Congruence, Similarity & Geometrical Proof (AQA GCSE Maths: Higher): Exam Questions

ABCD is a parallelogram.
ABP and QDC are straight lines.
Angle ADP = angle CBQ = 90^o.

Prove that triangle ADP is congruent to triangle CBQ.

Explain why AQ is parallel to PC.

$ABCD$ is a rhombus.

$M$ and $N$ are points on $BD$ such that $DN=MB.$

Prove that triangle $DNC$ is congruent to triangle $BMC$.

$PQ=PR$.
$S$ is the midpoint of $PQ$.
$T$ is the midpoint of $PR$.

Prove triangle $QTR$ is congruent to triangle $RSQ$.

$ABC$ and $AED$ are straight lines.
$BE$ and $CD$ are parallel.

$\begin{array}{l}BE=1.5\mathrm{cm}.\\ CD=6\mathrm{cm}.\\ AD=5\mathrm{cm}.\end{array}$

Calculate the length of $ED$.

ABC and EDC are straight lines.
EA is parallel to DB.

EC = 8.1 cm.
DC = 5.4 cm.
DB = 2.6 cm,

Work out the length of AE.

AC = 6.15 cm.

Work out the length of AB.

$ABCD$ is a parallelogram.

$E$ is the point where the diagonals $AC$ and $BD$ meet.

Prove that triangle $ABE$ is congruent to triangle $CDE$.

Circle the reason why these triangles are congruent.

Here are four triangles.

Which two triangles are congruent?
Circle two letters below.

$PRT$ and $QRS$ are similar triangles.

Which of these is equivalent to $\frac{QR}{PR}$?

Circle your answer.

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

• $BC$ bisects angle $ABQ$.

• $PBQ$ is a tangent to the circle.

Angle $CBQ=x$
Prove that $AC=BC$

The diagram shows two water towers in Kuwait.

The real height of tower $\mathbf{A}$ is $187\mathrm{m}$.
The real height of tower $\mathbf{B}$ is $147\mathrm{m}$.

Ahmed makes a scale model of both towers.

The height of tower $\mathbf{A}$ on the scale model is $90\mathrm{cm}$.

Work out the height of tower $\mathbf{B}$ on the scale model.
Give your answer correct to the nearest centimetre.

The diagram shows triangle ABC.
CD is parallel to AB.
A, C and E lie in a straight line.
Angles of size $a°,b°$and $c°$ are shown.

Insert  $a°,b°$° or $c°$ to make this statement true.
Give a reason for your answer.

Angle DCE = ......... because ....................................................................................................

Use the diagram and the answer to part (a) to show that the angles of a triangle add up to 180°.

Give a reason for each statement you make.

In the diagram AB is parallel to CD.
AED and BEC are straight lines.

Prove that triangle ABE is similar to triangle CDE.

The diagram below shows two triangles.

Prove that triangle ABC is congruent to triangle ACD.

Anna estimates the height of a tree.

Anna holds a ruler vertically so the height of the tree is exactly covered by the ruler.

She is 20 metres from the tree.
The ruler is 30cm long.
The horizontal distance from her eyes to the ruler is 60 cm.

Calculate an estimate of the height of the tree.

.......................... m

Give two reasons why this method may not be suitable to estimate the height of a very tall building.

ABCD is a parallelogram.

Prove that triangle ABD is congruent to triangle CDB.

$AOC$ and $BOD$ are diameters of a circle, centre $O$.

Prove that triangle $ABD$ and triangle $DCA$ are congruent.

$A,B,C$ and $D$ are four points on the circumference of a circle.

$AEC$ and $BED$ are straight lines.
Prove that triangle $ABE$ and triangle $DCE$ are similar.
You must give reasons for each stage of your working.

Steve has a photo and a rectangular piece of card.

The photo is 16 cm by 10 cm.
The card is 30 cm by 15 cm.

Steve cuts the card along the dotted line shown in the diagram below.

Steve throws away the piece of card that is 15 cm by $x$ cm.
The piece of card he has left is mathematically similar to the photo.

Work out the value of $x$.

Rectangle $ABCD$ is mathematically similar to rectangle $DAEF$.

$\begin{array}{l}AB=10\mathrm{cm}.\\ AD=4\mathrm{cm}.\end{array}$

Work out the area of rectangle $DAEF$.

PQR and PTS are straight lines.
Angle PTQ = Angle PSR = 90^o.
QT = 4 cm
RS = 12 cm
TS = 10 cm

Work out the area of the trapezium QRST.

Work out the length of PT.

The diagram shows a triangle $ABC$.

$LMNB$ is a parallelogram where
    $L$ is the midpoint of $AB$,
   $M$  is the midpoint of $AC$,
and   $N$ is the midpoint of $BC$.

Prove that triangle $ALM$ and triangle $MNC$ are congruent.
You must give reasons for each stage of your proof.

The diagram shows a triangle and a trapezium.

Prove that $a=b$.

In the diagram, AB and DC are parallel lines of equal length.

Prove that angle DAB = angle BCD.

ABCD is a quadrilateral.
AD = AB and CD = CB.

Prove that angle ADC is equal to angle ABC.

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

Prove that angle $ACB$ is 90°
You must not use any circle theorems in your proof.

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

Prove that angle $BOC$ is twice the size of angle $BAC$.

$PQRST$ is a regular pentagon.
$R,U$ and $T$ are points on a circle, centre $O$.
$QR$ and $PT$ are tangents to the circle.
$RSU$ is a straight line.

Prove that $ST=UT$.

The two triangles in the diagram are similar.

There are two possible values of $x$.

Work out each of these values.
State any assumptions you make in your working.

The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm.

The strut is one third of the way up the ladder.

The symmetrical cross section of the ladder shows two similar triangles.

Work out the vertical height, $h$ cm, of the ladder.

...........................cm

A and B are points on the circumference of a circle, centre O.
CA and CB are tangents to the circle.

Prove that triangle OAC is congruent to triangle OBC.

$C$ lies on a circle with diameter $AD$.
$B$ lies on $AC$ and $E$ lies on $AD$ such that $BE$ is parallel to $CD$.
$AB$ = 21 cm, $CD$ = 18 cm and $BE$ = 13.5 cm.

Work out the radius of the circle.

............................................ cm

 The diagonals of the cyclic quadrilateral ABCD intersect at X.  

Explain why triangle ADX is similar to triangle BCX.
Give a reason for each statement you make.

AD  = 10 cm, BC  = 8 cm, BX  = 5 cm, CX  = 7 cm.  

Calculate DX.

DX = ........................................... cm

A, B, C and D are points on the circle, centre O.
M is the midpoint of AB and N is the midpoint of CD.
OM = ON  

Explain, giving reasons, why triangle OAB is congruent to triangle OCD.

A solid metal cone has radius 10 cm and height 36 cm.

Calculate the volume of this cone.
[The volume, $V$, of a cone with radius $r$ and height $h$ is  $V=\frac{1}{3}\mathrm{\pi }{r}^{2}h$.]  

......................................... cm³

The cone is cut, parallel to its base, to give a smaller cone.

The volume of the smaller cone is half the volume of the original cone.

The smaller cone is melted down to make two different spheres.

The ratio of the radii of these two spheres is 1 : 2.

Calculate the radius of the smaller sphere.

[The volume, $V$, of a sphere with radius $r$ is  $V=\frac{4}{3}\mathrm{\pi }{r}^3$.]

.......................................... cm