In a triangle, does an angle bisector necessarily bisect the opposite side? Have a look at the triangle and tell me if $AD=DB$:
 A: I can understand your thinking: equal things (angles) should make equal things (lines).
But angles don't project linearly.  (Just think of a shadow across a slanted wall).  So this doesn't hold.
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Alternative but similar question (I'm just switching that the lines are equal but asking if the angles are?)

If those lines at the bottom are equal to the line at the side, does that mean the angles at the top are equal.  Hint 1: the first angle is 45 degrees.  Can the angles all be $45$ degrees?  Hint 2:  I can add equal lines forever.  Can I add equal angles forever?
And another image.

Here the angles are both $30^\circ$.  But because this is a $30-60-90$ triangle and a smaller $30-60-90$ triangle we can actually calculate all the sides.
You can see they are not equal.
A: AD is not necessarily equal to DB. They are only equal if a = b. This is because when a side of a triangle is cut into two pieces by an angle bisector, the ratio of the 2 segments is the same as the ratio of the 2 sides.
