Edexcel GCSE Maths: Angles, Polygons and Geometric Reasoning

$A$A, $B$B, $C$C and $D$D are four points on the circumference of a circle with centre $O$O. $BD$BD is a diameter of the circle.

Angle $ABD = 31^{\circ}$ABD=31∘ and angle $CDB = 24^{\circ}$CDB=24∘.

(a) Write down the size of angle $BAD$BAD, giving a reason for your answer. (2 marks)

(b) Work out the size of angle $BCD$BCD, and hence work out the size of angle $ADC$ADC. You must give reasons for your working. (3 marks)

The interior angle of a regular polygon is $156^{\circ}$156∘.

(a) Work out the number of sides of the polygon. (3 marks)

(b) Work out the sum of the interior angles of this polygon. (1 mark)

Two triangles, $PQR$PQR and $XYZ$XYZ, are similar. In triangle $PQR$PQR, $PQ = 6$PQ=6 cm and $QR = 9$QR=9 cm. In triangle $XYZ$XYZ, the side $XY$XY corresponds to $PQ$PQ and $XY = 15$XY=15 cm. The side $YZ$YZ corresponds to $QR$QR.

(a) Work out the length of $YZ$YZ. (2 marks)

(b) The area of triangle $PQR$PQR is $24\ \text{cm}^2$24 cm2. Work out the area of triangle $XYZ$XYZ. (2 marks)

$ABC$ABC is a straight line. $BD$BD is a line such that triangle $BCD$BCD has angle $BCD = 47^{\circ}$BCD=47∘ and angle $CDB = 68^{\circ}$CDB=68∘.

Work out the size of angle $ABD$ABD. Give reasons for each step of your working. (3 marks)

A line $PQ$PQ is parallel to a line $RS$RS. A straight line crosses $PQ$PQ at the point $A$A and crosses $RS$RS at the point $B$B. The angle between the crossing line and $PQ$PQ at $A$A, on the right of the line, is $72^{\circ}$72∘.

Write down the size of the co-interior (allied) angle at $B$B on the same side, and give a reason for your answer. (2 marks)

Write down the size of each exterior angle of a regular hexagon. (1 mark)