Question

proving a rectangle has congruent diagonals

given: abcd is a rectangle.
prove: abcd has congruent diagonals.

identify the steps that complete the proof

Explanation:

Step1: Properties of a parallelogram

In a parallelogram, opposite sides are congruent. Since \(ABCD\) is a rectangle (and a rectangle is a type of parallelogram), \(AB\cong CD\).

Step2: Side - Angle - Side (SAS) congruence

We know \(AD = DA\) (common side). \(\angle BAD\cong\angle CDA\) (from step 4) and \(AB\cong CD\) (from step 6). So, \(\triangle BAD\cong\triangle CDA\) by the SAS (Side - Angle - Side) congruence criterion.

Step3: Congruent triangles' corresponding parts

If \(\triangle BAD\cong\triangle CDA\), then their corresponding parts are congruent. So, \(AC\cong BD\) (diagonals of the rectangle \(ABCD\)).

Answer:

1. \(AB\cong CD\) (opposite sides of a parallelogram are congruent); 7. \(\triangle BAD\cong\triangle CDA\) (by SAS: \(AB = CD\), \(\angle BAD=\angle CDA\), \(AD = DA\)); 8. \(AC\cong BD\) (corresponding parts of congruent triangles are congruent)