IF angle is equal side is also equal ?

Question

I am having problem with this triangle type question. Someone please explain ty 
Image attached.
Case 1
1 Angle BAC = Angle ADC, now if this is given why wont it be BC and AC become equal since they are opposite equal angle? is it because no triangle formed ?

Does it happen only for isosceles triangle ?

Case 2
1. AD is angle bisector of Angle A. Why wont BD = DC ? since equal angle should have equal side ? or it other way round ?

a) Will equal angle always have equal side opposite to it ?
b) Will equal Side always have equal side angle to it ?

chetan2u · Answer

The reply to both your question is that you are comparing SAME angles but in two DIFFERENT triangles.
If two angles are EQUAL within same triangle answer is YES for (a) and (b)
But if you are looking at two different triangles, it may or may not.

Here you can surely say that ABC is similar to DAC as  \angle A=\angle D  ,  \angle B=\angle A  and  \angle C=\angle C , so we can say that  \frac{AB}{DA}=\frac{BC}{AC}=\frac{AC}{DC} .

GMATPanther · Answer

thanks was really wondering this for couple of days now

ccooley · Answer

chetan2u already answered your math questions, but I want to add something else! If you have questions like these, before you try looking up the answers, I recommend drawing out some extreme examples to see if you can understand the situation better that way.

For example, here's how I'd figure out your first question if I was studying Geometry. Try drawing the same diagram, but try making both angles extremely large, or extremely small. See if you can come up with a way to make the lengths obviously different, even though the angles are the same. Here's one I found, where BC and AC definitely aren't equal:

https://i.imgur.com/8NX2ehL.png

For your second question, draw some different triangles and check whether the angle bisector splits the opposite side in half. Here, it does:

https://i.imgur.com/4YEwF3x.png

But if the triangle is less symmetrical, it doesn't:

https://i.imgur.com/A6fAVhA.png

IF angle is equal side is also equal ?

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