Inscribed Angles and Circle Theorems

Now that you understand the parts of a circle — radius, diameter, chord, arc, central angle, and tangent — it is time to explore the angle relationships that form inside and on the circle. These theorems are among the most frequently tested concepts on the SAT math section and appear throughout geometry courses. The central idea is simple: where the vertex of an angle sits relative to the circle determines exactly how the angle and its intercepted arc are related.

Central Angle Review

A central angle has its vertex at the center of the circle, and its two sides are radii. The defining property is:

$\text{central angle} = \text{intercepted arc}$

If a central angle measures $80^\circ$, then the arc it cuts off (the intercepted arc) also measures $80^\circ$. This one-to-one relationship between central angles and arcs is the baseline against which every other circle angle theorem is compared.

The Inscribed Angle Theorem

An inscribed angle has its vertex on the circle, and its two sides are chords of the circle. The inscribed angle theorem is the single most important circle theorem:

$\text{inscribed angle} = \frac{1}{2} \times \text{intercepted arc}$

Equivalently, the intercepted arc is always twice the inscribed angle:

$\text{intercepted arc} = 2 \times \text{inscribed angle}$

Why does this work? Consider a central angle and an inscribed angle that both intercept the same arc. The central angle equals the arc. The inscribed angle, because its vertex is farther from the center (out on the circle), “sees” that same arc from a wider vantage point and measures exactly half as much. This 2-to-1 relationship holds regardless of where the inscribed angle’s vertex sits on the circle — as long as it intercepts the same arc.

SVG Diagram: Central Angle vs. Inscribed Angle

In the diagram above, both the green central angle (at center $O$) and the blue inscribed angle (at point $P$ on the circle) intercept the same amber arc $AB$. The central angle measures $80^\circ$ — equal to the arc. The inscribed angle measures $40^\circ$ — exactly half the arc. This 2-to-1 relationship is the inscribed angle theorem in action.

Corollary 1: Angle in a Semicircle (Thales’ Theorem)

If the two endpoints of the inscribed angle’s chords are the endpoints of a diameter, then the intercepted arc is a semicircle measuring $180^\circ$. Applying the inscribed angle theorem:

$\text{inscribed angle} = \frac{1}{2} \times 180^\circ = 90^\circ$

Any inscribed angle that intercepts a semicircle is a right angle. This result is known as Thales’ theorem, and it gives you a powerful shortcut: if one side of an inscribed triangle is a diameter, the angle opposite that side is always $90^\circ$.

This works in reverse too. If you see a right angle inscribed in a circle, the side opposite the right angle must be a diameter.

Corollary 2: Inscribed Angles on the Same Arc Are Equal

If two or more inscribed angles intercept the same arc, they all have the same measure — regardless of where their vertices sit on the circle (as long as they are on the same side of the chord).

$\text{If } \angle P \text{ and } \angle Q \text{ intercept the same arc, then } \angle P = \angle Q$

This follows directly from the inscribed angle theorem: each angle equals half the same arc, so they must be equal. This property is useful when you need to prove two angles are congruent without knowing their exact measures.

Tangent-Radius Relationship

Recall from circle properties that a tangent line touches the circle at exactly one point, and at that point the tangent is perpendicular to the radius:

$\text{tangent} \perp \text{radius at the point of tangency} \quad (90^\circ)$

This $90^\circ$ relationship is essential for solving many circle problems. When a tangent and a radius meet, you immediately have a right angle to work with — which often unlocks the rest of the problem through the Pythagorean theorem or angle sums.

Tangent-Chord Angle

A tangent-chord angle is formed by a tangent line and a chord drawn from the point of tangency. The measure of this angle equals half the intercepted arc:

$\text{tangent-chord angle} = \frac{1}{2} \times \text{intercepted arc}$

This formula mirrors the inscribed angle theorem. You can think of a tangent-chord angle as a special limiting case of an inscribed angle — as the vertex of an inscribed angle approaches the tangent point, the inscribed angle “becomes” a tangent-chord angle, and the half-arc relationship is preserved.

