Question

Proving a Rectangle Has Congruent Diagonals

Given: ABCD is a rectangle.
Prove: ABCD has congruent diagonals.

Statements:
1. ABCD is a rectangle (given).
2. Draw AC and BD (unique line postulate).
3. ∠BAD and ∠CDA are right angles (definition of a rectangle).
4. ∠BAD ≅ ∠CDA (all right angles are congruent).
5. ABCD is a parallelogram (definition of a rectangle).

Identify the steps that complete the proof.

6. (AB ∥ CD) (_________________________)
7. (AD ≅ AD) (_________________________)
8. (ABCD) is a parallelogram (_________________________)
9. (BD ≅ AC) (_________________________)

Answer 1

To prove that a rectangle has congruent diagonals, you can use the properties that define a rectangle and parallelogram, and follow geometric reasoning to establish that opposite sides of a rectangle are equal and parallel, and thus its diagonals are also congruent.

Step-by-step explanation:

To prove that a rectangle has congruent diagonals, one must use geometric principles and theorems. Given that ABCD is a rectangle, and AC and BD are its diagonals, by definition of a rectangle (statement 5), opposite sides are equal in length, thus AB ≅ CD and BC ≅ AD. For step 6, we state that opposite sides of a rectangle are parallel, which is a property of a rectangle. Step 7 is an application of the reflexive property, stating that a line segment is congruent to itself (AD ≅ AD). This step, while it may seem trivial, is an important part of the logical chain in geometric proofs. For step 8, restating that ABCD is a parallelogram is not necessary as it was already established in statement 5, so this step could be an elaboration that any rectangle is a parallelogram with congruent diagonals. Finally, step 9 concludes the proof by asserting that because ABCD is a parallelogram with congruent opposite sides, the diagonals must be congruent (BD ≅ AC).