A conjecture and the two-column proof used to prove the conjecture are shown.

 

Given: segment A B is congruent to segment B D, segment B D is congruent to segment C E, and segment C E is congruent to segment A C. Prove: triangle A B C is an isosceles triangle. Segment A D with endpoints D and A moving from left to right. Segment A D is diagonally down to the left from point A. B is the midpoint of segment A D. Segment A E shares endpoint at point A with segment A D. Segment A E is diagonally down to the right from point A. C is the midpoint of segment A E. Segment B C is horizontal between segment A D and segment A E.

 

Drag an expression or phrase to each box to complete the proof.

To prove that **triangle ABC** is an **isosceles triangle**, we need to apply the given information and relevant theorems in a **two-column proof** format. Let's walk through the reasoning step by step. I'll provide a structure to fill in the missing expressions or phrases.

### Given Information:
- \( \overline{AB} \cong \overline{BD} \)
- \( \overline{BD} \cong \overline{CE} \)
- \( \overline{CE} \cong \overline{AC} \)

We are tasked with proving that **triangle ABC** is isosceles.

### Two-column proof:

| **Statement** | **Reason** |
| --- | --- |
| 1. \( \overline{AB} \cong \overline{BD} \), \( \overline{BD} \cong \overline{CE} \), \( \overline{CE} \cong \overline{AC} \) | 1. **Given** |
| 2. \( \overline{AB} \cong \overline{AC} \) | 2. **Transitive property of congruence** (If \( \overline{AB} \cong \overline{BD} \), \( \overline{BD} \cong \overline{CE} \), and \( \overline{CE} \cong \overline{AC} \), then \( \overline{AB} \cong \overline{AC} \)) |
| 3. **Segment BC is shared between triangle ABC and BEC** | 3. **Common segment** |
| 4. **Triangle ABC has two congruent sides: \( \overline{AB} \cong \overline{AC} \)** | 4. **Definition of an isosceles triangle** (A triangle is isosceles if two sides are congruent) |
| 5. **Triangle ABC is an isosceles triangle** | 5. **Conclusion** (Based on step 4, \( \triangle ABC \) is isosceles.) |

### Key Steps Explained:
1. **Given:** The problem gives us that several segments are congruent.
2. **Transitive Property:** Using the transitive property of congruence, we deduce that \( \overline{AB} \cong \overline{AC} \).
3. **Common Segment:** \( BC \) is common in the diagram and doesn't affect the congruence of the sides of triangle \( ABC \).
4. **Isosceles Triangle:** By definition, if two sides of a triangle are congruent, it qualifies as an isosceles triangle.

### Conclusion:
By using the given information and the transitive property of congruence, we conclude that **triangle ABC** is isosceles since two of its sides, \( \overline{AB} \) and \( \overline{AC} \), are congruent.

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Do you need any more details or clarifications? Here are some related questions you can explore:

1. How does the transitive property work in geometry?
2. Can a triangle have all sides congruent and still be isosceles?
3. What is the definition of an equilateral triangle, and how does it differ from an isosceles triangle?
4. What is the importance of midpoint in triangle congruence proofs?
5. How can segment congruence be proven in different triangle types?

**Tip:** When solving geometry proofs, always label your diagram clearly to keep track of congruent sides and angles.

Given: segment AB is congruent to segment BD, segment BD is congruent to segment CE, and segment CE is congruent to segment AC. Prove: triangle ABC is an isosceles triangle.

Explore a two-column proof to demonstrate that triangle ABC is isosceles, using the transitive property of congruence and segment congruence. This solution is ideal for students in Grades 8-10 studying geometry and triangle properties.

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