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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
2 years, 7 months ago