Question

The figure below shows rectangle ABCD.

The two-column proof with missing statement proves that the diagonals of the rectangle bisect each other.


Statement

Reason

ABCD is a rectangle.

Given

and are parallel

Definition of a Parallelogram

and are parallel

Definition of a Parallelogram



Alternate interior angles theorem



Definition of a Parallelogram

∡ADB ≅ ∡CBD

Alternate interior angles theorem



Angle-Side-Angle (ASA) Postulate



CPCTC



CPCTC

bisects

Definition of a bisector



Which statement can be used to fill in the blank space?

∡ABD ≅ ∡DBC

∡CAD ≅ ∡ACB

∡BDA ≅ ∡BDC

∡CAB ≅ ∡ACB

Answer 1

Answer: We can find out the missing statement with help of below explanation.

Explanation:

We have a rectangle ABCD with diagonals AC and BD ( shown in given figure.)

We have to prove: Diagonals AC and BD bisect each other.

In triangles, AED and BEC.

`\angle ADB \cong \angle CBD` ( By alternative angle theorem)

`AD\cong BC` ( Because ABCD is a rectangle)

`\angle CAD\cong \angle ACB` ( By alternative angle theorem)

By ASA postulate,
`\triangle AED\cong \triangle BEC`

By CPCTC,
`BE\cong ED` and
`CE\cong EA`

⇒ BE= ED and CE=EA

By the definition of bisector, AC and BD bisect each other.

Answer 2

To Prove the diagonals of the rectangle bisect each other, Statement with reason is given:

Statement ABCD is a rectangle.

Reason:Given

Statement

:Opposite sides are parallel.(AB║DC)

Reason: Definition of a Parallelogram

Statement

:Opposite sides are parallel.(AD║BC)

Reason: Definition of a Parallelogram

Statement

:∠CAB ≅ ∠ ACB

Reason: Alternate interior angles theorem

Statement

: ∠ADB ≅ ∠CBD

Reason: Alternate interior angles theorem

Option D:∠CAB ≅ ∠ ACB

In ΔAED and ΔBEC

∠CAB ≅ ∠ ACB→→Alternate interior angle,as AB║DC.

∠ADB ≅ ∠CBD→→Alternate interior angle,as AD║BC.

AD=BC→→Opposite sides in a rectangle are equal.

ΔAED ≅ ΔBEC⇒⇒[ASA]

AE=EC→[CPCTC]

BE=ED→[CPCTC]