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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!

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  • $\begingroup$ A diagram would be great here. According to me, construction of an angle "inside another one" with the constructed angle's sides parallel to the first one renders exactly the same angle, but perhaps you mean something else. $\endgroup$
    – DonAntonio
    Commented Apr 12, 2016 at 22:16
  • $\begingroup$ You just need to draw the bisector and extend the two inner lines and play around with the angles $\endgroup$
    – user164550
    Commented Apr 12, 2016 at 22:19
  • $\begingroup$ Unfortunately, no picture was given. Ahmed's explanation makes sense to me. I was considering that. I'll give it a go! $\endgroup$
    – d3m0nxdrums
    Commented Apr 12, 2016 at 22:27
  • $\begingroup$ Always start out by drawing such a problem. $\endgroup$
    – user242559
    Commented Apr 12, 2016 at 22:29

1 Answer 1

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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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