The answer
(a) congruent by SAS
(b)(i) \(n = 18\)
(ii) angle \(BCA = 10^{\circ}\)
(iii) angle \(DEA = 30^{\circ}\)
O-Level E-Math 2016 Paper 2 Question 3 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 3 of the O-Level E-Math 2016 Paper 2. It tests sas congruence reasoning, in the Congruence & similarity / angles of polygons area. It is worth 7 marks: (a) 2, (b) 2 + 1 + 2. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) In a regular polygon all sides are equal, so \(AB = CD\) and \(BC = DE\); all interior angles are equal, so the included angles \(\angle ABC = \angle CDE\). With two pairs of equal sides and the equal included angle, \(\triangle ABC \equiv \triangle CDE\) by SAS.
(b)(i) Each interior angle of a regular \(n\)-gon is \(\dfrac{(n - 2) \times 180^{\circ}}{n}\). Setting this equal to \(160^{\circ}\): \(180n - 360 = 160n \Rightarrow 20n = 360 \Rightarrow n = 18\).
(ii) Triangle \(ABC\) is isosceles (\(AB = BC\)) with apex angle \(\angle ABC = 160^{\circ}\), so the base angles are equal: \(\angle BCA = \angle BAC = \dfrac{180^{\circ} - 160^{\circ}}{2} = 10^{\circ}\).
(iii) By the congruence in (a), \(\angle DEC = \angle BAC = 10^{\circ}\) and \(AC = CE\). The interior angle at \(C\) is \(\angle BCD = 160^{\circ}\), and \(\angle BCA = \angle DCE = 10^{\circ}\), so \(\angle ACE = 160^{\circ} - 10^{\circ} - 10^{\circ} = 140^{\circ}\). Triangle \(ACE\) is isosceles (\(AC = CE\)), so its base angles are \(\angle CEA = \angle CAE = \dfrac{180^{\circ} - 140^{\circ}}{2} = 20^{\circ}\). Hence \(\angle DEA = \angle DEC + \angle CEA = 10^{\circ} + 20^{\circ} = 30^{\circ}\).
Answer: (a) congruent by SAS
(b)(i) \(n = 18\)
(ii) angle \(BCA = 10^{\circ}\)
(iii) angle \(DEA = 30^{\circ}\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a sas congruence reasoning question from Congruence & similarity / angles of polygons, worth 7 marks: (a) 2, (b) 2 + 1 + 2.
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