Angles in Polygons: Eduqas GCSE Higher June 2017 Paper 1 Q10

Higher June 2017 Paper 1 Q10

EduqasHigherCurrent spec5 marksAngles in Polygons Show That & Proof

10.

(a) Show that the interior angle of a regular octagon is \(135^\circ\). [2]

(b)

Regular octagon with one interior angle of 135 degrees marked; a kite has been cut from the bottom, with vertex at the centre of the octagon. The angle of the kite at the centre is y degrees and the angle of the shaded shape where a kite edge meets an octagon side is x degrees

Diagram not drawn to scale

The shaded shape is formed by cutting a kite from a regular octagon as shown in the diagram.

Prove that \(y = 2x - 135\).
You must give reasons to support your statements. [3]

Mark scheme (a)

Scheme	Marks	Comments
(exterior angle =) \(\dfrac{360}{8}\) or \(45 \times 8 = 360\) (interior angle sum =) \(6 \times 180\)	M1	Accept equivalent methods
(interior angle =) \(180 - \dfrac{360}{8}\ (= 135)\) or \(\dfrac{6 \times 180}{8}\ (= 135)\)	A1	Do not allow \(1080 \div 8 = 135\) only, there must be evidence to support 1080 Allow SC2 for (exterior angle is) \(180 - 135 = 45\) and (exterior angle is) \(360 \div 8 = 45\) or for exterior angle sum is 360 oe (exterior angle is) \(180 - 135 = 45\) \(45 \times 8 = 360\)

Mark scheme (b)

Scheme	Marks	Comments
\(y + 180 - x + 180 - x + 135 = 360\) or \(x = \dfrac{y}{2} + \dfrac{135}{2}\) or \(\dfrac{y}{2} + 180 - x + \dfrac{135}{2} = 180\)	M1	Accept in any correct form
Convincing working leading to \(y = 2x - 135\)	A1	Given answer obtained without any wrong working seen If M0 then award SC1 for verification that \(y = 2x - 135\) leads to eg \(2x - 135 + 2(180 - x) + 135 = 360\)
One correct reason e.g. Angles on a straight line (add up to \(180^\circ\)), Angle sum of a quadrilateral (\(=360^\circ\)), Symmetry of kite or equivalent	E1	Allow for one correct statement Stated appropriately NB SC1 E1 is possible
	(5)