experts plshh give me the proof of all three theorems ,,, ??? faster i need it !!!

i mean theorem 9.1 , 9.2 and 9.3 ??? 

Theorem 9.2: Two triangles on the same base and between the same parallels are equal in area.

 

Given: Δ ABC and Δ PBC lie on the same base BC and between the same parallels l and m.

To prove: Area (ΔABC) = Area (ΔPBC)

Construction: Draw AD and PQ, the altitude drawn from A and P on BC respectively.

Proof :

AD = PQ [Perpendicular distance between two parallel line is always equal] .... (1)

Given, Area (ΔABC)  

and Area (ΔPBC) 

 

⇒ Area (ΔABC) = Area (ΔPBC)

[Hence proved]

 

Theorem 9.3: Two triangles having the same base (or equal bases) and equal areas lie between the same parallels

Given: ΔABC and ΔPBC lie on the same base BC. Area (ΔABC) = Area (ΔPBC). AD and PQ are the altitude drawn from A and P on BC respectively.

To prove: AP || BC

Proof :

Given, Area (ΔABC) = Area (ΔPBC)

⇒ AD = PQ

So, the altitudes AD and PQ drawn on side BC are equal. Therefore, the triangle must lie between the same parallel.

Thus, AP || BC (Distance between parallel lines is constant)

Hope you get it!!

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