AQA 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. The angle $\angle DBC = 32^{\circ}$∠DBC=32∘, and the angle $\angle ADB = 47^{\circ}$∠ADB=47∘.

(a) Write down, with a reason, the size of $\angle BCD$∠BCD. (2 marks)

(b) Work out the size of $\angle BDC$∠BDC, giving a reason. (2 marks)

(c) Work out the size of $\angle ABD$∠ABD. (1 mark)

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

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

(b) Write down the sum of the interior angles of this polygon. (1 mark)

Two straight lines $PQ$PQ and $RS$RS are parallel. A straight line crosses $PQ$PQ at the point $X$X and crosses $RS$RS at the point $Y$Y. At $X$X, on the upper side, the angle between the crossing line and $XQ$XQ (measured towards $Q$Q) is $3x + 10$3x+10 degrees. At $Y$Y, on the lower side, the co-interior (allied) angle between the crossing line and $YS$YS is $2x + 30$2x+30 degrees.

Work out the value of $x$x, giving reasons for each step of your working. (4 marks)

Triangle $ABC$ABC and triangle $PQR$PQR are mathematically similar. In triangle $ABC$ABC, $AB = 6$AB=6 cm and $BC = 9$BC=9 cm. In triangle $PQR$PQR, the side $PQ$PQ corresponds to $AB$AB and $PQ = 15$PQ=15 cm. The side $QR$QR corresponds to $BC$BC.

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

(b) In triangle $ABC$ABC, the angle $\angle ABC = 52^{\circ}$∠ABC=52∘. Write down the size of the corresponding angle $\angle PQR$∠PQR. (1 mark)

In triangle $ABC$ABC, the side $AB$AB is extended beyond $B$B to a point $D$D. The interior angle $\angle BAC = 64^{\circ}$∠BAC=64∘ and the interior angle $\angle ACB = 51^{\circ}$∠ACB=51∘. Work out the size of the exterior angle $\angle CBD$∠CBD, giving a reason. (2 marks)

Work out the size of one exterior angle of a regular hexagon. (1 mark)