﻿ Inscribed Angle of a Circle and the arc it forms. Formula explained with pictures and an interactive demonstration. An Inscribed angle is just...

# 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}$$### Interactive Inscribed Angle ### PracticeIdentifying the Inscribed Angles and their Intercepted Arcs If$$ 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} $$Every single inscribed angle in the picture on the left has the exact same measure, since each inscribed angle intercepts the exact same arc, which is$$ \overparen {AZ}$\$

• YZ2 = 52+122
• YZ = 13

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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