Circle Theorems — Angles, Tangents & Chords

Circle theorems are rules for finding angles in diagrams built from chords, tangents, radii, and cyclic quadrilaterals. Match the right theorem to the right condition and a tangled circle diagram collapses into one or two simple angle equations.

The Theorems And The Conditions That Trigger Them

Every circle theorem has a setup it requires. The angle relationships below only hold when that setup is genuinely present in the diagram.

Angle at the center is twice the angle at the circumference. If a central angle and an inscribed angle stand on the same arc, the central angle is twice the inscribed angle. So if the central angle is $\theta$, the inscribed angle on the same arc is $\dfrac{\theta}{2}$. Condition: both angles must stand on the same arc.

Angles in the same segment are equal. If two angles at the circumference stand on the same chord and lie in the same segment, they are equal. Condition: same chord, same segment.

Angle in a semicircle is $90^\circ$. If a triangle has a diameter as one side, the angle at the point on the circumference is a right angle. This is the center-angle theorem applied to a diameter: the central angle is $180^\circ$, and half of that is $90^\circ$.

Opposite angles in a cyclic quadrilateral add to $180^\circ$. If $A$ and $C$ are opposite angles of a quadrilateral whose four vertices all lie on the circle, then $A + C = 180^\circ$, and likewise for the other pair. Condition: all four vertices on the circle.

A radius and a tangent meet at $90^\circ$. A tangent touches the circle at exactly one point; the radius to that point is perpendicular to the tangent. So if $OA$ is a radius and the line at $A$ is tangent, the angle between them is $90^\circ$.

Tangent-chord theorem. If a tangent touches the circle at one end of a chord, the angle between the tangent and the chord equals the inscribed angle on that chord in the opposite segment. This turns a line-angle outside the circle into a familiar angle inside it.

Why The Center-Angle Rule Drives The Others

Several of these theorems are really one idea seen from different angles. The angle-at-the-center rule says an inscribed angle is half the central angle on the same arc. The semicircle right angle is just that rule with a central angle of $180^\circ$. "Angles in the same segment are equal" follows too: if every inscribed angle on a given arc equals half the same fixed central angle, they must all equal each other. Recognizing this shared root means you can rebuild the special cases instead of memorizing them in isolation.

Worked Example: Two Angles From One Central Angle

Let $O$ be the center of a circle, with chord $AB$ subtending a central angle $\angle AOB = 110^\circ$. Point $C$ lies on the circumference on the opposite arc from chord $AB$, and a tangent touches the circle at $A$. Find $\angle ACB$ and the angle between the tangent at $A$ and chord $AB$.

Start with the angle-at-the-center theorem. The inscribed angle on chord $AB$ is half the central angle:

Now use the tangent-chord theorem. The angle between the tangent at $A$ and chord $AB$ equals the angle in the opposite segment standing on chord $AB$, which is $\angle ACB$:

The key move is not the arithmetic. It is noticing that both unknown angles come from the same chord $AB$.

Practice And A Selection Checklist

To pick the right theorem on any diagram, run this checklist: Is there a marked central angle with a matching angle on the circle? Is one side a diameter? Is there a tangent touching at one point? Are all four vertices of a quadrilateral on the circle? Do two angles stand on the same chord? The first match usually names the theorem you need.

Now try your own: draw a circle with center $O$ and chord $PQ$ with $\angle POQ = 84^\circ$. Put a point $R$ on the circumference on the opposite arc from $PQ$, then draw a tangent at $P$. Find $\angle PRQ$, then the angle between the tangent at $P$ and chord $PQ$. (Using the same two theorems as above, both come out to $42^\circ$.)

Calculation Pitfalls To Watch

The most common slip is using the "twice" rule for angles that do not stand on the same arc; the central and inscribed angles must share an arc. Another is calling a line a tangent just because it looks like it touches the circle, when the tangent condition should actually be stated or established. Students also confuse "angles in the same segment are equal" with "opposite angles in a cyclic quadrilateral sum to $180^\circ$": one gives equality, the other a supplementary pair. A final trap is assuming any four-sided shape near a circle is cyclic, when all four vertices must truly lie on the circle.

Circle theorems show up in school geometry, angle-chasing proofs, and coordinate setups where a diagram carries more information than it first appears to, especially when you need to connect an outside tangent angle to an inside angle on the circle.