End of chapter exercises

$AOC$ is a diameter of the circle with centre $O$. $F$ is the mid-point of chord $EC$. $B\hat{O}C = C\hat{O}D$ and $\hat{B} = x$. Express the following angles in terms of $x$, stating reasons:

$D$, $E$, $F$ and $G$ are points on circle with centre $M$. $\hat{F}_1 = \text{7}\text{°}$ and $\hat{D}_2 = \text{51}\text{°}$. Determine the sizes of the following angles, stating reasons:

$O$ is a point on the circle with centre $M$. $O$ is also the centre of a second circle. $DA$ cuts the smaller circle at $C$ and $\hat{D}_1 = x$. Express the following angles in terms of $x$, stating reasons:

$O$ is the centre of the circle with radius $\text{5}$ $\text{cm}$ and chord $BC = \text{8}\text{ cm}$. Calculate the lengths of:

$PQ$ is a diameter of the circle with centre $O$. $SQ$ bisects $P\hat{Q}R$ and $P\hat{Q}S = x$.

$B\hat{O}D$ is a diameter of the circle with centre $O$. $AB = AD$ and $O\hat{C}D = \text{35}\text{°}$. Calculate the value of the following angles, giving reasons:

$O$ is the centre of the circle with diameter $AB$. $CD \perp AB$ at $P$ and chord $DE$ cuts $AB$ at $F$. Prove that:

$P$, $Q$, $R$ and $S$ are points on the circle with centre $M$. $PS$ and $QR$ are extended and meet at $T$. $PQ = PR$ and $P\hat{Q}R = \text{70}\text{°}$.

$POQ$ is a diameter of the circle with centre $O$. $QP$ is protruded to $A$ and $AC$ is a tangent to the circle. $BA \perp AQ$ and $BCQ$ is a straight line. Prove:

$TA$ and $TB$ are tangents to the circle with centre $O$. $C$ is a point on the circumference and $A\hat{T}B = x$. Express the following in terms of $x$, giving reasons:

$AOB$ is a diameter of the circle $AECB$ with centre $O$. $OE \parallel BC$ and cuts $AC$ at $D$.

$PQ$ and $RS$ are chords of the circle and $PQ \parallel RS$. The tangent to the circle at $Q$ meets $RS$ protruded at $T$. The tangent at $S$ meets $QT$ at $V$. $QS$ and $PR$ are drawn.

Let $T\hat{Q}S = x$ and $Q\hat{R}P = y$. Prove that:

The two circles shown intersect at points $F$ and $D$. $BFT$ is a tangent to the smaller circle at $F$. Straight line $AFE$ is drawn such that $DF = EF$. $CDE$ is a straight line and chord $AC$ and $BF$ cut at $K$. Prove that: