Given: In ∆ABC, ∠B=35°,∠C=65° and ∠BAX = ∠XAC

To find: Relation between AX, BX and CX in descending order.

In ∆ABC, by the angle sum property, we have

∠A + ∠B + ∠C = 180°

∠A + 35° + 65° = 180°

∠A + 100° = 180°

∴ ∠A = 80°

But ∠BAX = ∠A

= × 80° = 40°

Now in ∆ABX,

∠B = 35°

∠BAX = 40

And ∠BXA = 180° - 35° - 40°

= 105°

So, in ∆ABX,

∠B is smallest, so the side opposite is smallest, ie AX is smallest side.

∴ AX < BX …(1)

Now consider ∆AXC,

∠CAX = × ∠A

=× 80° = 40°

∠AXC = 180° - 40° - 65°

= 180° - 105° = 75°

Hence, in ∆AXC we have,

∠CAX = 40°, ∠C = 65°, ∠AXC =75°

∴∠CAX is smallest in ∆AXC

So the side opposite to ∠CAX is shortest

Ie CX is shortest

∴ CX <AX …. (2)

From 1 and 2 ,

BX > AX > CX

This is required descending order