Angles in Polygons & Parallel Lines (AQA GCSE Maths: Higher): Exam Questions

$ABC$ and $DEF$ are parallel lines.
$BEG$ is a straight line.
Angle $GEF=47°$.

Work out the size of the angle marked $x$.
Give reasons for your answer.

$ABC,PQR$ and $AQD$ are straight lines.
$ABC$ is parallel to $PQR$.

Angle $BAQ=35°$
Angle $BQA=90°$

Work out the size of the angle marked $x$.
Give reasons for each stage of your working.

$PRS$ and $TWY$ are parallel straight lines.
$QRWZ$ is a straight line.

Work out the value of $x$.
Give reasons for your answer.

$ABC,DEF$ and $PQRS$ are parallel lines.
$BEQ$ is a straight line.

Angle $ABE=60°$.
Angle $QER=80°$.

Work out the size of the angle marked $x$.
Give reasons for each stage of your working.

The diagram shows a regular octagon and a regular hexagon.

Find the size of the angle marked $x$.
You must show all your working.

$ABCDE$ and $PQRST$ are regular pentagons.

$SR$ is parallel to $DC$.
$AP=BQ=CR=DS=ET$

Work out the size of angle $SRC$.
You must show all your working.

$RS$ and $ST$ are 2 sides of a regular 12-sided polygon.
$RT$ is a diagonal of the polygon.

Work out the size of angle $STR$.
You must show your working.

$CDEF$ is a straight line.
$AB$ is parallel to $CF$.
$DE=AE$.

Work out the size of the angle marked $x$.
You must give reasons for your answer.

$ABCDE$ is a pentagon.

Angle $BCD=2\times$ angle $ABC$.

Work out the size of angle $BCD$.
You must show all your working.

What is the size of an exterior angle of a regular decagon?

Circle your answer.

The exterior angle of a regular polygon is 45°

Circle the name of the regular polygon.

Part of a regular polygon with 15 sides is shown.

Work out the size of an interior angle.

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

The sum of the angles in any quadrilateral is 360°

For example, in a rectangle 4 × 90° = 360°

Zak writes,

   5 × 90° = 450° so the sum of the angles in any pentagon must be 450°

Is he correct?

Tick a box.

      $\square$ Yes         $\square$ No

Show working to support your answer.

Here is a 10-sided polygon.

Work out the value of $x.$

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

Each interior angle of a regular polygon is 162°

Work out the number of sides the polygon has.

The diagram shows a triangle.

Work out the value of $x$.

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

The diagram shows a parallelogram $ABCD$ and an isosceles triangle $DEF$ in which $DE=DF$

$CDF$ and $ADE$ are straight lines.
Angle $BCD=58°$

Work out the size of angle $DEF.$
Give a reason for each stage of your working.

Work out the size of the exterior angle of a regular 12-sided polygon.

Use your answer to part (a) to write down the size of the interior angle of a regular 12-sided polygon.

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 an isosceles triangle, the size of each of the base angles is double the size of the third angle.

Work out the size of one base angle.

$AEC$ and $BED$ are straight lines.

Work out the size of $\angle AEB$.

Lines A and B are parallel.

Work out the size of both angles.

$ABCD$ is a parallelogram.

Angle $ADB=38°$.
Angle $BEC=41°$.
Angle $DAB=120°$.

Calculate the size of angle $x$.
You must give reasons for your answer.

$ABC$ is parallel to $EFGH$.

$GB=GF$
Angle $ABF=65°$

Work out the size of the angle marked $x$.
Give reasons for your answer.

$ABCD$ and $EFG$ are parallel lines.
$BC=CF$ Angle $BFE=70°$

Work out the size of the angle marked $x$.
Give reasons for each stage of your working.

$ABC$ and $DE$ are parallel lines.
$AEG$ and $BEF$ are straight lines.

Angle $AED=54°$
Angle $FEG=70°$

Work out the size of the angle marked $x$.
Give a reason for each stage of your working.

$ABC$ and $EDC$ are straight lines.
$AE$ and $BD$ are parallel.
Angle $ABD=125°$ Angle $BCD=30°$

Work out the size of the angle marked $x$.
Give reasons for your answer.

$ABCD$ is a parallelogram.
$EDC$ is a straight line.
$F$ is the point on $AD$ so that $BFE$ is a straight line.

Angle $EFD=35°$
Angle $DCB=75°$

Show that angle $ABF=70°$
Give a reason for each stage of your working.

