Answered

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

The figure below shows rectangle ABCD. The two-column proof with missing statement proves that the diagonals of the rectangle bisect each other. Statement Reaso class=

Verified Answer :

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

Step-by-step 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:

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]

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