Understand the problem

let O be a point inside a parallelogram ABCD such that \(\angle AOB+\angle COD =180\) prove that \(\angle OBC =\angle ODC\)

Source of the problem

C.M.I (Chennai mathematical institute UG-2019 entrance

Topic

Geometry

Difficulty Level

5 out of 10

Suggested Book

challenges and thrills of pre college mathemetics

Start with hints

Do you really need a hint? Try it first!

 

Draw a clear image of the given problem

 

translate ABCD along the vector AD SO A’ and D are the same , and so that B’ and C are the same

 

now , \(\angle COD +\angle CO’D=\angle COD+\angle A’O’D’ =180 \)

so OCO’D is cyclic . therefore \(\angle OO’C =\angle ODC\)

 

Also , vector BC and OO’ both equal AD so OBCO’ is parallelogram . therefore

\(\angle OBC =\angle OO’C=\angle ODC \)

 

 

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