BECE Mathematics 2019 Paper

ConceptsA triangle which has two of its sides equal is called an isosceles triangleProperties of Isosceles Triangle1. The two equal sides of an isosceles triangle are called the legs and the angle between them is called the vertex angle or apex angle.2. The side opposite the vertex angle is called the base and base angles are equal.3. The perpendicular from the vertex angle bisects the base and the vertex angle.Both triangle QPS and QSR are isosceles triangles (have two sides equal and hence 2 angles equal) For triangle QSR$\angle SQR = \angle QRS = 35^{\circ}$But angles in a triangle add up to $180^{\circ}$$\; \Rightarrow \angle SQR + \angle QRS + \angle RSQ = 180$$\; \Rightarrow 35 + 35 + \angle RSQ = 180$$\; 70 + \angle RSQ = 180$$\; \angle RSQ = 180 - 70$$\; \angle RSQ = 110^{\circ}$For triangle QPS$\angle RSQ + \angle QSP = 180$ (Angles on a straight line add up to $180^{\circ}$$\; \; \text{ But } \; \angle RSQ = 110^{\circ}$$\; \Rightarrow 110 + \angle QSP = 180$$\; \angle QSP = 180 - 110$$\; \angle QSP = 70$Angles in a triangle add up to $180^{\circ}$$\; \Rightarrow \angle SQP + \angle QSP + \angle QPS = 180$$\; \angle SQP = \angle QSP = 70$$\; \Rightarrow 70 + 70 + \angle QPS = 180$$\; \Rightarrow 140 + \angle QPS = 180$$\; \Rightarrow \angle QPS = 180 - 140$$\; \angle QPS = 40^{\circ}$