extended mathematics
posted by capriana .
In the diagram,<LMN=<ONM=90.P is the midpoint of MN.MN=2ML and MN=NO.prove that <OPN=<LNO

extended mathematics 
Reiny
You can't post diagrams in this forum

extended mathematics 
John
When you sketch out the diagram... two triangles. Since MN=2ML, and MN=N0, then NO=2ML and it also provides that angle LMN is 90 degrees and since it is an isosceles triange, angle LMN = angle LNM and they are both 45 degrees. Triangle LMN has a similar triange sitting on top of it (you can't see it without the diagram), but it contains angle PNL as it's end point. Since angle M = Angle P and they are 90 degrees and since the angle MLN equals the angle ONL from Parallel lines Alternate angles, and the angle zPN (that is in the small similar triangle, that is sitting on top of triangle LMN), is equal to MLN, we thus know that angle OPN=angle LNO and they are 45 degrees. QED : )
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