Question

given: abcd is a rectangle. prove: abcd has congruent diagonals. identify the steps that complete the proof.

Explanation:

Step1: Recall properties of parallelogram

In a parallelogram, opposite sides are congruent. Since \(ABCD\) is a parallelogram (step 5), for step 6, the reason for \(\overline{AB}\cong\overline{CD}\) is the parallelogram - side theorem.

Step2: Prove triangles congruent

In \(\triangle BAD\) and \(\triangle DCB\), we have \(\overline{AB}\cong\overline{CD}\) (step 6), \(\angle BAD\cong\angle CDA\) (step 4), and \(\overline{AD}\cong\overline{AD}\) (step 7). By the Side - Angle - Side (SAS) congruence postulate, \(\triangle BAD\cong\triangle DCB\). So the reason for step 8 is the SAS congruence postulate.

Step3: Use congruent - triangle property

Since \(\triangle BAD\cong\triangle DCB\), corresponding parts of congruent triangles are congruent. So \(\overline{BD}\cong\overline{AC}\). The reason for step 9 is CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. parallelogram side theorem
2. SAS congruence postulate
3. CPCTC