Mathematics
The given figure shows a parallelogram ABCD. Points M and N lie in diagonal BD such that DM = BN. Prove that:
(i) △DMC ≅ △BNA and so CM = AN.
(ii) △AMD ≅ △CNB and so AM = CN.
(iii) ANCM is a parallelogram.
Quadrilaterals
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Answer
(i) Given:
ABCD is a parallelogram.
To prove:
△DMC ≅ △BNA and so CM = AN
Proof:
In triangle DMC and BNA,
CD = AB (opposite sides of parallelogram)
DM = BN (Given)
∠ CDM = ∠ ABN (alternate angles)
So, by Side Angle Side congruency,
△DMC ≅ △BNA
By using Corresponding Parts of Congruent Triangles,
CM = AN
Hence, △DMC ≅ △BNA and CM = AN.
(ii)To prove:
△AMD ≅ △CNB and so AM = CN.
Proof:
In triangle AMD and CNB,
AD = BC (opposite sides of parallelogram)
DM = BN (Given)
∠ ADM = ∠ CBN (alternate angles)
So, by Side Angle Side congruency,
△AMD ≅ △CNB
By using Corresponding Parts of Congruent Triangles,
AM = CN
Hence, △AMD ≅ △CNB and so AM = CN.
(iii)To prove:
ANCM is a parallelogram.
Proof:
CM = AN (proved)
AM = CN (proved)
Hence, ANCM is a parallelogram.
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