EMT 725

Philippa M. Rhodes

Solution I

Since the three angle bisectors of a triangle are concurrent, we can create the bases of two triangles that would have the unknown vertex as the third angle.

Now, by constructing the angle bisectors of the two available angles of each triangle,

we have two points on the angle bisector of the unknown angle. So, we can draw the angle bisector.

Solution II

In an isosceles triangle, the bisector of the angle opposite to the base is the perpendicular bisector of the base. Therefore, we can find the angle bisector of the angle formed by two nonintersecting segments by drawing a line that will intersect both of the segments so that the interior angles on a given side of the line are congruent. Click here to see this construction.

This means that segment QR is the base of an isosceles triangle. So, the perpendicular bisector is the bisector of the unknown angle.

Construction

Draw a line parallel to one of the segments through a point on the other segment.

Draw the line that bisects VRJ.

Since segment AX is parallel to line VR, AXR VRW. Since VRW WRJ and since WRJ and CRX are vertical angles, CRX AXR.