Tangents, Secants & Chords: Angle Relationships

When lines cross a circle — chords, secants, tangents, in any combination — the angles they form are controlled by the arcs they cut off. There are really only three rules, and which one you use depends on where the vertex of the angle sits: on the circle, inside it, or outside it.

Vertex on the circle: half the intercepted arc. Vertex inside the circle: half the sum of the two arcs. Vertex outside the circle: half the difference of the two arcs. Locate the vertex first, and the rest is arithmetic.

Vertex on the circle: half the arc

You already know one case — an inscribed angle is half its intercepted arc. A tangent and a chord meeting at the point of tangency follow the same pattern: the angle equals half the intercepted arc.

So if a tangent and a chord form an angle that cuts off a $140^\circ$ arc, the angle measures $70^\circ$ .

Vertex inside: half the sum

When two chords intersect inside a circle, each angle they form equals half the sum of the two arcs it and its vertical angle intercept: $m\angle = \dfrac{1}{2}(\text{arc}_1 + \text{arc}_2)$ .

The two arcs are the ones directly across from the angle — the arc it opens toward and the arc its vertical angle opens toward. The angle averages them.

Vertex outside: half the difference

When two secants, two tangents, or a secant and a tangent meet at a point outside the circle, the angle at that external point equals half the difference of the intercepted arcs: $m\angle P = \dfrac{1}{2}(\text{far arc} - \text{near arc})$ .

The far arc is the bigger one, farther from the vertex; the near arc is the smaller one tucked between the two lines close to the vertex. Always subtract near from far — an angle can't be negative.

A memory hook: inside the circle the lines pinch the arcs together, so you add. Outside, the vertex pulls away from the circle, so you subtract.

The figure shows the outside case: two secants from $P$ cut a near arc (between $A$ and $C$ ) and a far arc (between $B$ and $D$ ).

Worked examples

Example 1: two chords inside

Chords $\overline{AD}$ and $\overline{BC}$ intersect at $E$ inside a circle. Arc $AB$ measures $80^\circ$ and arc $CD$ measures $40^\circ$ . Find $m\angle AEB$ .

Vertex inside: half the sum of the arcs

m\angle AEB = \dfrac{1}{2}\left(m\overset{\frown}{AB} + m\overset{\frown}{CD}\right)

Substitute

m\angle AEB = \dfrac{1}{2}(80^\circ + 40^\circ)

Simplify

m\angle AEB = 60^\circ

Answer: $m\angle AEB = 60^\circ$

Example 2: two secants outside

Two secants from external point $P$ intercept arcs of $100^\circ$ and $40^\circ$ . Find $m\angle P$ .

Vertex outside: half the difference of the arcs

m\angle P = \dfrac{1}{2}(\text{far arc} - \text{near arc})

Substitute

m\angle P = \dfrac{1}{2}(100^\circ - 40^\circ)

Simplify

m\angle P = 30^\circ

Answer: $m\angle P = 30^\circ$

Example 3: tangent and chord

A tangent and a chord meet at the point of tangency, intercepting an arc of $150^\circ$ . Find the angle between them.

Vertex on the circle: half the intercepted arc

m\angle = \dfrac{1}{2}\,m\overset{\frown}{\text{arc}}

Substitute

m\angle = \dfrac{1}{2}(150^\circ)

Simplify

m\angle = 75^\circ

Answer: $75^\circ$

Try one yourself

Common questions

How do I decide between half the sum and half the difference?

Find the vertex. If the crossing point is inside the circle, use half the sum. If the lines meet outside the circle, use half the difference. If the vertex is on the circle itself, the angle is just half of one arc.

Which two arcs go into the formula for chords crossing inside?

The arc the angle opens toward and the arc its vertical angle opens toward — the two arcs directly opposite each other at the crossing. The other pair of arcs belongs to the other pair of vertical angles.

For the outside case, how do I tell the far arc from the near arc?

The near arc is the small one squeezed between the two lines right next to the external point. The far arc is the one on the opposite side of the circle. Far minus near keeps the answer positive.

Do these rules change for two tangents instead of two secants?

No. Two tangents, two secants, or one of each — if they meet outside the circle, the angle is half the difference of the intercepted arcs. For two tangents, the two arcs together make the whole circle, so they add to $360^\circ$ .

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Tangents, Secants & Chords: Angle Relationships

Vertex on the circle: half the arc

Vertex inside: half the sum

Vertex outside: half the difference

Worked examples

Example 1: two chords inside

Example 2: two secants outside

Example 3: tangent and chord

Try one yourself

Common questions

Keep learning

Inscribed Angles & the Inscribed-Angle Theorem

Central Angles & Arc Measure

Segment-Length Relationships in Circles

Circle Vocabulary: Center, Radius, Chord, Secant, Tangent