LECTURE 2 : CIRCLE THEOREMS
SAMUEL OPOKU BINEY (FIIFI)
GOLDEN SUNBEAM INTERNATIONAL SCHOOL
September 24, 2024
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL) September 24, 2024 1 / 26
Introduction to Circle Theorems
Overview
In this lecture, we will cover the six important circle theorems, providing
diagrams, examples, and solutions for each:
The Angle at the Center Theorem
The Angle in a Semicircle Theorem
Angles in the Same Segment Theorem
Opposite Angles of a Cyclic Quadrilateral
Tangent-Radius Theorem
Alternate Segment Theorem
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL) September 24, 2024 2 / 26
Theorem 1: Angle at the Center
Theorem 1: Angle at the Center
Statement: The angle subtended by an arc at the center of a circle is
twice the angle subtended by the same arc at any point on the
circumference.
θ 2θ
O
B
Example: In the figure above, if ∠ACB = 30◦ , find ∠AOB.
Solution: By the theorem,
∠AOB = 2 × ∠ACB = 2 × 30◦ = 60◦ .September 24, 2024
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MENSURATION IISCHOOL) 3 / 26
Theorem 2: Angle in a Semicircle
Theorem 2: Angle in a Semicircle
Statement: The angle subtended by a diameter at the circumference is
always a right angle (90◦ ).
B
90◦ A
Example: If AB is the diameter of the circle, what is ∠ACB?
Solution: Since AB is the diameter, ∠ACB = 90◦ as per the theorem.
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Theorem 3: Angles in the Same Segment
Theorem 3: Angles in the Same Segment
Statement: Angles subtended by the same arc or chord at the
circumference are equal.
A
θ
θ
D
C
Example: If ∠ACB = 40◦ , find ∠ADB.
Solution: Since ∠ACB and ∠ADB are subtended by the same arc AB,
∠ADB = 40◦ .
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MENSURATION IISCHOOL) September 24, 2024 5 / 26
Theorem 4: Opposite Angles of a Cyclic Quadrilateral
Theorem 4: Opposite Angles of a Cyclic Quadrilateral
Statement: The opposite angles of a quadrilateral inscribed in a circle
sum up to 180◦ .
α
A
D
C
β
Example: In the figure above, if α = 70◦ , find β.
Solution: Since α + β = 180◦ ,
β = 180◦ − 70◦ = 110◦ .
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MENSURATION IISCHOOL) September 24, 2024 6 / 26
Theorem 5: Tangent-Radius Theorem
Theorem 5: Tangent-Radius Theorem
Statement: A tangent to a circle is perpendicular to the radius at the
point of contact.
O
90◦
A B
Example: In the figure, if OA = 5 cm, what is the angle between OA and
the tangent at A?
Solution: The angle is 90◦ as the radius is perpendicularSeptember
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL)
to the24,tangent
2024
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Theorem 6: Alternate Segment Theorem
Theorem 6: Alternate Segment Theorem
Statement: The angle between a tangent and a chord through the point
of contact is equal to the angle subtended by the chord in the alternate
segment.
θ
C
D
A B
θ
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MENSURATION IISCHOOL) September 24, 2024 8 / 26
Theorem 6: Alternate Segment Theorem
Example: If ∠BCD = 40◦ , find the angle between the tangent at A and
the chord AB.
Solution: By the Alternate Segment Theorem,
∠BCD = ∠ADB = 40◦ .
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL) September 24, 2024 9 / 26
Theorem 6: Alternate Segment Theorem
Introduction
Circle Theorems describe properties of angles and lines in and around
circles. These theorems are fundamental to understanding geometry and
are widely tested in examinations.
In this lecture, we will explore:
Angle relationships in circles.
The behavior of tangents and chords.
Proofs and practical applications of circle theorems.
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MENSURATION IISCHOOL) September 24, 2024 10 / 26
Theorem 6: Alternate Segment Theorem
Theorem 1: Angle in a Semicircle
Statement: The angle subtended by a diameter at the circumference of a
circle is a right angle (90°).
C
90◦
B A
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MENSURATION IISCHOOL) September 24, 2024 11 / 26
Theorem 6: Alternate Segment Theorem
Example 1: Angle in a Semicircle (WASSCE)
Question: In the diagram below, AB is the diameter of the circle with
center O, and C lies on the circumference of the circle. If ∠ACB = 90◦ ,
find the value of ∠OCB.
C
90◦
B O A
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MENSURATION IISCHOOL) September 24, 2024 12 / 26
Theorem 6: Alternate Segment Theorem
Theorem 2: Angles at the Center and Circumference
Statement: The angle subtended by an arc at the center of a circle is
twice the angle subtended by the same arc at the circumference.
2θ
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MENSURATION IISCHOOL) September 24, 2024 13 / 26
Theorem 6: Alternate Segment Theorem
Example 2: Angles at the Center and Circumference
(WASSCE)
Question: In the diagram, ∠AOB = 60◦ . Find ∠ACB.
C
60◦
B A
◦ 60◦
Solution: Since ∠AOB = 60 , ∠ACB =
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL = 30◦ .
