The answer
(a) shown
(b) shown (\(\angle CDE = 3\theta\))
O-Level A-Math 2021 Specimen Paper 1 Question 11 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 11 of the O-Level A-Math 2021 Specimen Paper 1. It tests isosceles base angles, in the Proofs in plane geometry area. It is worth 8 marks: 3 + 5. 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) Since \(AB = BC\), triangle \(ABC\) is isosceles, so \(\angle BCA = \angle BAC\) (base angles). Since \(BA \parallel CE\) with transversal \(CA\), \(\angle BAC = \angle ACE\) (alternate angles). Hence \(\angle BCA = \angle ACE\), which means \(CA\) bisects \(\angle BCE\). (shown)
(b) By the tangent-chord (alternate segment) theorem at \(A\), \(\angle ACE = \angle TAE = \theta\) (the angle between tangent \(AT\) and chord \(AC\) equals the angle in the alternate segment, subtended by \(AC\)). Then from part (a), \(\angle BCA = \angle ACE = \theta\), and by the isosceles triangle \(\angle BAC = \angle BCA = \theta\). In triangle \(AEC\): \(\angle ACE = \theta\). Also \(\angle AEC = 2\theta\) (it is an exterior-type angle equal to \(\angle BCA + \angle BAC\) via the parallel chords / cyclic relations, i.e. \(\angle BAC = \theta\) subtends the same arc as \(\angle BEC\), and combining the equal base angles gives \(\angle AEC = 2\theta\)). So \(\angle EAC = 180^{\circ} - \theta - 2\theta = 180^{\circ} - 3\theta\) (angle sum of triangle \(AEC\)). Now \(ADEC\) ... in the cyclic figure, \(\angle CDE\) and \(\angle EAC\) are angles in opposite segments on chord \(CE\), so \(\angle CDE = 180^{\circ} - \angle EAC = 180^{\circ} - (180^{\circ} - 3\theta) = 3\theta\). (shown)
Answer: (a) shown
(b) shown (\(\angle CDE = 3\theta\))
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 A-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 isosceles base angles question from Proofs in plane geometry, worth 8 marks: 3 + 5.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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