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 $\angle{AEM}=\angle{BFM}$.