ ΔDCE | 3. SAS Postulate |

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## Problem 2

Given: JK ≅ LK; JM LM
Prove: ΔKJM ΔKLM

Looking at this diagram:
- We're given two pairs of congruent sides
- KM is shared by both triangles (it's a common side)
- By the Reflexive Property, KM ≅ KM

So we have:
- Side: JK ≅ LK (given)
- Side: JM ≅ LM (given)
- Side: KM ≅ KM (reflexive property)

This is SSS (Side-Side-Side) postulate - all three sides are congruent.

Completed Proof:

| Statements | Reasons |
|------------|---------|
| 1. JK ≅ LK; JM ≅ LM | 1. Given |
| 2. KM ≅ KM | 2. Reflexive Prop. |
| 3. ΔKJM ≅ ΔKLM | 3. SSS Postulate |

---

## Problem 3

Given: ∠G ≅ ∠I; FH bisects ∠GFI
Prove: ΔGFH ΔIFH

Let me think through this:
- We're given that ∠G ∠I
- FH bisects GFI, which means it divides ∠GFI into two equal angles
- So ∠GFH ≅ ∠IFH (by definition of angle bisector)
- FH is shared by both triangles, so FH ≅ FH (reflexive property)

We have:
- Angle: ∠G ≅ ∠I (given)
- Angle: ∠GFH ≅ ∠IFH (definition of angle bisector)
- Side: FH FH (reflexive property)

Wait, I need to check the order. For AAS, we need two angles and a non-included side. Let me verify:
- ∠G and ∠I are one pair
- ∠GFH and ∠IFH are another pair
- FH is the side opposite to ∠G and I

Yes, this is AAS (Angle-Angle-Side).

Completed Proof:

| Statements | Reasons |
|------------|---------|
| 1. ∠G ≅ ∠I; FH bisects ∠GFI | 1. Given |
| 2. ∠GFH ≅ ∠IFH | 2. Def. of angle bisector |
| 3. FH ≅ FH | 3. Reflexive Prop. |
| 4. ΔGFH ΔIFH | 4. AAS Theorem |

---

## Final Answer:

Problem 1:
- Statement 1: BC ≅ DC; AC ≅ EC
- Reason 3: SAS Postulate

Problem 2:
- Statement 1: JK LK; JM ≅ LM
- Reason 3: SSS Postulate

Problem 3:
- Reason 1: Given
- Reason 2: Def. of angle bisector
- Statement 3: FH ≅ FH
- Reason 4: AAS Theorem