4.9: Intro To 2-Column Proofs (Answer Key) | PDF - Free Printable

Problem Description:

The task involves constructing two-column proofs for the congruence of triangles. The image contains two problems:

1. Problem 1: Prove that \( \triangle ABC \cong \triangle DCE \).
2. Problem 2: Prove that \( \triangle FGH \cong \triangle FHI \).

We will solve each problem step by step.

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Problem 1: Prove \( \triangle ABC \cong \triangle DCE \)

#### Given:
- \( C \) is the midpoint of line segment \( AE \).
- \( C \) is the midpoint of line segment \( BD \).

#### To Prove:
\( \triangle ABC \cong \triangle DCE \).

#### Proof:

| Statements | Reasons |
|------------------------------|---------------------------------------|
| 1. \( C \) is the midpoint of \( AE \). | Given. |
| 2. \( AC = CE \). | Definition of midpoint. |
| 3. \( C \) is the midpoint of \( BD \). | Given. |
| 4. \( BC = CD \). | Definition of midpoint. |
| 5. \( \angle ACB = \angle DCE \). | Vertical angles are congruent. |
| 6. \( \triangle ABC \cong \triangle DCE \). | Side-Angle-Side (SAS) Congruence Postulate. |

#### Explanation:
1. Step 1: We are given that \( C \) is the midpoint of \( AE \). By definition, this means \( AC = CE \).
2. Step 2: Similarly, we are given that \( C \) is the midpoint of \( BD \). By definition, this means \( BC = CD \).
3. Step 3: The angles \( \angle ACB \) and \( \angle DCE \) are vertical angles. Vertical angles are always congruent.
4. Step 4: With \( AC = CE \), \( BC = CD \), and \( \angle ACB = \angle DCE \), we can apply the Side-Angle-Side (SAS) Congruence Postulate to conclude that \( \triangle ABC \cong \triangle DCE \).

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Problem 2: Prove \( \triangle FGH \cong \triangle FHI \)

#### Given:
- \( FGHI \) is a kite.
- \( FG = FI \).
- \( GH = HI \).

#### To Prove:
\( \triangle FGH \cong \triangle FHI \).

#### Proof:

| Statements | Reasons |
|------------------------------|---------------------------------------|
| 1. \( FGHI \) is a kite. | Given. |
| 2. \( FG = FI \). | Definition of kite. |
| 3. \( GH = HI \). | Definition of kite. |
| 4. \( FH = FH \). | Reflexive property of congruence. |
| 5. \( \triangle FGH \cong \triangle FHI \). | Side-Side-Side (SSS) Congruence Postulate. |

#### Explanation:
1. Step 1: We are given that \( FGHI \) is a kite. By definition, in a kite, two pairs of adjacent sides are congruent.
2. Step 2: From the properties of a kite, we know \( FG = FI \) and \( GH = HI \).
3. Step 3: The side \( FH \) is common to both triangles \( \triangle FGH \) and \( \triangle FHI \). By the reflexive property, \( FH = FH \).
4. Step 4: With \( FG = FI \), \( GH = HI \), and \( FH = FH \), we can apply the Side-Side-Side (SSS) Congruence Postulate to conclude that \( \triangle FGH \cong \triangle FHI \).

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Final Answers:

1. For Problem 1:
\[
\boxed{\triangle ABC \cong \triangle DCE \text{ by SAS}}
\]

2. For Problem 2:
\[
\boxed{\triangle FGH \cong \triangle FHI \text{ by SSS}}
\]