Notera

Lesson 10.4: Inscribed Angles

What You'll Learn

In this lesson you'll discover the relationship between an inscribed angle and its intercepted arc, and apply it to angles in semicircles and inscribed quadrilaterals.
Definition

Inscribed Angle

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of the angle is the intercepted arc.
OABCarc AC∠B
Point BB is on the circle. Rays BA‾\overline{BA} and BC‾\overline{BC} are chords. Arc ACAC (not containing BB) is the intercepted arc.
The measure of an inscribed angle is half the measure of its intercepted arc.

m∠B=12 mAC⌢m\angle B = \frac{1}{2}\,m\overset{\frown}{AC}
Inscribed angle = ½ × intercepted arc. Always.
Ex

Example — Finding an Inscribed Angle

In ⊙O\odot O, mAC⌢=130°m\overset{\frown}{AC} = 130°. Find m∠Bm\angle B.
Solution Steps
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

If ∠B\angle B and ∠D\angle D both intercept AC⌢\overset{\frown}{AC}, then ∠B≅∠D\angle B \cong \angle D.
ABC∠C = 90° (semicircle)
When the intercepted arc is a semicircle (180°), the inscribed angle is 12(180°)=90°\frac{1}{2}(180°) = 90°.
An inscribed angle that intercepts a semicircle is a right angle (90°90°).

If AB‾\overline{AB} is a diameter, any point CC on the circle (other than AA or BB) makes ∠ACB=90°\angle ACB = 90°.
Diameter as chord ⟹ inscribed angle = 90°.
Ex

Example — Angles in a Semicircle

AB‾\overline{AB} is a diameter of ⊙O\odot O. CC is on the circle and m∠BAC=35°m\angle BAC = 35°. Find m∠ACBm\angle ACB and m∠ABCm\angle ABC.
Solution Steps
ABCDαβα + β = 180°
When all four vertices of a quadrilateral lie on a circle, opposite angles are supplementary.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

m∠A+m∠C=180°m\angle A + m\angle C = 180°
m∠B+m∠D=180°m\angle B + m\angle D = 180°
Cyclic quadrilateral ⟹ opposite angles add to 180°.
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Practice Problem
easy
In a circle, inscribed angle ∠P\angle P intercepts an arc of 96°96°. What is m∠Pm\angle P?
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Practice Problem
easy
QR‾\overline{QR} is a diameter. SS is on the circle and m∠RQS=28°m\angle RQS = 28°. Find m∠QSRm\angle QSR.
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Practice Problem
medium
Quadrilateral ABCDABCD is inscribed in a circle. If m∠A=72°m\angle A = 72° and m∠B=95°m\angle B = 95°, find m∠Cm\angle C and m∠Dm\angle D.
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Practice Problem
easy
Two inscribed angles ∠X\angle X and ∠Y\angle Y intercept the same arc of 140°140°. True or false: m∠X=m∠Ym\angle X = m\angle Y.
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