When you inscribe a quadrilateral inside a circle, a very interesting property exists in the angles of the inscribed quadrilateral. Take a look at the following inscribed quadrilateral.
What can we say about the measures of angles A, B, C, and D? Work through the following slideshow to learn more about these angles.
Proof Step 1
Let's take a close look at the angles and arcs of a quadrilateral that is inscribed in a circle. Based on the Inscribed Angle Theorem, we can say that:
m∠A = \(\mathsf{\frac{1}{2}}\) m\(\small\mathsf{ \overset{\displaystyle\frown}{BCD} }\) Proof Step 2
Now, we can write the following two equations:
m\(\small\mathsf{ \overset{\displaystyle\frown}{BCD} }\) + m\(\small\mathsf{ \overset{\displaystyle\frown}{BAD} }\) = 360° Proof Step 3
Now, we can do a final substitution:
\(\mathsf{\frac{1}{2}}\)m\(\small\mathsf{ \overset{\displaystyle\frown}{BCD} }\) + \(\mathsf{\frac{1}{2}}\)\(\small\mathsf{ \overset{\displaystyle\frown}{BAD} }\) = 180° Proof Step 4
We can follow the same process for angles B and D. What is the result? m∠B = \(\mathsf{\frac{1}{2}}\)m\(\small\mathsf{ \overset{\displaystyle\frown}{ADC} }\) and m∠D = \(\mathsf{\frac{1}{2}}\)m\(\small\mathsf{ \overset{\displaystyle\frown}{ABC} }\) |
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This is called the Inscribed Quadrilateral Theorem:
Inscribed Quadrilateral Theorem
A quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
