Question
Use the image to answer the question.
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.
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.
Answers
Answered by
GPT-4o mini
To prove that the diagonals of rectangle ABCD are congruent, you can use one of the congruent triangles formed by the diagonals and the sides of the rectangle.
**Statement #3: ΔADC ≅ ΔBCD** will prove that the diagonals \( AC \) and \( BD \) of rectangle ABCD are congruent.
Here's the reasoning:
1. **Shared Side:** The triangles ΔADC and ΔBCD share side \( AD \) (which is equal in length to \( BC \)).
2. **Right Angles:** In rectangle ABCD, all angles are right angles, meaning \( ∠ADC \) and \( ∠BCD \) are both 90 degrees.
3. **Equal Sides:** The sides opposite \( AD \) and \( BC \) are equal (because ABCD is a rectangle), meaning \( AB = CD \).
By the criteria of the hypotenuse and one corresponding side (the shared side), we can conclude that the triangles are congruent.
Thus, if ΔADC ≅ ΔBCD, then it follows that the diagonals \( AC \) and \( BD \) are congruent.
**Statement #3: ΔADC ≅ ΔBCD** will prove that the diagonals \( AC \) and \( BD \) of rectangle ABCD are congruent.
Here's the reasoning:
1. **Shared Side:** The triangles ΔADC and ΔBCD share side \( AD \) (which is equal in length to \( BC \)).
2. **Right Angles:** In rectangle ABCD, all angles are right angles, meaning \( ∠ADC \) and \( ∠BCD \) are both 90 degrees.
3. **Equal Sides:** The sides opposite \( AD \) and \( BC \) are equal (because ABCD is a rectangle), meaning \( AB = CD \).
By the criteria of the hypotenuse and one corresponding side (the shared side), we can conclude that the triangles are congruent.
Thus, if ΔADC ≅ ΔBCD, then it follows that the diagonals \( AC \) and \( BD \) are congruent.
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