Finding the Measure of Circumscribed Angles: Relationships with Central Angles

This lesson explores the relationships between central and circumscribed angles within quadrilaterals. It defines central angles as those with vertices at the circle's center and circumscribed angles as those with vertices outside the circle. The lesson explains how to use properties of quadrilaterals and kites to find angle measures, demonstrating that the sum of a central angle and a circumscribed angle is 180 degrees. An example problem is solved using these concepts, showing that a circumscribed angle with an intersected arc of 60 degrees measures 120 degrees.

What is a central angle?

• An angle whose vertex is outside the circle and rays are tangent to the circle

• An angle that measures exactly 90 degrees

• An angle whose vertex is at the center of the circle and sides intersect the circle

• An angle formed by two intersecting chords

What is the relationship between a central angle and a circumscribed angle?

• One is always double the other

• They are always equal

• Their measures sum to 180 degrees

• Their measures sum to 360 degrees

In the quadrilateral OJPS, which angles are replaced with 90 degrees?

• Angle OSP and angle PJO

• Angle JOS and angle OSP

• Angle JOS and angle SPJ

• Angle SPJ and angle PJO

If the central angle measures 60 degrees, what is the measure of the circumscribed angle?

• 60 degrees

• 90 degrees

• 120 degrees

• 180 degrees

How do you determine the measure of a circumscribed angle if the intersected arc is known?

• Double the arc measure

• Subtract the arc measure from 180 degrees

• Add the arc measure to 180 degrees

• Subtract the arc measure from 360 degrees