Angles in Polygons & Parallel Lines (OCR 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.

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

[2]

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

[1]

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 ....................................................................................................

[2]

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.

[3]

Here is a 10-sided polygon.

Work out the value of $x.$

$x=$................................................. [4]

Each interior angle of a regular polygon is 162°

Work out the number of sides the polygon has.

[3]

The diagram shows a triangle.

Work out the value of $x$.

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

[4]

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.

[5]

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

Circle your answer.

[1]

The exterior angle of a regular polygon is 45°

Circle the name of the regular polygon.

[1]

Part of a regular polygon with 15 sides is shown.

Work out the size of an interior angle.

....................................................degrees [2]

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.

 [2]

Three regular polygons meet at a point.

One is a regular pentagon and another is a square.

Find the number of sides of the third polygon.

You must show your working.

$ABCD$ 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.

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.

[2]

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

Triangle 1: ............°, .............°, .............°
Triangle 2: ............°, .............°, .............° [4]

Here is a quadrilateral.

$a=90°$ and $a:b=5:3$

$x:y=1:3$

Show that $b=x$

[3]

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 $ST=UT$.

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°.

[3]

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

Is Imran correct?
Show your reasoning.

[3]

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.

[5]

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

Show that $x=y$.

[4]

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

Work out the size of the angle$x$.

[4]

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

Work out the size of angle $w$.

..............................degrees [3]

$AB,CD$ 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 [4]

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.

[3]

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

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

[4]