Key takeaway: An inscribed angle is formed by two chords that meet at a point on the circle, and its measure is exactly half the measure of the arc between its sides.
Grade 10 – GeometryG-C.A.2Geometry and Measurement
Definition of Inscribed Angle
An inscribed angle has its corner on the circle itself, with both sides running along chords. The stretch of circle caught between those sides is the intercepted arc.
The theorem behind it says the opening is half that arc, which also makes it half of the angle at the center that shares the same arc. Two consequences follow: every such opening over one arc is equal, and one standing over a semicircle must be a right angle.
Formula and notation
For an inscribed angle \(\angle ACB\) intercepting arc \(\overset{\frown}{AB}\):
\[ m\angle ACB = \tfrac{1}{2}\, m\overset{\frown}{AB} \]
If \(O\) is the center, the central angle satisfies \(m\angle AOB = 2\, m\angle ACB\).
Key facts
- Its measure is half the arc, never equal to it.
- Openings that face the same arc are congruent.
- One standing over a diameter is always 90 degrees.
- Opposite corners of a four-sided figure drawn inside a circle total 180 degrees.
Inscribed Angle example
A stage light at a point on a circular track opens \((5x – 3)^{\circ}\) and spans a stretch of track measuring \((8x + 6)^{\circ}\). Find \(x\), the opening and the stretch.
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The opening is half the stretch, so the stretch is twice the opening: \(2(5x – 3) = 8x + 6\).
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Expand and solve: \(10x – 6 = 8x + 6\), so \(2x = 12\) and \(x = 6\).
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Angle: \(5(6) – 3 = 27^{\circ}\). Arc: \(8(6) + 6 = 54^{\circ}\).
\(x = 6\); the opening is \(27^{\circ}\) and the stretch of track is \(54^{\circ}\).
Common mistake
Watch out
Seth sets \(5x – 3 = 8x + 6\), treating the opening as equal to its stretch, and gets \(x = -3\).
Why it is wrong
Equal measures hold only for an angle at the center. An inscribed angle is half of what it faces. A negative \(x\) here would give negative angles, a clue that the setup is wrong. Use \(2(5x – 3) = 8x + 6\).
Related terms and lessons
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