10.2 Angles and segments in circles

Angles and Segments in Circles

This section covers how angles behave when their vertices sit at different positions relative to a circle: at the center, on the circle, inside the circle, or outside the circle. Each position produces a different relationship between the angle and the arcs it intercepts. Mastering these relationships is the key to solving most circle problems in this unit.

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Angles in Circles

Central and Inscribed Angles

A central angle has its vertex at the center of the circle and is formed by two radii. The measure of a central angle equals the measure of its intercepted arc. If a central angle measures 80°, its intercepted arc also measures 80°.

An inscribed angle has its vertex on the circle and is formed by two chords. An inscribed angle always measures half the intercepted arc. So if the intercepted arc is 80°, the inscribed angle is 40°.

A circumscribed angle (formed by two tangents from the same external point) has its vertex outside the circle. Its measure equals half the difference of the two intercepted arcs. Since the two arcs together make a full 360°, if the minor arc is $x$, the major arc is $360 - x$, and the angle equals $\frac{1}{2}((360 - x) - x) = 180 - x$.

Inscribed Angle Theorem

The Inscribed Angle Theorem states that an inscribed angle is half the intercepted arc:

$m\angle ABC = \frac{1}{2} \cdot m\overset{\frown}{AC}$

Since a central angle equals its intercepted arc, this also means:

$m\angle ABC = \frac{1}{2} \cdot m\angle AOC$

where $O$ is the center and both angles intercept arc $\overset{\frown}{AC}$.

To apply this in problems:

1. Identify the intercepted arc of the inscribed angle (the arc that lies in the interior of the angle).
2. If you know the arc (or the central angle on that arc), divide by 2 to get the inscribed angle.
3. If you know the inscribed angle, multiply by 2 to get the arc or central angle.

For example, if a central angle measures 120°, any inscribed angle intercepting the same arc measures 60°. If an inscribed angle measures 45°, the intercepted arc is 90°.

Key Corollaries of the Inscribed Angle Theorem

• Congruent inscribed angles: All inscribed angles that intercept the same arc are congruent. No matter where you place the vertex on the major arc, the angle stays the same.
• Semicircle theorem: An inscribed angle that intercepts a semicircle (a 180° arc, meaning the chord connecting the endpoints of the angle is a diameter) is always 90°. This is one of the most commonly tested facts on circle problems. The converse is also true: if an inscribed angle is 90°, then the chord it subtends must be a diameter.
• Opposite angles in an inscribed quadrilateral: If a quadrilateral is inscribed in a circle (a cyclic quadrilateral), each pair of opposite angles is supplementary. That's because opposite angles intercept arcs that together make 360°, so their measures add to $\frac{1}{2}(360°) = 180°$.

Angles Formed by Chords, Secants, and Tangents

Where the vertex of the angle sits determines which formula you use. There are three cases beyond the central angle.

Case 1: Vertex inside the circle (two chords intersecting)

When two chords intersect inside a circle, the angle formed equals half the sum of the two arcs it intercepts (the arc in front of the angle and the arc in front of its vertical angle):

$m\angle = \frac{1}{2}(m\overset{\frown}{arc_1} + m\overset{\frown}{arc_2})$

For example, if the two intercepted arcs measure 70° and 110°, the angle is $\frac{1}{2}(70 + 110) = 90°$. Notice that the inscribed angle formula is actually a special case of this: when the vertex is on the circle, one of the two "arcs" shrinks to 0°, so the formula reduces to half of one arc.

Case 2: Vertex on the circle

This is the Inscribed Angle Theorem: the angle equals half the intercepted arc. A tangent-chord angle (formed by a tangent and a chord meeting at the point of tangency) also follows this same rule:

$m\angle = \frac{1}{2} \cdot m\overset{\frown}{intercepted\ arc}$

For a tangent-chord angle, the intercepted arc is the arc that lies inside the angle, between the chord and the tangent line.

Case 3: Vertex outside the circle

Whether the angle is formed by two secants, a secant and a tangent, or two tangents, the formula is the same: half the difference of the intercepted arcs.

$m\angle = \frac{1}{2}(m\overset{\frown}{far\ arc} - m\overset{\frown}{near\ arc})$

The "far arc" is the larger arc (farther from the vertex) and the "near arc" is the smaller one (closer to the vertex). For example, if two secants from an external point intercept arcs of 140° and 40°, the angle is $\frac{1}{2}(140 - 40) = 50°$.