Quadrilaterals Worksheet With Answers Page 38
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50This theorem can be stated as “ Either diagonal of a rhombus bisects a pair of opposite angles.”
We can summarize the properties of a rhombus as follows:
Properties of a Rhombus
1. Opposite sides of a rhombus are parallel.
2. Opposite sides of a rhombus are congruent.
3. Opposite angles of a rhombus are congruent.
4. Consecutive angles of a rhombus are supplementary.
5. The diagonals of a rhombus bisect each other.
6. The diagonals of a rhombus are perpendicular.
7. Either diagonal of a rhombus bisects a pair of
opposite angles.
To show that a parallelogram is a rhombus, the converse of the following theorem can be used.
Theorem: If a parallelogram is a rhombus, then its diagonals are perpendicular.
Converse: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a
rhombus.
The proof of the converse is given below.
Given:
ABCD
D
C
AC ⊥
DB
Prove:
ABCD is a rhombus
M
A
B
Proof:
∠CMD and ∠CMB are right angles because
AC ⊥
. ∠CMD ≅ ∠CMB since all right angles
DB
≅
MC ≅
are congruent. The diagonals of a parallelogram bisect each other and
DM
MB .
MC
by the reflexive property. Now we have ∆DMC ≅ ∆BMC by the SAS congruence postulate. By
DC ≅
CPCTC,
. Parallelogram ABCD is a rhombus by the definition of a rhombus.
BC
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