Geometry - Limiting Parallel Rays

[xcr]

Homework Statement

Let us assume that ray XY is a limiting parallel to line l, with P*X*Y. Prove that ray PY is a limiting parallel to line l.

Homework Equations

Steps will go something like this: Show that ray PZ meets line lat a point V. Pick a point S such that P is between S and Z. Show that SX meets line PQ in a point U such that P is between U and Q. Choose any W such that X is between U and W and show that ray XW is between ray XY and ray XR so that ray XW meets line l at point T.

The Attempt at a Solution

Since we know that ray XY is a limiting parallel ray, we can get that ray XW interests line l at the point T because it is between ray XY and ray XY. I'm not really sure where to go from here.

 

[mighty2000]

Any help would be appreciated.



First, let's define some points for clarity. Let P be the point where ray PY intersects line l, and let V be the point where ray PZ intersects line l. We can also define point S as the point on line l such that P is between S and Z, and point U as the point where ray SX intersects line PQ such that P is between U and Q.

Now, let's consider the triangle PVU. By the transitive property of parallel lines, we can deduce that PV is parallel to ray XY, and PU is parallel to ray XW. Since ray XY is a limiting parallel to line l, we know that ray XW also intersects line l at point T.

Next, we can use the Parallel Postulate to show that the alternate interior angles of parallel lines are congruent. This means that angle PUV is congruent to angle XYW.

Since angle XYW is an exterior angle of triangle PVU, it must be larger than angle PUV. This means that ray XW is closer to line l than ray PY, making it a limiting parallel to line l.

Therefore, we have proven that ray PY is a limiting parallel to line l with point P as its intersection point.

I hope this helps in your understanding. Keep up the good work in your studies of geometry!