Proof that diagonals of a rectangle are congruent using triangle congruence.

Problem Analysis:

We are tasked with proving that the diagonals of a rectangle are congruent. The given information is that \(ABCD\) is a parallelogram, and angles \(\angle ADC\) and \(\angle BCD\) are right angles. We need to show that the diagonals \(\overline{DB}\) and \(\overline{CA}\) are congruent.

Proof Strategy:

1. Use the properties of a parallelogram:
- Opposite sides of a parallelogram are congruent.
- Opposite angles of a parallelogram are congruent.

2. Use the given information:
- \(\angle ADC\) and \(\angle BCD\) are right angles, which means \(ABCD\) is a rectangle (a special type of parallelogram with right angles).

3. Prove triangle congruence:
- Show that \(\triangle ACD \cong \triangle BDC\) using an appropriate congruence criterion.
- Once the triangles are proven congruent, corresponding parts of congruent triangles (CPCTC) will imply that the diagonals \(\overline{DB}\) and \(\overline{CA}\) are congruent.

Step-by-Step Proof:

#### Statements and Reasons:

1. Given:
- \(ABCD\) is a parallelogram.
- \(\angle ADC\) and \(\angle BCD\) are right angles.

2. Opposite sides of a parallelogram are congruent:
- \(\overline{AD} \cong \overline{BC}\) (Reason: Opposite sides of a parallelogram are congruent).

3. Right angles are congruent:
- \(\angle ADC \cong \angle BCD\) (Reason: All right angles are congruent).

4. Reflexive property:
- \(\overline{DC} \cong \overline{DC}\) (Reason: Reflexive Property of Congruence).

5. Prove \(\triangle ACD \cong \triangle BDC\):
- We have:
- \(\overline{AD} \cong \overline{BC}\) (from step 2).
- \(\angle ADC \cong \angle BCD\) (from step 3).
- \(\overline{DC} \cong \overline{DC}\) (from step 4).
- By the Side-Angle-Side (SAS) Congruence Postulate:
- \(\triangle ACD \cong \triangle BDC\) (Reason: SAS Congruence).

6. Corresponding parts of congruent triangles are congruent (CPCTC):
- Since \(\triangle ACD \cong \triangle BDC\), the corresponding parts are congruent.
- Therefore, \(\overline{DB} \cong \overline{CA}\) (Reason: CPCTC).

Final Answer:

\[
\boxed{\overline{DB} \cong \overline{CA}}
\]

Explanation of the Solution:

- We started by using the properties of a parallelogram and the given right angles to establish congruence between certain sides and angles.
- We then applied the SAS Congruence Postulate to prove that \(\triangle ACD \cong \triangle BDC\).
- Finally, we used the Corresponding Parts of Congruent Triangles (CPCTC) to conclude that the diagonals \(\overline{DB}\) and \(\overline{CA}\) are congruent.

This completes the proof.