Question

Given that quadrilateral ABCD is a rectangle, which of the following statements, once proven, will prove that the diagonals of rectangle ABCD are congruent? Statement #1: △ADC≅△CBA Statement #2: △BCD≅△DAB Statement #3: △ADC≅△BCD (1 point) Statement #, once proven, will prove that the diagonals of rectangle ABCD are congruent.

Answer 1

In a rectangle, the diagonals are congruent if we can show that the two triangles formed by one of the diagonals are congruent. Among the statements you've provided, we can analyze them one by one:

1. Statement #1: △ADC≅△CBA
   This statement suggests that triangles ADC and CBA are congruent. However, to use this to conclude that the diagonals are congruent, you would need to relate this directly to the diagonals AC and BD, which this statement does not do.

2. Statement #2: △BCD≅△DAB
   This statement suggests that triangles BCD and DAB are congruent. If these two triangles are congruent, then their corresponding sides, which include the diagonals (BD and AC), must be equal. Therefore, proving this statement will indeed confirm that the diagonals are congruent.

3. Statement #3: △ADC≅△BCD
   Similar to statement #1, while this statement might suggest some relationships, it does not directly imply anything about the diagonals of the rectangle.

Based on this analysis, the correct choice is:

Statement #2: △BCD≅△DAB

Proving statement #2 will show that the diagonals of rectangle ABCD are congruent.