Inscribed Angle of a Circle and its intercepted arc

Theorems and examples

An Inscribed Angle's

  • vertex lies somewhere on the circle
  • sides are chords from the vertex to another point in the circle
  • creates an arc , called an intercepted arc
  • The measure of the inscribed angle is half of measure of the intercepted arc (This only works for the most frequently studied case when the vertex point such as B is not within arc AC.)

Look at the picture above

  • $$ \angle $$ ABC is the inscribed angle
  • $$\overline{BC}$$ and $$\overline{AC}$$ are the chords
  • $$\overparen{AC}$$ is the intercepted arc
  • Formula: $ \angle ABC = \frac{1}{2} \overparen{AC}$$

Interactive Inscribed Angle

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Identifying the Inscribed Angles and their Intercepted Arcs

Problem 1

Identify the inscribed angles and their intercepted arcs

Inscribed Angle Diagram

If $$ m\angle XYZ =40^{o} $$, what is measure of $$ \overparen{XZ} $$ ?

Measure of inscribed angle = ½ measure of the intercepted arc. Therefore, $$ \overparen{XZ} = 2 \cdot 40^{o}= 80^{o} $$

Picture of inscribed angles

Every single inscribed angle in the picture on the left has the exact same measure, since each inscribed angle intercepts the exact same arc, which is $$ \overparen {AZ} $$

Problem 2

What is the length of YZ?

Inscribed Angle Example
  • YZ2 = 52+122
  • YZ = 13
Problem 3

What is the length of YX?

Problem 4

Use your knowledge of the properties of inscribed angles and arcs to determine what is erroneous about the picture below.

Find the error in this inscribed angle problem

The error is that 992 +1322 ≠ 1642. Since the Pythagorean theorem does not hold, the X is not a right angle and the measure of arc ≠ 180°.

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