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MathChapter 19: Circles

Notes

Inscribed Angles Notes

Connect an inscribed angle to its intercepted arc and apply three high-value circle corollaries.

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An inscribed angle has its vertex on the circle. Its sides are chords, and the arc lying between those sides—opposite the vertex—is the intercepted arc.

The inscribed-angle theorem

Inscribed angle and intercepted arcAn inscribed angle with vertex on the circle intercepts a 120-degree arc and measures 60 degrees.
Inscribed angle and intercepted arcAn inscribed angle with vertex on the circle intercepts a 120-degree arc and measures 60 degrees.120° arc60°OABC
Inscribed angle and intercepted arc
Angle–arc relationship
\[m\angle A=\frac12m\widehat{BC}\]

Equivalently, the intercepted arc is twice the inscribed angle.

Worked example

Arc from an inscribed angle

An inscribed angle measures \(47^\circ\). Find its intercepted arc.

  1. Double

    \(m\widehat{BC}=2(47^\circ)=94^\circ\).

\(94^\circ\).

Three SAT corollaries

Inscribed-angle corollaries
ConditionConclusionFast recognition
Same intercepted arcInscribed angles are congruentTrace both angle sides to the same endpoints
Intercepted arc is a semicircleInscribed angle is \(90^\circ\)One side-to-side chord is a diameter
Quadrilateral inscribed in a circleOpposite angles total \(180^\circ\)All four vertices lie on the circle
Inscribed angles intercepting the same arcTwo inscribed angles share the same 96-degree intercepted arc, so both measure 48 degrees.
Inscribed angles intercepting the same arcTwo inscribed angles share the same 96-degree intercepted arc, so both measure 48 degrees.96° arcOABCD
Inscribed angles intercepting the same arc
Angle inscribed in a semicircleThe angle opposite a diameter is marked as a right angle.
Angle inscribed in a semicircleThe angle opposite a diameter is marked as a right angle.OABC
Angle inscribed in a semicircle
Cyclic quadrilateralA quadrilateral has all four vertices on a circle; opposite angles shown are 72 and 108 degrees and are supplementary.
Cyclic quadrilateralA quadrilateral has all four vertices on a circle; opposite angles shown are 72 and 108 degrees and are supplementary.72°108°OABCDNot to scale
Cyclic quadrilateral

Figure not drawn to scale.

Worked example

Cyclic quadrilateral algebra

Opposite angles in a cyclic quadrilateral are \(3x+6\) and \(5x-10\) degrees. Find x.

  1. Supplement

    \((3x+6)+(5x-10)=180\), so \(8x=184\).

\(x=23\).
Mini check

Diameter check

Triangle ABC is inscribed in a circle and AC is a diameter. If \(\angle A=34^\circ\), find \(\angle B\) and \(\angle C\).

Show answer and explanation

Answer: \(\angle B=90^\circ\), \(\angle C=56^\circ\).

The angle opposite diameter AC is right; triangle angles then total 180 degrees.

Key takeaways

Key takeaways

  • An inscribed angle is half its intercepted arc.
  • Inscribed angles intercepting the same arc are congruent.
  • An angle inscribed in a semicircle is right.
  • Opposite angles of a cyclic quadrilateral are supplementary.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.