MENSURATION IISCHOOL) September 24, 2024 14 / 26
Theorem 6: Alternate Segment Theorem
Theorem 3: Tangent-Chord Angle
Statement: The angle between a tangent and a chord through the point
of contact is equal to the angle in the alternate segment.
Tangent
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MENSURATION IISCHOOL) September 24, 2024 15 / 26
Theorem 6: Alternate Segment Theorem
Example 3: Tangent-Chord Angle (WASSCE)
Question: Find ∠ABC if the angle between the tangent at point A and
the chord AC is 45◦ .
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MENSURATION IISCHOOL) September 24, 2024 16 / 26
Theorem 6: Alternate Segment Theorem
Question 1
Question: In the diagram, O is the center of the circle, and ∠ABC = 50◦ .
Calculate ∠AOC .
A
C
B
O
Solution:
∠AOC = 2 × ∠ABC = 2 × 50◦ = 100◦
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MENSURATION IISCHOOL) September 24, 2024 17 / 26
Theorem 6: Alternate Segment Theorem
Question 2
Question: In the diagram, ABCD is a cyclic quadrilateral. If
∠BCD = 80◦ and ∠DAB = 60◦ , calculate ∠ABC .
A
B
C D
Solution:
∠DAB + ∠BCD = 180◦ ⇒ 60◦ + 80◦ = 180◦
Hence, ∠ABC = 100◦ .
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MENSURATION IISCHOOL) September 24, 2024 18 / 26
Theorem 6: Alternate Segment Theorem
Question 3
Question: In the diagram, O is the center of the circle, and AB is a
tangent at A. If ∠AOB = 110◦ , find ∠OAB.
A
Tangent at A
B
O
Solution: The angle between a tangent and a radius is 90◦ . Hence:
∠AOB 110◦
∠OAB = 90◦ − = 90◦ − = 35◦
2 2
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MENSURATION IISCHOOL) September 24, 2024 19 / 26
Theorem 6: Alternate Segment Theorem
Question 4
Question: In a cyclic quadrilateral PQRS, if ∠PQR = 70◦ and
∠PSR = 40◦ , find ∠PRQ.
Solution: Using the property of opposite angles in a cyclic quadrilateral:
∠PQR +∠PSR = 180◦ ⇒ 70◦ +∠PRQ = 180◦ ⇒ ∠PRQ = 110◦
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MENSURATION IISCHOOL) September 24, 2024 20 / 26
Theorem 6: Alternate Segment Theorem
Question 5
Question: If O is the center of a circle and ∠ABC = 60◦ , find ∠AOC .
Solution: The angle subtended at the center is twice the angle subtended
at the circumference. Hence:
∠AOC = 2 × ∠ABC = 2 × 60◦ = 120◦
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL) September 24, 2024 21 / 26
Theorem 6: Alternate Segment Theorem
Question 6
Question: In the diagram, ABCD is a cyclic quadrilateral. If
∠ABC = 90◦ and ∠ADC = 110◦ , find ∠BCD.
Solution: Using the opposite angles of a cyclic quadrilateral property:
∠ABC + ∠ADC = 180◦ ⇒ 90◦ + 110◦ = 180◦
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MENSURATION IISCHOOL) September 24, 2024 22 / 26
Theorem 6: Alternate Segment Theorem
Question 7
Question: In the diagram, O is the center of the circle. If ∠AOB = 130◦ ,
find ∠ACB.
Solution: The angle subtended at the center is twice that at the
circumference. Hence:
∠AOB 130◦
∠ACB = = = 65◦
2 2
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MENSURATION IISCHOOL) September 24, 2024 23 / 26
Theorem 6: Alternate Segment Theorem
Question 8
Question: AB is a tangent to a circle at point A. If ∠ABC = 60◦ and O
is the center, find ∠OAC .
Solution: The angle between the tangent and the radius is 90◦ . Therefore:
∠OAC = 90◦ − ∠ABC = 90◦ − 60◦ = 30◦
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MENSURATION IISCHOOL) September 24, 2024 24 / 26
Theorem 6: Alternate Segment Theorem
Question 9
Question: In the diagram, ABCD is a cyclic quadrilateral. If
∠ABC = 80◦ and ∠BCD = 70◦ , find ∠DAB.
Solution: Using the sum of opposite angles in a cyclic quadrilateral:
∠ABC + ∠DAB = 180◦ ⇒ 80◦ + ∠DAB = 180◦
Thus, ∠DAB = 100◦ .
SAMUEL OPOKU BINEY (FIIFI) (GOLDEN SUNBEAM INTERNATIONAL
MENSURATION IISCHOOL) September 24, 2024 25 / 26
Theorem 6: Alternate Segment Theorem
Question 10
Question: In the diagram, O is the center of the circle, and
∠AOB = 120◦ . Find ∠ACB.
Solution: By the angle at the center theorem:
1 1
∠ACB = × ∠AOB = × 120◦ = 60◦
2 2
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MENSURATION IISCHOOL) September 24, 2024 26 / 26
