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Inscribed Angles in Circles

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What if you had a circle with two chords that share a common endpoint? How could you use the arc formed by those chords to determine the measure of the angle those chords make inside the circle? After completing this Concept, you'll be able to use the Inscribed Angle Theorem to solve problems like this one.

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Inscribed Angles in Circles CK-12

Guidance

An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle. The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.

The Inscribed Angle Theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

m\angle ADC =\frac{1}{2} m \widehat{AC} \text{ and } m\widehat{AC}  = 2 m\angle ADC

Inscribed angles that intercept the same arc are congruent. This is called the Congruent Inscribed Angles Theorem and is shown below.

\angle ADB and \angle ACB intercept \widehat{AB} , so m\angle ADB = m\angle ACB . Similarly, \angle DAC and \angle DBC intercept \widehat{DC} , so m\angle DAC = m\angle DBC .

An angle intercepts a semicircle if and only if it is a right angle ( Semicircle Theorem ). Anytime a right angle is inscribed in a circle, the endpoints of the angle are the endpoints of a diameter and the diameter is the hypotenuse.

\angle DAB intercepts a semicircle, so m\angle DAB = 90^\circ . \angle DAB is a right angle, so \widehat{DB} is a semicircle.

Example A

Find m \widehat{DC} and m\angle ADB .

From the Inscribed Angle Theorem:

m \widehat{DC} & = 2 \cdot 45^\circ=90^\circ \\m\angle ADB & = \frac{1}{2}\cdot 76^\circ=38^\circ

Example B

Find m\angle ADB and m\angle ACB .

The intercepted arc for both angles is \widehat{AB} . Therefore,

m\angle ADB & = \frac{1}{2} \cdot 124^\circ=62^\circ\\m\angle ACB & = \frac{1}{2} \cdot 124^\circ=62^\circ

Example C

Find m\angle DAB in \bigodot C .

C is the center, so \overline{DB} is a diameter. \angle DAB 's endpoints are on the diameter, so the central angle is 180^\circ .

m\angle DAB & = \frac{1}{2} \cdot 180^\circ=90^\circ.

Inscribed Angles in Circles CK-12

Guided Practice

1. Find m\angle PMN, \ m\widehat{PN}, \ m\angle MNP, and m\angle LNP .

2. Fill in the blank: An inscribed angle is ____________ the measure of the intercepted arc.

3. Fill in the blank: A central angle is ________________ the measure of the intercepted arc.

Answers:

1. Use what you've learned about inscribed angles.

m\angle PMN & = m\angle PLN=68^\circ \quad \ \ \text{by the Congruent Inscribed Angles Theorem}.\\m\widehat{PN} & = 2 \cdot 68^\circ=136^\circ \quad \ \ \ \text{from the Inscribed Angle Theorem.}\\m\angle MNP & =90^\circ \qquad \qquad \qquad \ \text{by the Semicircle Theorem}.\\m\angle LNP & = \frac{1}{2} \cdot 92^\circ=46^\circ \qquad \ \text{from the Inscribed Angle Theorem.}

2. half

3. equal to

Practice

Fill in the blanks.

  1. An angle inscribed in a ________________ is 90^\circ .
  2. Two inscribed angles that intercept the same arc are _______________.
  3. The sides of an inscribed angle are ___________________.
  4. Draw inscribed angle \angle JKL in \bigodot M . Then draw central angle \angle JML . How do the two angles relate?

Find the value of x and/or y in \bigodot A .

Solve for x .

  1. Fill in the blanks of the Inscribed Angle Theorem proof.

Given : Inscribed \angle ABC and diameter \overline{BD}

Prove : m\angle ABC = \frac{1}{2} m \widehat{AC}

Statement Reason

1. Inscribed \angle ABC and diameter \overline{BD}

m\angle ABE = x^\circ and m\angle CBE = y^\circ

1.
2. x^\circ + y^\circ = m\angle ABC 2.
3. 3. All radii are congruent
4. 4. Definition of an isosceles triangle
5. m\angle EAB = x^\circ and m\angle ECB = y^\circ 5.
6. m\angle AED = 2x^\circ and m\angle CED = 2y^\circ 6.
7. m\widehat{AD}= 2x^\circ and m \widehat{DC} = 2y^\circ 7.
8. 8. Arc Addition Postulate
9. m\widehat{AC} = 2x^\circ + 2y^\circ 9.
10. 10. Distributive PoE
11. m\widehat{AC} = 2m\angle ABC 11.
12. m\angle ABC=\frac{1}{2} m \widehat{AC} 12.

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