TITLE: Proving that two angles have the same bisector in Geometry
QUESTION [1 upvotes]: Inside a given angle, another angle is constructed such that its sides are parallel to the sides of the given one and are the same distance away from them. Prove that the bisector of the constructed angle lies on the bisector of the given angle.
The section is on quadrilaterals and parallel lines. 
We are given two angles (I called them angle ABC and angle KLM). BC is parallel to ML and AB is parallel to KL. We are also given that the sides of the constructed angle are the same distance away from the sides of the given angle.
I do not know how to go about solving this problem. Some properties we learned about parallel lines were the Z, F, and C properties. I'm thinking they may play a role in determining that each angle has the same bisector? Any ideas would be appreciated!

REPLY [3 votes]: If angle KLM is inside angle ABC, and the distance between AK is the same as the distance between CM, then the point L is equidistant from both sides AB and BC. This is the definition of angle bisector; any point on the bisector is at equal distances from each side of the angle. So point L is on the bisector of angle ABC.
Since any bisector must go through the vertex of the angle, point L lies on the bisector of KLM as well.
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