$AE,DBG$ and $CF$ are parallel.
$DA=DB=DC$.
Angle $EAB=$ angle $BCF=38°$

Work out the size of the angle marked $x$.
You must show your working.

The diagram shows a pattern using four identical rhombuses.

Work out the size of the angle marked $a$.
You must show your working.

$ABCDEFGH$ is a regular octagon.
$BCKFGJ$ is a hexagon.

$JK$ is a line of symmetry of the hexagon.
Angle $BJG$ $=$ angle $CKF=140°$

Work out the size of angle $KFE$.
You must show all your working.

The diagram shows part of a pattern made from tiles.

The pattern is made from two types of tiles, tile A and tile B.

Both tile A and tile B are regular polygons.

Work out the number of sides tile A has.

$ABCDE$ is a regular pentagon.
$ACFG$ is a square.

Work out the size of angle $DCF$.
You must show all your working.

$ABCDEFGH$ is a regular octagon.
$KLQFP$ and $MNREQ$ are two identical regular pentagons.

Work out the size of the angle marked $x$.
You must show all your working.

In the diagram, $AB$, $BC$ and $CD$ are three sides of a regular polygon P.

Show that polygon P is a hexagon.
You must show your working.

Here is a parallelogram.

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

The diagram shows a square and 4 regular pentagons.

Work out the size of the angle marked $x$.

$AGC$ and $DEF$ are parallel lines.
$ADB$ and $GE$ are parallel lines. $BEC$ is a straight line.

Angle $DBE=95°$
Angle $CGE=55°$

Work out the size of the angle marked $x$.
Give reasons for each stage of your working.

$ABCDE$ is a regular pentagon.
$BCF$ and $EDF$ are straight lines.

Work out the size of angle $CFD$.
You must show how you got your answer.

Here is a quadrilateral.

$a=90°$ and $a:b=5:3$

$x:y=1:3$

Show that $b=x$

An interior angle of an isosceles triangle is p° and an exterior angle is q °.

It is given that q = 5p .
Write the ratio p : q in its simplest form.

Work out the two different possible sets of angles for the isosceles triangle.

Triangle 1: ............°, .............°, .............°
Triangle 2: ............°, .............°, .............°

The diagram shows a regular pentagon.
$AB$ and $CD$ are two of the lines of symmetry of the pentagon.

Work out the size of the angle marked $x$.
You must show all your working.

Here is a pentagon $ABCDE$.

$AB=BC=BD$
$ABDE$ is a kite.

Angle $AED=40°$
Angle $EDB=130°$
Angle $BDC=72°$

Work out the size of angle $ACB$.

$ABCDEFGHI$ is a regular 9-sided polygon.

The vertices $B$ and $E$ are joined with a straight line.

Work out the size of angle $BEF$.
You must show how you get your answer.

The diagram shows a hexagon.
The hexagon has one line of symmetry.

$FA=BC$ $EF=CD$
Angle $ABC=117°$

Angle $BCD=2\times$ angle $CDE$.

Work out the size of angle $AFE$.
You must show all your working.

$PQRST$ 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 $ST=UT$.

$AB,BC,CD$ and $DE$are four of the sides of a regular decagon.

Work out the size of angle $w$.

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

$AB,CD$ and $EF$ are straight lines.

Ava assumes that $AB$and $CD$ are parallel.

What answer should she get for the size of angle $y$?

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

In fact,

$AB$ and $CD$ are not parallel angle$w$ is 60°

What effect does this have on the size of angle $y$?
Tick a box.

Show working to support your answer.

$AB,CD$ and $YZ$ are straight lines.
All angles are in degrees.

Show that $AB$ is parallel to $CD$.

The diagram shows a regular 10-sided polygon, $ABCDEFGHIJ$

Show that $x=y$.

The diagram shows a regular octagon $ABCDEFGH$ and a regular pentagon $ABIJK$

Work out the size of the angle$x$.

Imran joins two tiles together as shown below.
One tile is a regular hexagon and the other tile is a regular pentagon.

Show that angle $\mathrm{a}$ is 132°.

Imran thinks that another tile in the shape of a regular polygon will fit exactly into angle $a$.

Is Imran correct?
Show your reasoning.

Angle BAE is part of a regular 18-sided polygon.
Angle CAD is part of a regular 10-sided polygon.
The dashed line through A is a line of symmetry of both polygons.

Work out angle BAC.