Circle angles, cyclic quadrilaterals and tangents — 0580
Match the angle theorem to the same chord and the specified arc.
Extended · Syllabus E4.7
Notes and key methods
The angle at the centre subtended by a chord is twice the angle at the circumference on the same arc configuration. Angles subtended by the same chord in the same segment are equal. Opposite interior angles of a cyclic quadrilateral sum to 180°.
The angle between a tangent and a chord equals the angle in the alternate segment. Keep track of minor and reflex central angles: the appropriate doubled angle depends on which arc contains the circumference point. A radius to a tangent point is perpendicular to the tangent.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Circle angles, cyclic quadrilaterals and tangents
AB is a chord, O is the centre, and C and D are points on the major arc AB. The angle ACB is 56°. A tangent touches at A. Separately, P,Q,R,S in this order form a cyclic quadrilateral with interior angle \(\displaystyle P=76\)°.
Find the minor angle AOB in degrees.
Find angle OAB in degrees.
Find the acute angle between the tangent at A and chord AB, in degrees.
Find interior angle R of PQRS in degrees.
Show the reasoning
(a)
Use the centre-angle theorem with C on the major arc.
The same chord gives twice the circumference angle: 112°.
Answer: \(112\).
(b)
Use \(\displaystyle OA=OB\) and the triangle angle sum.
In isosceles triangle AOB, each base angle is \(\displaystyle \frac{180- 112}{2}= 34\)°.
Answer: \(34\).
(c)
Use the tangent-chord theorem with the appropriate acute angle.
The alternate-segment angle equals ACB=56°.
Answer: \(56\).
(d)
Identify P and R as opposite angles in the cyclic quadrilateral.
Opposite cyclic angles sum to 180°, giving 104°.
Answer: \(104\).
Example 2 · Circle angles, cyclic quadrilaterals and tangents
AB is a chord, O is the centre, and C and D are points on the major arc AB. The angle ACB is 57°. A tangent touches at A. Separately, P,Q,R,S in this order form a cyclic quadrilateral with interior angle \(\displaystyle P=77\)°.
Find the minor angle AOB in degrees.
Find angle OAB in degrees.
Find the acute angle between the tangent at A and chord AB, in degrees.
Find interior angle R of PQRS in degrees.
Show the reasoning
(a)
Use the centre-angle theorem with C on the major arc.
The same chord gives twice the circumference angle: 114°.
Answer: \(114\).
(b)
Use \(\displaystyle OA=OB\) and the triangle angle sum.
In isosceles triangle AOB, each base angle is \(\displaystyle \frac{180- 114}{2}= 33\)°.
Answer: \(33\).
(c)
Use the tangent-chord theorem with the appropriate acute angle.
The alternate-segment angle equals ACB=57°.
Answer: \(57\).
(d)
Identify P and R as opposite angles in the cyclic quadrilateral.
Opposite cyclic angles sum to 180°, giving 103°.
Answer: \(103\).
Example 3 · Circle angles, cyclic quadrilaterals and tangents
AB is a chord, O is the centre, and C and D are points on the major arc AB. The angle ACB is 58°. A tangent touches at A. Separately, P,Q,R,S in this order form a cyclic quadrilateral with interior angle \(\displaystyle P=78\)°.
Find the minor angle AOB in degrees.
Find angle OAB in degrees.
Find the acute angle between the tangent at A and chord AB, in degrees.
Find interior angle R of PQRS in degrees.
Show the reasoning
(a)
Use the centre-angle theorem with C on the major arc.
The same chord gives twice the circumference angle: 116°.
Answer: \(116\).
(b)
Use \(\displaystyle OA=OB\) and the triangle angle sum.
In isosceles triangle AOB, each base angle is \(\displaystyle \frac{180- 116}{2}= 32\)°.
Answer: \(32\).
(c)
Use the tangent-chord theorem with the appropriate acute angle.
The alternate-segment angle equals ACB=58°.
Answer: \(58\).
(d)
Identify P and R as opposite angles in the cyclic quadrilateral.
Opposite cyclic angles sum to 180°, giving 102°.
Answer: \(102\).
Common mistakes
Keep intermediate values exact until the final step.