Quadrilaterals Worksheet With Answers Page 31
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50The results of this exploration lead us to the following theorem.
Theorem
The diagonals of a rectangle are congruent.
The proof of this theorem is as follows:
Given: Rectangle ABCD with diagonals AC and BD
Right angle DAB
Prove: AC ≅ BD
D
C
A
B
Proof:
AD ≅
We have
BC
since a rectangle is a parallelogram and the opposite sides of
a parallelogram are congruent. ∠ DAB and ∠ CBA are supplementary because
consecutive angles of a parallelogram are supplementary. We can now write
m ∠ DAB + m ∠ CBA = 180 by definition of supplementary angles. Using
substitution, the equation becomes 90 + m ∠ CBA = 180. Solving this equation ,
m ∠ CBA = 90. We have ∠ DAB ≅ ∠ CBA since all right angles are congruent.
AB ≅
by the reflexive property. Now ∆ DAB ≅ ∆ CBA by the SAS
AB
congruence postulate. AC ≅ BD by CPCTC.
The converse of this theorem can be stated as follows. The proof of this theorem
will be left as an exercise.
Theorem
If the diagonals of a parallelogram are
congruent, then the parallelogram is
a rectangle.
This theorem can be used to prove a parallelogram is a rectangle.
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