Question

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

Explanation:

Step1: Reason for \(\overline{AB}\cong\overline{CD}\)

Since \(ABCD\) is a parallelogram (step 5), in a parallelogram, opposite sides are congruent. So the reason for statement 6 (\(\overline{AB}\cong\overline{CD}\)) is "opposite sides of a parallelogram are congruent".

Step2: Reason for \(\triangle BAD\cong\triangle DCB\)

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 criterion, \(\triangle BAD\cong\triangle DCB\). So the reason for statement 8 (\(\triangle BAD\cong\triangle DCB\)) is "SAS (Side - Angle - Side)".

Step3: Reason for \(\overline{BD}\cong\overline{AC}\)

Since \(\triangle BAD\cong\triangle DCB\) (step 8), corresponding parts of congruent triangles are congruent. So the reason for statement 9 (\(\overline{BD}\cong\overline{AC}\)) is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

1. opposite sides of a parallelogram are congruent; 8. SAS (Side - Angle - Side); 9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)