Answer:
Step-by-step explanation:To prove that the rectangle ABCD has congruent diagonals, we can follow the following steps:
1. Draw diagonal AC in the rectangle ABCD.
2. By definition, a rectangle is a parallelogram with four right angles.
3. By the Parallelogram Side Theorem (♣), opposite sides of a parallelogram are congruent.
4. Since ABCD is a rectangle, its opposite sides are congruent, which means AB is congruent to CD and AD is congruent to BC.
5. Now, consider the triangles ABD and CBD.
6. In triangle ABD, side AB is congruent to side AD, and in triangle CBD, side CB is congruent to side BC.
7. By the Side-Side-Side (SSS) congruence theorem (♦), if two triangles have three pairs of corresponding sides congruent, then the triangles are congruent.
8. Therefore, triangle ABD and triangle CBD are congruent.
9. By the Congruent Triangles Diagonal Test (♠), if two triangles are congruent, then their corresponding parts, including diagonals, are congruent.
10. Hence, diagonal AC is congruent to diagonal BD.
11. Therefore, we have proven that the rectangle ABCD has congruent diagonals.
In summary, the steps that complete the proof are: 2, ♣, 4, 7, ♦, ♠.