Geometry & Measures
Grade 7-9Circle Theorems
Circle theorems are a set of angle facts specific to circles, and Higher tier questions expect you to name the correct theorem as a reason alongside every calculation. This lesson covers the angle in a semicircle, the angle at the centre versus the circumference, angles in the same segment, cyclic quadrilaterals, tangent properties, and the alternate segment theorem.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Angle in a semicircle and angle at the centre
The angle in a semicircle is always 90 degrees, whenever the triangle is formed using a diameter as one side and a third point anywhere on the circle. The angle at the centre of a circle is always twice the angle at the circumference, when both angles are subtended by the same arc.
If AB is a diameter and angle ABC is 35 degrees, then angle ACB is 90 degrees, so angle BAC is 180 minus 90 minus 35, which is 55 degrees. If angle ACB at the circumference is 70 degrees, the angle AOB at the centre subtended by the same arc AB is double that, 140 degrees.
Angles in the same segment and cyclic quadrilaterals
Angles subtended by the same arc, from the same side of that arc, are always equal to each other, a fact known as angles in the same segment. In a cyclic quadrilateral (one with all four vertices on the circle), opposite angles always sum to 180 degrees.
If angle ADB is 70 degrees, then angle ACB, in the same segment, is also 70 degrees. In a cyclic quadrilateral with angle DAB equal to 95 degrees, angle BCD is 180 minus 95, which is 85 degrees.
Tangent properties
A tangent meets a radius at exactly 90 degrees at the point of contact. Two tangents drawn from the same external point to a circle are always equal in length, and the line from that external point to the centre bisects the angle between the two tangents.
If TA is a tangent to a circle at A, with O the centre, then angle OAT is 90 degrees. If TA and TB are both tangents from the same external point T, then TA and TB are equal in length.
The alternate segment theorem
The angle between a tangent and a chord drawn from the point of contact is always equal to the angle in the alternate segment, meaning the angle subtended by that same chord from the other side of the circle.
If the angle between tangent TA and chord AB is 60 degrees, then angle ACB, where C is a point on the major arc on the other side of AB, is also 60 degrees, by the alternate segment theorem.
Worked Examples
Three exam-style questions, fully solved.
AB is a diameter of the circle with centre O. Angle ABC = 35 degrees. Find angle BAC, giving reasons for your answer.
Easy- 1.State the angle in a semicircle: angle ACB = 90°
- 2.Use the angle sum of a triangle: 180 - 90 - 35
Answer: Angle BAC = 55°
ABCD is a cyclic quadrilateral. Angle DAB = (3x + 15) degrees and angle BCD = (2x + 25) degrees. Find x, then find the size of each angle.
Medium- 1.State that opposite angles in a cyclic quadrilateral sum to 180°: (3x + 15) + (2x + 25) = 180
- 2.Simplify and solve: 5x + 40 = 180, so x = 28
- 3.Substitute back to find each angle
Answer: Angle DAB = 99°, angle BCD = 81°
TA is a tangent to the circle at A. O is the centre, and OA and OB are radii, with angle AOB = 130 degrees. Find angle TAB, giving full reasons for your answer.
Hard- 1.State that OA = OB (radii of the same circle), so triangle OAB is isosceles, giving angle OAB = angle OBA = (180 - 130) ÷ 2 = 25°
- 2.State that angle OAT = 90° (a tangent is perpendicular to the radius at the point of contact)
- 3.Subtract to find angle TAB: 90 - 25
Answer: Angle TAB = 65°
Avoid These
The most common mistakes students make.
Not stating the name of the circle theorem used as a reason, which usually costs a mark even when the numerical answer is correct.
Confusing the angle at the centre (twice the angle at the circumference) with angles in the same segment (always equal to each other).
Forgetting that the angle in a semicircle theorem only applies when one side of the triangle is a diameter, not just any chord.
Using the alternate segment theorem with the wrong pair of angles, rather than matching the tangent-chord angle to the angle in the segment on the far side of that chord.
Forgetting that a tangent meets a radius at exactly 90 degrees, and instead treating the angle between them as unknown.
FAQ
Questions parents and students ask.
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