The figure below shows rectangle ABCD:
The following two-column proof with a missing statement proves that the diagonals of the rectangle bisect each other:
Statement Reason
ABCD is a rectangle. Given
Line segment AB and Line segment CD are parallel . Definition of a Parallelogram
Line segment AD and Line segment BC are parallel. Definition of a Parallelogram
? Alternate interior angles theorem
Line segment BC is congruent to line segment AD. Definition of a Parallelogram
∠ADB ≅ ∠CBD Alternate interior angles theorem
ΔADE ≅ ΔCBE Angle-Side-Angle (ASA) Postulate
Line segment BE is congruent to line segment DE. CPCTC
Line segment AE is congruent to line segment CE. CPCTC
Line segment AC bisects Line segment BD. Definition of a bisector
Answers
Given : rectangle ABCD:
To Find : complete proof to prove that the diagonals of the rectangle bisect each other:
Solution:
Statement Reason
ABCD is a rectangle. Given
Line segment AB and Line segment CD are parallel Definition of a Parallelogram
Line segment AD and Line segment BC are parallel. Definition of a Parallelogram
∠CAD ≅ ∠ACB Alternate interior angles theorem
Line segment BC is congruent to line segment AD. Definition of a Parallelogram
∠ADB ≅ ∠CBD Alternate interior angles theorem
ΔADE ≅ ΔCBE Angle-Side-Angle (ASA) Postulate
Line segment BE is congruent to line segment DE. CPCTC
Line segment AE is congruent to line segment CE. CPCTC
Line segment AC bisects Line segment BD. Definition of a bisector
QED
Hence proved
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