Geometric Proof (AQA GCSE Further Maths): Exam Questions

$P,Q,R$ and $S$ are points on the circumference of a circle.

$w:y=7:5$

$PQ=PS$

Work out the size of angle $x$.

.......................degrees

$A,B$and $C$ are points on the circumference of a circle, centre $O$.

$ECD$ is a tangent to the circle at $C$. Angle $AOB=2x+46°$ Angle $OBC=37°$ Angle $ACD=3x$

Work out the value of $x$.

............................degrees

$A,B,C$and $D$ are points on a circle.

$\angle BCA=x$ $\angle ACD=2x$ $\angle CAD=3x$ $\angle CAB=4x$

Prove that $AC$ is a diameter.

$A,B$ and $C$ are points on a circle, centre $O$.

$ACD$ is a straight line. Angle $BCD=w$

Prove that   $w=x+90°$.

$A,B$, $C$ and $D$ are points on a circle, centre $O$.

The line $AB$ is a diameter of the circle.

The line $DC$ is parallel to the line $AB$.

Angle $DAB=x$

The line $DB$ is shown.

Work out the size of angle $DBC$ in terms of $x$.

$F,H,K$ and $J$ are points on a circle.

Chords $HJ$and $KF$ intersect at $L$. $EFG$ is a tangent to the circle. $FH$ and $JK$are parallel.

Angle $FHJ$ = 2$x$

Give reasons why angle $FKJ$ and angle $HJK$ are also equal to 2$x$.

Angle $FKJ$......................................

Angle $HJK$...................................

Work out the values of$x$and $y$.

You must show your working. Do not use trial and improvement.

$x=...................y=...................$

 $A,B,C$ and $D$ are points on a circle.

$D,E$ and $F$ are points on a different circle, centre $C$.

$DCE,ADF$ and $BCF$ are straight lines.

angle $DEF=x$

Prove that angle $BAD=2x$.

In the case when $AB$ is parallel to $DE$, work out the size of angle $x$.

...........................degrees

$D$, $E$, $F$ and $S$ are points on a circle.

$RST$ is a tangent.

The straight line $EDT$ is parallel to $FS$.

$DS=DT$

Prove that $FD$ is parallel to $RST$.

Use angle $DTS$ as $x$ to help you.