Prove parallelogram properties

NY.Math: GEO‑G.CO.11

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Problem

Bao wants to prove that if parallelogram \[ABCD\] has a right angle, then its diagonals are congruent.

\[A\]\[B\]\[C\]\[D\]\[E\]

Bao says: "I can prove this using the fact that opposite sides in a parallelogram are congruent, and by establishing the congruence of a single pair of triangles."

**Which pair of triangles is Bao referring to, and which criterion should he use for establishing congruence?**

Choose 1 answer:

Choose 1 answer:

(Choice A)   

\[\triangle ABC\] and \[\triangle CDA\] by angle-side-angle

A

\[\triangle ABC\] and \[\triangle CDA\] by angle-side-angle

(Choice B)   

\[\triangle ABC\] and \[\triangle CDA\] by side-angle-side

B

\[\triangle ABC\] and \[\triangle CDA\] by side-angle-side

(Choice C)   

\[\triangle CDA\] and \[\triangle DCB\] by angle-side-angle

C

\[\triangle CDA\] and \[\triangle DCB\] by angle-side-angle

(Choice D)   

\[\triangle CDA\] and \[\triangle DCB\] by side-angle-side

D

\[\triangle CDA\] and \[\triangle DCB\] by side-angle-side

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Let's analyze the problem and prove the parallelogram properties step by step.

### Given:
- Parallelogram \(ABCD\) has a right angle.
- We need to prove that the diagonals of the parallelogram \(AC\) and \(BD\) are congruent.

### Bao's Strategy:
Bao plans to use the fact that opposite sides in a parallelogram are congruent and establish the congruence of a single pair of triangles.

### Analyzing the Diagram:
1. **Properties of Parallelograms**:
   - Opposite sides are congruent: \( \overline{AB} \cong \overline{CD} \) and \( \overline{AD} \cong \overline{BC} \).
   - If one angle is a right angle, then all angles are right angles (since the adjacent angles in a parallelogram are supplementary).

2. **Focus on Triangles**:
   - Consider triangles \( \triangle ABC \) and \( \triangle CDA \).
   - Since \(ABCD\) is a parallelogram, the sides \( \overline{AB} \cong \overline{CD} \) and \( \overline{AD} \cong \overline{BC} \).
   - \( \angle ABC \) and \( \angle CDA \) are right angles, which implies they are congruent (\(90^\circ\)).

### Establishing Triangle Congruence:
- **Side-Angle-Side (SAS) Criterion**:
  - In \( \triangle ABC \) and \( \triangle CDA \), we have:
    - \( \overline{AB} \cong \overline{CD} \) (opposite sides of parallelogram).
    - \( \angle ABC \cong \angle CDA \) (both are right angles).
    - \( \overline{BC} \cong \overline{DA} \) (opposite sides of parallelogram).
  - By the **SAS** (Side-Angle-Side) Congruence Postulate, \( \triangle ABC \cong \triangle CDA \).

### Conclusion:
- Since \( \triangle ABC \) is congruent to \( \triangle CDA \), their corresponding parts are congruent, including the diagonals \( \overline{AC} \) and \( \overline{BD} \).
- Therefore, the diagonals \( \overline{AC} \cong \overline{BD} \).

### Correct Answer:
- **(B) \( \triangle ABC \) and \( \triangle CDA \) by Side-Angle-Side**.

Bao is referring to the triangles \( \triangle ABC \) and \( \triangle CDA \) and should use the **SAS** (Side-Angle-Side) criterion for establishing their congruence.

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

1. How does the congruence of triangles relate to the properties of parallelograms?
2. Why does the presence of a right angle in a parallelogram affect the diagonals?
3. What is the significance of the SAS Congruence Postulate in geometric proofs?
4. How do the properties of a rectangle relate to those of a parallelogram?
5. What other methods could be used to prove the diagonals of a parallelogram are congruent?

**Tip:** When working with parallelograms, always consider the properties of their angles and sides, as they often lead to useful congruences.

Learn how to prove the properties of a parallelogram with right angles. Understand Bao's approach using triangle congruence and the SAS criterion to establish that diagonals are congruent. Suitable for high school geometry students.

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