Mathematics
Prove that bisectors of any two opposite angles of a parallelogram are parallel.
Rectilinear Figures
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Answer
Let the parallelogram be ABCD as shown in the figure below:
In parallelogram ABCD we have,
∠A = ∠C (Opposite angles are equal)
so,
∠DAR = ∠QCB (As AR bisects ∠A and QC bisects ∠C and ∠A = ∠C)
In △ADR and △CBQ,
⇒ ∠DAR = ∠QCB (Proved above)
⇒ AD = BC (Opposite sides of a || gm)
⇒ ∠D = ∠B (Opposite angles of a || gm)
Hence, △ADR ≅ △CBQ by ASA axiom.
∴ ∠DRA = ∠BQC (By C.P.C.T.) …….(i)
Also,
∠RAQ = ∠DRA (Alternate angles are equal) ………(ii)
From (i) and (ii) we get,
∠RAQ = ∠BQC (These are also corresponding angles)
Since, corresponding angles are equal, we can say that
AR || QC.
Hence, proved that bisectors of any two opposite angles of a parallelogram are parallel.
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