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in the figure below line segment joining made point M and N of sides AB and CD quadrilateral ABCD is perpendicular to both. Prove that other sides of quadrilateral are are equal?

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**Answer:**

### Given:

- line segment joining made point M and N of sides AB and CD quadrilateral ABCD is perpendicular to both.

### To prove:

- Prove that other sides of quadrilateral are are equal

### Solution:

Construction Join - CM and DM

In ΔCMN and ΔDMN

MN=MN ( Common)

∠CNM = ∠DNM = 90° (MN Is perpendicular to DC)

CN = DN (Since N is the mid point of DC)

By SAS congruency Δ CMN ≅Δ DMN

Therefore,

CM = DM (CPCT)

∠CMN = ∠ DMN (CPCT)

∠AMN = ∠BMN = 90 (Since MN is perpendicular to AB)

So, ∠AMN − ∠CMN = ∠BMN − ∠ DMN (Since ∠CMN = ∠ DMN )

∠AMD = ∠ BMC

In Δ AMD and Δ BMC DM = CM (Proved above)

∠AMD = ∠ BMC (Proved above)

AM = BM (M is the mid point of AB)

By SAS congruency

Δ AMD ≅Δ BMC

Therefore, AD = BC (CPCT)

### Hence Proved

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