WAEC Mathematics 2022 Theory — Question 4

Model answer

Since BCDE is a cyclic quadrilateral, opposite angles are supplementary: (2y+10)+(2x+40)=180 → 2y+2x+50=180 → x+y=65 ... (1) Also, given ∠BAD=5x−35°, and using the exterior angle/other relationships: x+y=65° ...(1) ∠BAD=5x−35°, and using the relevant angle properties in the diagram (angle at centre = 2 × angle at circumference): (5x−35)=2(2y+10) → 5x−35=4y+20 → 5x−4y=55 ...(2) Solving (1) and (2): from (1), y=65−x. Substituting into (2): 5x−4(65−x)=55 → 5x−260+4x=55 → 9x=315 → x=35, y=30. ∠ABC+(2x+40)+(5x−35)+40=360 (sum of angles in cyclic quadrilateral) ∠ABC+110+140+40=360 ∠ABC=360−290=70°.