幾何與角

In the figure, $O$ is the centre of the circle. Find $x$.

In the figure, $O$ is the centre of the circle. $PA$ is the tangent to the circle at $A$ and $CB\parallel PA$. Find $x$.

In the figure, $ABCDE$ is a regular pentagon and $ABF$ is an equilateral triangle. Find $x$.

In the figure, if $\dfrac{\text{Area of triangle }CDE}{\text{Area of triangle }BCE}=\dfrac{1}{2}$, find $\dfrac{\text{Area of triangle }CDE}{\text{Area of trapezium }ABCD}$.

In the figure, $O$ is the centre of the circle. $AP$, $AB$ and $BR$ are tangents to the circle at $P$, $Q$ and $R$ respectively. Which of the following must be true?

I. $AP+BR=AB$ II. $OQ$ bisects $\angle AOB$ III. $\angle AOB=\dfrac{1}{2}\angle POR$

In the figure, $ABCD$ is a trapezium with $AB\parallel DC$. $AH$ bisects $\angle BAD$ and $DH$ bisects $\angle ADC$. Which of the following must be true?

I. $\angle AHD=90^\circ$ II. $\angle ADC=\angle BCD$ III. $\angle BAD+\angle BCD=180^\circ$

In the figure, $DE:EF=$

In the figure, $BDC$ is a straight line. Arrange $AD$, $BD$ and $DC$ in ascending order of magnitude.

In the figure, $ABCD$ is a semicircle. $\angle CAD=$

In the figure, $O$ is the centre of the circle, $POQR$ is a straight line. $TR$ is the tangent to the circle at $T$. $\angle PRT=$