Triangle Proofs Worksheet - Free Printable

You’ve uploaded a worksheet titled “Triangle Congruence Proofs Practice” with three proof problems. I’ll solve each one step by step, filling in the missing statements and reasons.

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## 🔹 Problem 1

Given: C is the midpoint of BE and AD.
Prove: ΔABC ≅ ΔDEC

Solution:

| Statement | Reason |
|------------------------------|----------------------------------|
| 1. C is midpoint of BE and AD | Given |
| 2. BC ≅ EC | Definition of a midpoint |
| 3. AC ≅ DC | Definition of a midpoint |
| 4. ∠ACB ≅ ∠DCE | Vertical angles are congruent |
| 5. ΔABC ≅ ΔDEC | SAS (Side-Angle-Side) Congruence |

✔ Explanation:
- Since C is the midpoint of both segments, it divides them into two equal parts → BC ≅ EC and AC ≅ DC.
- Angles ∠ACB and ∠DCE are vertical angles (formed by intersecting lines), so they are congruent.
- With two sides and the included angle congruent, we use SAS to prove triangle congruence.

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## 🔹 Problem 2

Given: BC ≅ DA and AC bisects ∠BCD
Prove: ΔABC ≅ ΔCDA

Solution:

| Statement | Reason |
|------------------------------|----------------------------------|
| 1. BC ≅ DA | Given |
| 2. AC bisects ∠BCD | Given |
| 3. ∠BCA ≅ ∠DCA | Definition of angle bisector |
| 4. AC ≅ AC | Reflexive Property |
| 5. ΔABC ≅ ΔCDA | SAS (Side-Angle-Side) Congruence |

✔ Explanation:
- We’re given that BC ≅ DA.
- Since AC bisects ∠BCD, it splits it into two equal angles → ∠BCA ≅ ∠DCA.
- Side AC is common to both triangles → reflexive property.
- So again, we have two sides and the included angle → SAS congruence.

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## 🔹 Problem 3

Given: JK ≅ FH and KJ || FH
Prove: ΔJKG ≅ ΔHFG

*(Note: Based on the diagram, points J, K, G form one triangle; F, H, G form another. The shared point is G — likely the intersection point of diagonals or transversal.)*

Solution:

| Statement | Reason |
|------------------------------|----------------------------------|
| 1. JK ≅ FH | Given |
| 2. KJ || FH | Given |
| 3. ∠JKG ≅ ∠FHG | Alternate Interior Angles (from parallel lines & transversal) |
| 4. ∠KGJ ≅ ∠FGH | Vertical Angles are congruent |
| 5. ΔJKG ≅ ΔHFG | AAS (Angle-Angle-Side) Congruence |

✔ Explanation:
- We’re given side JK ≅ FH.
- Since KJ || FH, and line JG (or HG) acts as a transversal, then alternate interior angles are congruent → ∠JKG ≅ ∠FHG.
- Angles at point G (∠KGJ and ∠FGH) are vertical angles → congruent.
- Now we have two angles and a non-included side → AAS congruence.

*(Note: You could also use ASA if you consider the side between the two angles, but since the side given is not between the two angles we identified, AAS is more appropriate here.)*

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## ✔ Final Answers Summary:

Problem 1:

- Statements 3–5 filled as above → SAS

Problem 2:

- Statements 3–5 filled as above → SAS

Problem 3:

- Statements 3–5 filled as above → AAS

Let me know if you’d like these drawn out or explained with diagrams!