Question

Help Me!

In the quadrilateral ABCD shown below, the sides AB and CD are parallel. M is the Mid point of the side BC.

The lines DM and AB extended, meet at N.


`\large\sf\color{Aqua}\underline{Questions}`

i) Are the areas of ∠DCM and ∠BMN equal?why?

ii) What is the relation between the areas of the quadrilateral and the triangle ADN?​

Answer 1

I think I have proved it before you asked this question before also.

Explanation:

SEE the image for solution.

HOPE it helps

Have a great day

Answer 2

Given:

• DC ║ AB
• CM = MB as M is midpoint of BC

i) Since DN and BC are transversals, we have:

• ∠DCM ≅ ∠NBM and
• ∠CDM ≅ ∠BNM as alternate interior angles

As two angles and one side is congruent, the triangles are also congruent:

• ΔDCM ≅ ΔNBM (according to AAC postulate)

So their areas are same.

ii)

The quadrilateral has area of:

• A(ADCB) = A(ADMB) + A(DCM)

And the triangle has area of:

• A(ADN) = A(ADMB) + A(NBM)

Since the areas of triangles DCM and NBM are same, the quadrilateral ADCB has same area as triangle ADN.