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A quadrilateral upper A upper B upper C upper D. Two diagonals connect vertices upper A to upper C and upper B to upper D. The point of intersection is upper O.

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.

1 answer

To prove that the diagonals of rectangle ABCD are congruent, we can use the fact that triangles formed by the diagonals have equal lengths.

Given that ABCD is a rectangle, the diagonals AC and BD intersect at point O, and we can consider the triangles formed:

  1. Triangle AOD (formed by the diagonal AC)
  2. Triangle BOC (formed by the diagonal BD)

In a rectangle:

  • The diagonals bisect each other.
  • Triangle AOD and triangle BOC can be proven to be congruent by using the properties of rectangles.

Among the statements given:

  • Statement #1: △ADC≅△CBA
  • Statement #2: △BCD≅△DAB
  • Statement #3: △ADC≅△BCD

Out of these, Statement #3: △ADC≅△BCD would directly help in proving that the diagonals are congruent. If triangles ADC and BCD are congruent, this would imply that the diagonals AC and BD are equal in length, since both triangles share a common side (OB or OA), both have right angles (because ABCD is a rectangle), and also the remaining sides AD and BC are equal (due to the properties of a rectangle).

Thus, the answer is:

Statement #3