If the diagonals of a quadrilateral bisect the angles, is the quadrilateral always a parallelogram? Explain your answer.

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THATGUY260 THATGUY260

19-05-2018
Mathematics

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If the diagonals of a quadrilateral bisect the angles, is the quadrilateral always a parallelogram? Explain your answer.

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sqdancefan sqdancefan · Answer 1 · 2018-05-23T18:18:08+02:00

Yes. If the diagonals bisect the angles, the quadrilateral is always a parallelogram, specifically, a rhombus.

Consider quadrilateral ABCD. If diagonal AC bisects angles A and C, then ΔACB is congruent to ΔACD (ASA). Hence AB=AD and BC=CD (CPCTC).

Likewise, if diagonal BD bisects angles B and D, triangles BDA and BDC are congruent, thus AB=BC and AD=CD. (CPCTC again). Now, we have AB=BC=CD=AD, so the figure is a rhombus, hence a parallelogram.