幾何與角

In the figure, the two circles touch each other at $C$. The diameter $AB$ of the bigger circle is tangent to the smaller circle at $D$. If $DE$ bisects $\angle ADC$, find $\theta$.

In the figure, $EB$ and $EC$ are the angle bisectors of $\angle ABC$ and $\angle ACD$ respectively. If $\angle A=40^\circ$, find $\angle E$.

In the figure, $O$ is the centre of the circle. If the diameter $AOB$ rotates about $O$, which of the following is/are constant?

I. $\theta+\phi$ II. $AC+BD$ III. $AC\times BD$

In the figure, $AB=16$, $CD=8$, $BF=9$, $GD=4$, $EG=2$. Find $GC$.

In the figure, $ABCD$ is a square of side $a$ and $BDEF$ is a rhombus. $CEF$ is a straight line. Find the length of the perpendicular from $B$ to $DE$.

In the figure, $TB$ touches the semi-circle at $B$. $TA$ cuts the semi-circle at $P$ such that $TP=PA$. If the radius of the semi-circle is $2$, find the area of the shaded region.

In the figure, $O$ is the centre of the circle. Find $a+c$.

In the figure, $O$ is the centre of the circle $BCD$. $ABC$ and $EDC$ are straight lines, $BC=DC$ and $\angle AED=70^\circ$. Find $\angle BOD$.

In the figure, $ABCDE$ and $ABXYZ$ are two identical regular pentagons. Find $\angle AEZ$.

In the figure, $TPA$ and $TQB$ are tangents to the circle at $P$ and $Q$ respectively. If $PQ=PR$, which of the following must be true?

I. $\angle APR=\angle QRP$ II. $\angle QTP=\angle QPR$ III. $\angle QPR=\angle APR$