Question

given abcd is a rectangle. prove abcd has congruent diagonals. identify the steps that complete the proof. statements: 1. abcd is a rectangle. 2. draw ac and bd. 3. ∠bad and ∠cda are right angles. 4. ∠bad≅∠cda. 5. abcd is a parallelogram. 6. ab≅cd. 7. ad≅ad. 8. △bad≅△dcb. 9. bd≅ac. reasons: 1. given. 2. unique line possible. 3. def of a rectangle. 4. all right angles are congruent. 5. def of a rectangle. 7. reflexive property.

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite sides are congruent. Since $ABCD$ is a parallelogram (step 5), $AB\cong CD$ because of the property that opposite - sides of a parallelogram are congruent.

Step2: Determine triangle congruence criterion

We have $\angle BAD\cong\angle CDA$ (step 4), $AB\cong CD$ (step 6), and $AD\cong AD$ (step 7). By the Side - Angle - Side (SAS) congruence criterion, $\triangle BAD\cong\triangle DCA$.

Step3: Use CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Since $\triangle BAD\cong\triangle DCA$, then the corresponding sides $BD\cong AC$.

Answer:

1. Opposite sides of a parallelogram are congruent
2. SAS
3. CPCTC