UPSC Prelims 2001 Q148: Science & Technology – Quantitative Aptitude
In the above figure, ABCD is a cyclic quadrilateral, AB = BC and angle BAC = 70°, then angle
ADC is
- (a) 40°
- (b) 80°
- (c) 110°
- (d) 140°
Answer: (d) 140°
Explanation
Angle ADC in a cyclic quadrilateral with AB = BC and angle BAC = 70°
| Step | Working |
|---|---|
| Given | ABCD is a cyclic quadrilateral, AB = BC, ∠BAC = 70° |
| Isosceles triangle ABC | AB = BC → ∠BCA = ∠BAC = 70° |
| Angle ABC | 180° − 70° − 70° = 40° |
| Angle ADC | Opposite angles of a cyclic quadrilateral add to 180° → 180° − 40° = 140° |
Given: ABCD is a cyclic quadrilateral with AB = BC and angle BAC = 70°. Triangle ABC is isosceles, so angle BCA is also 70° and angle ABC is 40°. Opposite angles of a cyclic quadrilateral are supplementary, so angle ADC = 140°.
Answer: (d) 140°.
Why each option is right or wrong
- (a) Incorrect. 40° is angle ABC, not angle ADC.
- (b) Incorrect. 80° does not follow from the given angles.
- (c) Incorrect. 110° would need angle ABC to be 70°.
- (d) CORRECT. 180° − 40° = 140° for angle ADC.
Key point
Cyclic quadrilateral: opposite angles sum to 180°; an exterior angle equals the interior opposite angle.
Remember
∠ABC = 40°, so ∠ADC = 180° − 40° = 140°.
Common mistakes
- Stopping at angle ABC (40°) without finishing.
- Forgetting that AB = BC makes the base angles equal.
- Using 180° − 70° = 110° by mistake.
Exam tip
Find the angle opposite the one asked, then subtract from 180°.
Sources
- NCERT Class IX — Mathematics, chapter 'Circles' (cyclic quadrilaterals)
- NCERT Class IX — Mathematics, chapter 'Triangles' (isosceles triangles)