# Converse : Triangles with Equal areas and Equal Bases have Equal Corresponding Altitudes.

**Given:** Area of two triangles ABC and DBC are on the same base BC and are equal. AE and DF are altitudes of triangles ABC and DBC respectively.

**To Prove: **AE = DF

**Proof :**ar Î” ABC = ar Î” DBC

Ã— AP Ã— BC = Ã— DQ Ã— BC

AP = DQ

âˆ´ Triangles with equal areas and equal bases have equal corresponding altitudes.

Triangles ABC and DBC are on the same base BC with A, D on the opposite sides of the line BC, such that the ar(Î”ABC) = ar(Î”DBC). Show that BC bisects AD.

**Given : **Î”ABC and Î”BCD are on the same base BC. A and D are on opposite sides of BC and ar(Î”ABC) = ar(Î”BCD).

**To prove:** AO = OD.

**Construction:** Draw perpendiculars AM and DN on BC from A and D respectively.

**Proof:**

ar(Î”ABC) = ar(Î”BCD) ...(given)

BC x AM = BC x DN

.^{.}. AM = DN

In the Î”AMO and the Î”DNO,

âˆ AMO = âˆ DNO ...(each 90Â°)

âˆ AOM = âˆ DON ...(Vertically opposite angle)

AM = DN (Proved above)

.^{.}. Î”AMO â‰… Î”DNO

.^{.}. AO = OD