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9.4: Inscribed Angles

Difficulty Level: At Grade Created by: CK-12

Learning Objective

  • Find the measure of inscribed angles and the arcs they intercept

Inscribed Angle, Intercepted Arc

An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. An inscribed angle is said to intercept an arc of the circle. We will prove shortly that the measure of an inscribed angle is half of the measure of the arc it intercepts.

Notice that the vertex of the inscribed angle can be anywhere on the circumference of the circle--it does not need to be diametrically opposite the intercepted arc.

Measure of Inscribed Angle

The measure of a central angle is twice the measure of the inscribed angle that intercepts the same arc.

Proof.

\angle{COB} and \angle{CAB} both intercept \widehat{CB}. \angle{COB} is a central angle and angle \angle{CAB} is an inscribed angle.

We draw the diameter of the circle through points A and O, and let m\angle{CAO} = x^\circ and m\angle{BAO} = y^\circ.

We see that \triangle AOC is isosceles because \overline{AO} and \overline{AC} are radii of the circle and are therefore congruent.

From this we can conclude that m\angle{ACO} = x^\circ.

Similarly, we can conclude that m\angle{ABO} = y^\circ.

We use the property that the sum of angles inside a triangle equals 180^\circ to find that:

m\angle{AOC} = 180^\circ - 2x and m\angle{AOB} = 180^\circ - 2y.

Then,

m\angle{COD} =180^\circ - m\angle{AOC}=180^\circ-(180^\circ-2x) = 2x and m\angle{BOD} =180^\circ - m\angle{AOB}=180^\circ-(180^\circ-2y) = 2y.

Therefore

m\angle{COB}=2x+2y=2(x+y)=2 (m\angle{CAB}). \blacklozenge

Inscribed Angle Corollaries a-d

The theorem above has several corollaries, which will be left to the student to prove.

a. Inscribed angles intercepting the same arc are congruent

b. Opposite angles of an inscribed quadrilateral are supplementary

c. An angle inscribed in a semicircle is a right angle

d. An inscribed right angle intercepts a semicircle

Here are some examples the make use of the theorems presented in this section.

Example 1

Find the angle marked x in the circle.

The m\angle{AOB} is twice the measure of the angle at the circumference because it is a central angle.

Therefore, m\angle{OAB}=35^\circ.

This means that x = 180^\circ - 35^\circ = 145^\circ.

Example 2

Find the angles marked x in the circle.

m\widehat{ADC} &= 2 \times 95^\circ = 190^\circ \Rightarrow m\widehat{ABC} = 360^\circ - 190^\circ = 170^\circ\\m\widehat{AB} &= 2 \times 35^\circ = 70^\circ \Rightarrow m\widehat{BC} = 170^\circ -70^\circ = 100^\circ

So, x=50^\circ.

Example 3

Find the angles marked x and y in the circle.

First we use \triangle ABD to find the measure of angle x.

x + 15^\circ + 32^\circ = 180^\circ \Rightarrow x = 133^\circ

Therefore, m\angle{CBD} = 180^\circ - 133^\circ = 47^\circ.

m\angle{BCE} = m\angle{BDE} because they are inscribed angles and intercept the same arc \Rightarrow m\angle{BCE} = 15^\circ.

In \triangle BFC, y + 47^\circ + 15^\circ = 180^\circ \Rightarrow y = 118^\circ.

Lesson Summary

In this section we learned about inscribed angles. We found that an inscribed angle is half the measure of the arc it intercepts. We also learned some corollaries related to inscribed angles and found that if two inscribed angles intercept the same arc, they are congruent.

Review Questions

  1. In \bigodot A, m\widehat{PO} = 90^\circ, m\widehat{ON} = 95^\circ and m\angle{MON} = 60^\circ. Find the measure of each angle:
    1. m\angle{PMO}
    2. m\angle{PNO}
    3. m\angle{MPN}
    4. m\angle{PMN}
    5. m\angle{MPO}
    6. m\angle{MNO}
  2. Quadrilateral ABCD is inscribed in \bigodot O such that m\widehat{AB} = 70^\circ, m\widehat{BC} = 85^\circ, m\widehat{AD} = 130^\circ. Find the measure of each of the following angles:
    1. m\angle{A}
    2. m\angle{B}
    3. m\angle{C}
    4. m\angle{D}
  3. In the following figure, m\widehat{IF} = 5x + 60^\circ, m\angle{IGF} = 3x + 25^\circ and m\widehat{HG} = 2x + 10^\circ. Find the following measures:
    1. m\angle{I}
    2. m\angle{F}
    3. m\angle{H}
    4. m\widehat{IH}
  4. Prove the inscribed angle theorem corollary a.
  5. Prove the inscribed angle theorem corollary b.
  6. Prove the inscribed angle theorem corollary c.
  7. Prove the inscribed angle theorem corollary d.
  8. Find the measure of angle x.
  9. Find the measure of the angles x and y.
  10. Suppose that \overline{AB} is a diameter of a circle centered at O, and C is any other point on the circle. Draw the line through O that is parallel to \overline{AC}, and let D be the point where it meets \widehat{BC}. Prove that D is the midpoint of \widehat{BC}.

Review Answers

    1. 45^\circ
    2. 45^\circ
    3. 60^\circ
    4. 92.5^\circ
    5. 107.5^\circ
    6. 72.5^\circ
    1. 80^\circ
    2. 102.5^\circ
    3. 100^\circ
    4. 77.5^\circ
    1. 15^\circ
    2. 15^\circ
    3. 55^\circ
    4. 70^\circ
  1. Proof
  2. Proof
  3. Proof
  4. Proof
    1. 110^\circ
    2. 100^\circ
    3. 32.5^\circ
    4. 30^\circ
    1. x = 74^\circ, y = 106^\circ
    2. x = 35^\circ, y = 35^\circ
  5. Hint: \overline{AC} \parallel \overline{OD}, so \angle{CAB} \cong \angle{DOB}.

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