Angle Relationships in Circles
Central Angles
Move point C around the circle and notice the relationship between the measure of the central angle and the measure of arc CD
Central Angles
Inscribed Angles
An inscribed angle in a circle is an angle in a circle where the vertex is on the circle and its sides are chords.
Is the measure of the inscribed angle the same as the measure of arc CD?
What can you do to the measure of arc CD to get the value for the measurement of the inscribed angle?
What is the relationship between the measurement of an inscribed angle and the intercepted arc?
Move point D in the circle below and determine if the relationship that you predicted is true in all cases.
Angles Inside a Circle when Secant Lines Intersect
Is the measure of the angle in the diagram below the same as the measurement of arc CF?
Now find the average of arc CF and arc ED. What do you notice?
Move point C to a new location, and find the average of arc CF and arc ED.
What is a rule that we could create to describe this relationship?
Secants Intersecting Inside the Circle
Tangent Lines and Secant Lines
Move point D so that arc CFD is 180 degrees. What is the measure of the angle formed by the tangent line and the secant line?
Move point D so that arc CFD is 120 degrees. What is the measure of the angle formed by the tangent line and the secant line?
What do you think the relationship between the arc and the angle is in this scenario?
Move point D around to see if your prediction is correct.
Tangent Lines and Secant Lines
Secant Lines Intersecting Outside the Circle
In the diagram below you can see that the angle outside the circle is less than either arc DE or arc FG.
What number do you get if you subtract the measurement of arc GF from the measurement of arc DE. (round to the nearest whole number)
What can we do to this number to get 25?
Now move point d so that the arc measurements change.
Now subtract the measurement of arc GF from the measurement of arc DE again.
What can you do to this number to get the measure of the angle outside the circle?
What is a formula that we can create that will work every time in this scenario?
Quadrilateral Inscribed Inside a Circle
In the circle below, what do you get if you add up the measurements of angle F and angle D?
What do you get if you add up the measurements of angle C and angle E?
Now move point C and D to new positions so that all the angles change.
What value do you get if you add up the opposite angles in the quadrilateral?
Will this always occur?