Prove parallelogram properties
NY.Math: GEO‑G.CO.11
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Problem
Consider this diagram of quadrilateral 
\[ABCD\], which is not drawn to scale.
Quadrilateral A B C D. There is one line from point A to point C and another line from point B to point D. These lines intersect at a point E. Segment A E is congruent to segment C E. Segment B E is congruent to segment D E.
\[A\]
\[B\]
\[C\]
\[D\]
\[E\]
Which two statements must be true based on the information indicated by the diagram?
Choose 2 answers:
Choose 2 answers:
(Choice A)   
\[\angle CDE\cong\angle ECD\]
A
\[\angle CDE\cong\angle ECD\]
(Choice B)   
\[\angle ABE\cong\angle CDE\]
B
\[\angle ABE\cong\angle CDE\]
(Choice C)   
 
 
\[\overline{AC}\cong\overline{BD}\]
C
 
 
\[\overline{AC}\cong\overline{BD}\]
(Choice D)   
 
 
\[\overline{AE}\cong\overline{CE}\]
D
 
 
\[\overline{AE}\cong\overline{CE}\]
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Let's break down the problem and prove the properties step by step.

### Given:
- Quadrilateral \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(E\).
- \( \overline{AE} \cong \overline{CE} \) (Segments \(AE\) and \(CE\) are congruent).
- \( \overline{BE} \cong \overline{DE} \) (Segments \(BE\) and \(DE\) are congruent).

### To Prove:
We need to determine which two statements must be true based on this information.

### Analyzing the Diagram:
1. **Triangles \( \triangle ABE \) and \( \triangle CDE \):**
   - Since \( \overline{AE} \cong \overline{CE} \) and \( \overline{BE} \cong \overline{DE} \), the triangles \( \triangle ABE \) and \( \triangle CDE \) share congruent sides \( AE = CE \) and \( BE = DE \).
   - Additionally, these triangles share the common side \( \overline{AB} \cong \overline{CD} \).

2. **Congruency of Triangles:**
   - By **SSS (Side-Side-Side) Congruence** Postulate, \( \triangle ABE \cong \triangle CDE \).

### Conclusion Based on Congruency:
- Since \( \triangle ABE \cong \triangle CDE \), corresponding angles must also be congruent.
- This implies that:
  - \( \angle ABE \cong \angle CDE \)
  - \( \angle CDE \cong \angle ECD \)

### Answer Choices:
- **(A) \( \angle CDE \cong \angle ECD \)**: True, since the triangles are congruent.
- **(B) \( \angle ABE \cong \angle CDE \)**: True, as corresponding angles of congruent triangles.
- **(C) \( \overline{AC} \cong \overline{BD} \)**: This statement is not necessarily true based on the information provided.
- **(D) \( \overline{AE} \cong \overline{CE} \)**: This is given information, not something that needs to be proven from the congruency of triangles.

### Correct Statements:
- **(A) \( \angle CDE \cong \angle ECD \)**
- **(B) \( \angle ABE \cong \angle CDE \)**

These two statements must be true based on the information provided.

---

Would you like more details, or do you have any further questions? Here are some related questions to consider:

1. How does the SSS Congruence Postulate help in proving triangle congruency?
2. What is the significance of corresponding angles in congruent triangles?
3. How can we use the properties of parallelograms to prove other geometric relationships?
4. What is the relationship between the diagonals of a parallelogram?
5. How does the congruence of opposite sides contribute to the definition of a parallelogram?

**Tip:** When proving geometric properties, always look for congruent triangles, as they often provide the key relationships needed for the proof.

Explore how to prove parallelogram properties using congruent triangles. Solve the problem involving quadrilateral ABCD with diagonals AC and BD intersecting at point E. Understand the SSS Congruence Postulate and its application in geometry.

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