Chord, Secant, and Tangent Angle Relationships
Calculate angle measures formed by two chords, two secants, or tangent-secant pairs from inside and outside a circle. Apply the arc-angle relationships to set up and solve equations.
A lesson of Geometry, in Circles, about 10 minutes.
By the end of this lesson, you will be able to find angle measures formed by two chords, two secants, or secant-tangent combinations using the chord-angle and secant-angle theorems, and apply segment length products for intersecting chords and secants.
The Big Idea
When lines intersect in relation to a circle, the angles formed are always related to arcs — but the formula depends on whether the intersection is inside the circle, on the circle, or outside the circle. Three cases, three formulas. Once you identify the case, the calculation is straightforward.
Case 1: Intersection INSIDE the Circle (Two Chords)
Chord-Chord Angle Theorem:
$$m\angle 1 = \frac{1}{2}(m\overset{\frown}{AC} + m\overset{\frown}{BD})$$
The angle equals half the sum of the two intercepted arcs.
Chord-Chord Segment Product:
$$AE \cdot EC = BE \cdot ED$$
(Products of the two segments of each chord are equal.)
Case 2: Intersection ON the Circle (Inscribed or Tangent-Chord)
Already covered: inscribed angle = ½ arc.
Tangent-chord angle = ½ intercepted arc.
Case 3: Intersection OUTSIDE the Circle
Secant-Secant, Secant-Tangent, or Tangent-Tangent:
$$m\angle = \frac{1}{2}(\text{far arc} - \text{near arc})$$
The angle equals half the difference of the intercepted arcs (the "far" minus the "near" arc).
Secant-Secant product: $PA \cdot PB = PC \cdot PD$ (whole × whole = whole × whole from external point)
Secant-Tangent product: $PA^2 = PC \cdot PD$ (tangent² = external × whole secant)
Example 1 — Two Chords (Inside)
Chords $\overline{AC}$ and $\overline{BD}$ intersect inside circle $O$. $\overset{\frown}{AB} = 70°$ and $\overset{\frown}{CD} = 110°$. Find the angle at the intersection.
$$m\angle = \frac{1}{2}(70 + 110) = \frac{1}{2}(180) = 90°$$
Example 2 — Chord Products
Chords intersect inside a circle: $AE = 4$, $EC = 9$, $BE = 6$. Find $ED$.
$$4 \cdot 9 = 6 \cdot ED \implies ED = 6$$
Example 3 — Two Secants from Outside
From external point $P$: near arc = 40°, far arc = 140°. Find $m\angle P$.
$$m\angle P = \frac{1}{2}(140 - 40) = \frac{1}{2}(100) = 50°$$
Example 4 — Tangent-Secant from Outside
External point $P$: tangent $PA$ touches at $A$, secant passes through $B$ and $C$ (near). $\overset{\frown}{AC}$ (far) $= 200°$, $\overset{\frown}{AB}$ (near) $= 80°$.
$$m\angle P = \frac{1}{2}(200 - 80) = 60°$$