Congruent Triangles Proof Worksheet-1 Answers | PDF | Euclid ... - Free Printable

You've uploaded a Geometry proof activity focused on proving triangles congruent using postulates like SSS, SAS, ASA, AAS, and HL. Let’s go through each problem step by step, correcting any errors, filling in missing blanks, and explaining the reasoning clearly.

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## 🔢 Problem 1

Given:

- $\overline{BC} \cong \overline{DC}$
- $\overline{AC} \cong \overline{EC}$

Prove:

$\triangle BCA \cong \triangle DCE$

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✔ Corrected & Explained Proof:

| Statements | Reasons |
|-------------------------------------|-----------------------------|
| 1. $\overline{BC} \cong \overline{DC}$, $\overline{AC} \cong \overline{EC}$ | 1. Given |
| 2. $\angle BCA \cong \angle DCE$ | 2. Vertical Angles Theorem |
| 3. $\triangle BCA \cong \triangle DCE$ | 3. SAS (Side-Angle-Side) |

> ✔ Explanation:
> - We are given two pairs of congruent sides: $BC \cong DC$ and $AC \cong EC$.
> - The angle between these sides — $\angle BCA$ and $\angle DCE$ — are vertical angles, which are always congruent.
> - Therefore, we have two sides and the included angle congruent → SAS Congruence Postulate.

> 📌 Note: In your image, you wrote “SAS” correctly for #3 — that’s perfect!

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## 🔢 Problem 2

Given:

- $\overline{JK} \cong \overline{LK}$
- $\overline{JM} \cong \overline{LM}$

Prove:

$\triangle KJM \cong \triangle KLM$

*(Note: The diagram shows triangle KJM and KLM sharing side KM.)*

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✔ Corrected & Explained Proof:

| Statements | Reasons |
|-------------------------------------|-----------------------------|
| 1. $\overline{JK} \cong \overline{LK}$, $\overline{JM} \cong \overline{LM}$ | 1. Given |
| 2. $\overline{KM} \cong \overline{KM}$ | 2. Reflexive Property |
| 3. $\triangle KJM \cong \triangle KLM$ | 3. SSS (Side-Side-Side) |

> ✔ Explanation:
> - We are given two pairs of sides: $JK \cong LK$ and $JM \cong LM$.
> - The third side, $KM$, is shared by both triangles → reflexive property → $KM \cong KM$.
> - All three sides are congruent → SSS Congruence Postulate.

> 📌 Note: In your image, you wrote “△KML ≅ △KMJ” — this is incorrect notation. The order matters! It should be △KJM ≅ △KLM to match corresponding vertices. Also, the reason “SSS” is correct.

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## 🔢 Problem 3

Given:

- $\angle G \cong \angle I$
- $\overline{FH}$ bisects $\angle GFI$

Prove:

$\triangle GFH \cong \triangle IFH$

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✔ Corrected & Explained Proof:

| Statements | Reasons |
|-------------------------------------|-----------------------------|
| 1. $\angle G \cong \angle I$, $\overline{FH}$ bisects $\angle GFI$ | 1. Given |
| 2. $\angle GFH \cong \angle IFH$ | 2. Definition of Angle Bisector |
| 3. $\overline{FH} \cong \overline{FH}$ | 3. Reflexive Property |
| 4. $\triangle GFH \cong \triangle IFH$ | 4. AAS (Angle-Angle-Side) |

> ✔ Explanation:
> - We’re given $\angle G \cong \angle I$ (first angle).
> - Since $\overline{FH}$ bisects $\angle GFI$, it creates two equal angles: $\angle GFH \cong \angle IFH$ (second angle).
> - Side $\overline{FH}$ is common to both triangles → reflexive property.
> - So we have two angles and a non-included side congruent → AAS Congruence Postulate.

> 📌 Note: In your image, you wrote “△HGF ≅ △HIF” — this is not standard notation. It should be △GFH ≅ △IFH to match vertex correspondence. Also, you wrote “bisect” — that’s fine, but the full phrase is “Definition of Angle Bisector”. And “AAS” is correct!

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## 🧩 Summary of Corrections:

| Problem | Issue in Your Answer | Correction |
|---------|---------------------------------------------|--------------------------------------------|
| 1 | None | Perfect! |
| 2 | Triangle notation: △KML ≅ △KMJ | Should be △KJM ≅ △KLM |
| 3 | Triangle notation: △HGF ≅ △HIF | Should be △GFH ≅ △IFH |
| 3 | Reason 2: “Def. of bisect” | Should be “Definition of Angle Bisector” |

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## 🎯 Final Answers (Clean Version):

1.

Proof:
1. $\overline{BC} \cong \overline{DC}, \overline{AC} \cong \overline{EC}$ — Given
2. $\angle BCA \cong \angle DCE$ — Vertical Angles Theorem
3. $\triangle BCA \cong \triangle DCE$ — SAS

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2.

Proof:
1. $\overline{JK} \cong \overline{LK}, \overline{JM} \cong \overline{LM}$ — Given
2. $\overline{KM} \cong \overline{KM}$ — Reflexive Property
3. $\triangle KJM \cong \triangle KLM$ — SSS

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3.

Proof:
1. $\angle G \cong \angle I, \overline{FH}$ bisects $\angle GFI$ — Given
2. $\angle GFH \cong \angle IFH$ — Definition of Angle Bisector
3. $\overline{FH} \cong \overline{FH}$ — Reflexive Property
4. $\triangle GFH \cong \triangle IFH$ — AAS

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✔ You did great overall! Just pay attention to vertex order in triangle names and use complete terminology (like “Definition of Angle Bisector”) for full credit.

Let me know if you want help drawing the diagrams with tick marks/arcs too!