Quadrilaterals Worksheet With Answers Page 13
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50c) Measure the lengths of the segments AM, MC, DM, and MB of diagonals
AC and DB as shown in figure 3 below. Compare these results to the
conjecture made previously. The figure below is representative of the
possible results. Measurements will vary according to the parallelogram
drawn.
figure 3
These results can be summarized in the following theorem about the diagonals
of a parallelogram.
Theorem
The diagonals of a parallelogram bisect
each other.
.
This can be proven using other properties of a parallelogram as follows.
Given:
ABCD with diagonals AC and DB
Prove: AC and DB bisect each other
D
C
M
A
B
Proof:
By the definition of a parallelogram, we have DC
AB . It follows that
∠ CDB ≅ ∠ ABD since alternate interior angles are congruent when two
parallel lines are cut by a transversal. ∠ DMC ≅ ∠ BMA because vertical
angles are congruent. DC ≅ DC by the reflexive property. Now we have
∆ AMB ≅ ∆ CMD by the AAS theorem. By CPCTC,
DM ≅
MB
and
AM ≅
MC
. Therefore, AC and DB bisect each other.
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