Summary of Circle Angle Relationships

Worked Examples

Example 1: Central angle to inscribed angle

A central angle intercepts an arc of $100^\circ$. An inscribed angle intercepts the same arc. Find the inscribed angle.

Step 1: The central angle equals its intercepted arc, so the arc is $100^\circ$.

Step 2: Apply the inscribed angle theorem.

$\text{inscribed angle} = \frac{1}{2} \times 100^\circ = 50^\circ$

Answer: The inscribed angle is $50^\circ$.

Example 2: Inscribed angle to arc

An inscribed angle measures $35^\circ$. Find the measure of its intercepted arc.

Step 1: Apply the inscribed angle theorem in reverse.

$\text{intercepted arc} = 2 \times \text{inscribed angle} = 2 \times 35^\circ = 70^\circ$

Answer: The intercepted arc is $70^\circ$.

Example 3: Triangle inscribed in a semicircle

Triangle $ABC$ is inscribed in a circle where $\overline{AC}$ is a diameter. Prove that $\angle B = 90^\circ$.

Step 1: Since $\overline{AC}$ is a diameter, the arc $AC$ (going the long way, through the side where $B$ sits) is a semicircle measuring $180^\circ$.

Step 2: $\angle B$ is an inscribed angle that intercepts this semicircular arc.

Step 3: By Thales’ theorem:

$\angle B = \frac{1}{2} \times 180^\circ = 90^\circ$

Answer: $\angle B = 90^\circ$. Any triangle inscribed in a semicircle with one side as the diameter has a right angle opposite that diameter.

Example 4: Two inscribed angles on the same arc

Inscribed angle $\angle P$ intercepts arc $XY$ and measures $48^\circ$. Inscribed angle $\angle Q$ also intercepts arc $XY$. Find $\angle Q$.

Step 1: Both angles intercept the same arc $XY$.

Step 2: By the same-arc corollary, inscribed angles intercepting the same arc are equal.

$\angle Q = \angle P = 48^\circ$

Answer: $\angle Q = 48^\circ$. It does not matter where $Q$ sits on the circle (as long as it is on the same side of chord $XY$ as $P$) — the angle is the same.

Example 5: Tangent-chord angle

A tangent line touches a circle at point $T$, and a chord $\overline{TA}$ is drawn from $T$. The arc $TA$ (on the side of the angle) measures $130^\circ$. Find the angle between the tangent and the chord.

Step 1: Identify the intercepted arc. The tangent-chord angle intercepts arc $TA$ measuring $130^\circ$.

Step 2: Apply the tangent-chord formula.

$\text{tangent-chord angle} = \frac{1}{2} \times 130^\circ = 65^\circ$

Answer: The angle between the tangent and the chord is $65^\circ$.

Check: The arc on the other side of chord $TA$ is $360^\circ - 130^\circ = 230^\circ$. The angle on the other side of the tangent-chord pair would be $\frac{1}{2} \times 230^\circ = 115^\circ$. And $65^\circ + 115^\circ = 180^\circ$, which makes sense because the two tangent-chord angles on either side of the chord are supplementary (they form a straight line along the tangent).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• A central angle equals its intercepted arc; an inscribed angle equals half the intercepted arc
• Thales’ theorem: An inscribed angle on a diameter is always $90^\circ$ — this creates right triangles instantly
• Same-arc corollary: All inscribed angles intercepting the same arc are equal, no matter where the vertex sits on the circle
• A tangent-chord angle also equals half its intercepted arc — think of it as an inscribed angle at the tangent point
• The tangent is perpendicular to the radius at the point of tangency ($90^\circ$)
• These theorems appear frequently on the SAT — memorize the half-arc relationship and Thales’ theorem

Return to Geometry for more topics in this section.