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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.

Exercises

\(1\). In \(\triangle{ABC}\), \(\angle{A}=90^{\circ}\), \(E\) is the foot of the perpendicular from \(A\) to \(BC\), \(D\) is a point on \(BC\) such that \(BD=DC\) and \(F\) is a point on \(BC\) such that \(\angle{BAF}=\angle{CAF}\). Show that \(\angle{DAF}=\angle{FAE}\).

\(2\). Let \(ABCD\) be a parallelogram. Let \(P\) be a point in the interior of \(ABCD\) such that \(\angle{BAP}=\angle{PCB}\). Show that \(\angle{ABP}=\angle{ADP}\).

\(3\). Let the angle bisectors of \(\triangle{ABC}\), \(AD\), \(BE\), \(CF\) intersect at \(O\). Let \(G\) be the foot of the perpendicular from \(O\) to \(BC\). Show that \(\angle{BOD}=\angle{COG}\).

\(4\). Let \(ABCD\) be a quadrilateral such that \(AD=BC\). Let \(M\), \(N\) be the respective midpoints of \(AB\) and \(DC\). Extend \(AD\) and \(MN\) to meet at \(E\), \(BC\) and \(MN\) to meet at \(F\). Show that \(\angle{AEM}=\angle{BFM}\).

Note by Victor Loh
3 years, 2 months ago

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