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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 need to show that triangles formed by the diagonals and the sides of the rectangle are congruent.

Given the options:

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

Since a rectangle has opposite sides that are equal and all angles are right angles, we can use the properties of these triangles to establish congruence.

Analysis:

  • Statement #1: △ADC≅△CBA

    • This statement compares triangles that do not share any common sides with the diagonals; hence, it doesn't effectively help in proving the congruence of the diagonals.

  • Statement #2: △BCD≅△DAB

    • This compares triangles located on either side of diagonal BD. However, this does not directly reference diagonal AC either.

  • Statement #3: △ADC≅△BCD

    • This statement compares the triangles formed by both diagonals AC and BD. Since diagonals divide the rectangle into two congruent triangles, if this statement is proven, it confirms that both triangles share side AD = BC, and both contain right angles at points of intersection (O) making them congruent triangles, which would lead to the conclusion that AC = BD (the diagonals are congruent).

Conclusion:

Statement #3, once proven, will prove that the diagonals of rectangle ABCD are congruent.