How to show two angles are equal

There are many ways to show two angles are equal. We usually try the first few techniques listed below, before the last couple.

  • Using parallel lines;

  • Using congruent triangles;

  • Using isosceles triangles;

  • Using parallelograms;

  • Using similar triangles;

  • Circle properties: Angles subtended by the same chord, external angles of cyclic quadrilaterals, alternate segment theorem;

  • Via a third (or fourth) angle;

  • Showing the two angles are the sum, difference, twice or half of other equal angles.


11. In ABC\triangle{ABC}, A=90\angle{A}=90^{\circ}, EE is the foot of the perpendicular from AA to BCBC, DD is a point on BCBC such that BD=DCBD=DC and FF is a point on BCBC such that BAF=CAF\angle{BAF}=\angle{CAF}. Show that DAF=FAE\angle{DAF}=\angle{FAE}.

22. Let ABCDABCD be a parallelogram. Let PP be a point in the interior of ABCDABCD such that BAP=PCB\angle{BAP}=\angle{PCB}. Show that ABP=ADP\angle{ABP}=\angle{ADP}.

33. Let the angle bisectors of ABC\triangle{ABC}, ADAD, BEBE, CFCF intersect at OO. Let GG be the foot of the perpendicular from OO to BCBC. Show that BOD=COG\angle{BOD}=\angle{COG}.

44. Let ABCDABCD be a quadrilateral such that AD=BCAD=BC. Let MM, NN be the respective midpoints of ABAB and DCDC. Extend ADAD and MNMN to meet at EE, BCBC and MNMN to meet at FF. Show that AEM=BFM\angle{AEM}=\angle{BFM}.

Note by Victor Loh
6 years, 11 months ago

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