Geometry · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. What is the angle subtended at the center by a chord if the angle subtended at the circle is 40 degrees?
A80 degrees
B40 degrees
C20 degrees
D140 degrees
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Answer: A) 80 degrees
The central angle is twice the inscribed angle: 2×40=80 degrees.
2. Angles in the same segment of a circle are:
AEqual
BSupplementary
CComplementary
DIn the ratio 2:1
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Answer: A) Equal
Angles in the same segment of a circle are equal, because they all subtend the same arc and each equals half the central angle of that arc.
3. Two equal chords of a circle intersect within the circle. The line joining the point of intersection to the centre makes __________ angles with the two chords.
AEqual
BComplementary
CRight
DSupplementary
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Answer: A) Equal
If two equal chords intersect within a circle, the line joining the point of intersection to the centre makes equal angles with the chords, since the chords are equidistant from the centre.
4. In a circle with centre O, the chord AB subtends an angle of 130 degrees at the centre. D is a point lying on the minor arc AB. Find angle ADB.
A115 degrees
B65 degrees
C130 degrees
D50 degrees
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Answer: A) 115 degrees
A point on the minor arc looks at the chord across the major arc, so the relevant angle at the centre is the reflex angle AOB = 360 - 130 = 230 degrees. Halving it gives angle ADB = 230 / 2 = 115 degrees. Taking half of 130 gives 65 degrees, which is the angle from a point on the MAJOR arc. As a check, 65 + 115 = 180, which is correct because those two points and A and B form a cyclic quadrilateral.
5. ABCD is a cyclic quadrilateral. Its side AB is produced beyond B to a point P, and angle CBP = 65 degrees. It is also given that angle BCD = 95 degrees. Find angle BAD.
A115 degrees
B95 degrees
C65 degrees
D85 degrees
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Answer: D) 85 degrees
Angle BAD and angle BCD are opposite angles of the cyclic quadrilateral, so they are supplementary: angle BAD = 180 - 95 = 85 degrees. The extra fact fits neatly: angle CBP is the exterior angle at B, so it equals the interior opposite angle ADC = 65 degrees, and then angle ABC = 180 - 65 = 115 degrees. All four angles check out: 85 + 115 + 95 + 65 = 360 degrees. Answering 115 degrees gives angle ABC rather than angle BAD.