How do I prove the sides equal in this question? If the angular bisector of an angle of a triangle bisects the opposite side, prove that the other two sides are equal.
This was the question written in my math book. I know there is a theorem related to this, but I can't understand that.
I am a 9th grader so use simple language which I can understand
 A: Suppose $AD$ is the angle bisector of $\angle A$ in $\triangle ABC$ where $D$ is on $BC$, then $$\frac{AB}{AC} = \frac{BD}{CD}$$ so if $D$ is the midpoint of $BC$ we must have $AB=AC$. You can try to prove this important property of angle bisectors.
A: This can be taken as a self evident truth (axiom). It is possible the question in your book takes a different starting point of self evident truths and wants you to use logic to derive the equality of the two lengths.
Why can this be taken as a self evident truth?
Imagine walking x steps on a straight line and then rotating 90 degrees using your left hand and then taking y steps. OK, now go back to your starting point and facing the same direction take x steps and then rotate 90 degrees using your right hand and take y steps.
Now ask yourself what is the distance you traveled in both of these journeys. Well, you should feel no more tired in either case.
The attached picture should help.
Hint: By the way you phrased the question, you are dealing with an isosceles triangle. I was forced to draw the picture the way I did. You will certainly find this helpful:
The Isosceles Decomposition Theorem

