OCR GCSE Maths: Angles, Polygons and Geometric Reasoning

In the diagram, $JK$JK is parallel to $LM$LM. A straight line crosses $JK$JK at the point $A$A and crosses $LM$LM at the point $B$B, so that $B$B lies below $A$A. On line $JK$JK the point $K$K is to the right of $A$A, and on line $LM$LM the points $L$L and $M$M lie with $L$L to the left of $B$B and $M$M to the right of $B$B. The acute angle between the crossing line and the ray $AK$AK, measured on the side below $JK$JK, is $68^{\circ}$68∘.

(a) Find the size of $\angle ABL$∠ABL, giving a reason for your answer. (2 marks)

(b) Find the size of $\angle ABM$∠ABM, giving a reason for your answer. (2 marks)

(c) Write down the size of the angle vertically opposite to the $68^{\circ}$68∘ angle at $A$A. (1 mark)

Each interior angle of a regular polygon is $162^{\circ}$162∘.

(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)

$A$A, $B$B, $C$C and $D$D are four points on the circumference of a circle, in that order, so that $ABCD$ABCD is a cyclic quadrilateral. The angle $\angle ABC = 84^{\circ}$∠ABC=84∘ and the angle $\angle BCD = 97^{\circ}$∠BCD=97∘.

(a) Find the size of $\angle ADC$∠ADC, giving a reason for your answer. (2 marks)

(b) Find the size of $\angle DAB$∠DAB, giving a reason for your answer. (2 marks)

In triangle $ABC$ABC, the sides $AB$AB and $AC$AC are equal, so the triangle is isosceles. The angle at the apex, $\angle BAC = 48^{\circ}$∠BAC=48∘.

Work out the size of the base angle $\angle ABC$∠ABC, giving a reason for your answer. (3 marks)

Three straight lines $OA$OA, $OB$OB and $OC$OC all meet at the point $O$O, dividing the full turn around $O$O into three angles. Going around the point, $\angle AOB = 135^{\circ}$∠AOB=135∘ and $\angle BOC = 102^{\circ}$∠BOC=102∘.

Work out the size of the third angle $\angle COA$∠COA, giving a reason for your answer. (2 marks)

Work out the size of one exterior angle of a regular decagon (a regular polygon with $10$10 sides). (1 mark)