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In the diagram above:
An inscribed angle is an angle formed by two chords with the vertex located on the circle. The sides of the inscribed angle are chords that connect the vertex to two other points on the circle.
In the diagram above:
When they subtend the same arc, the central angle is twice the inscribed angle. The diagram below shows this relationship.
In the relationship diagram above, the central angle is . The inscribed angle intercepting the same minor arc is therefore:
Draw the diameter through and the center . The two resulting isosceles triangles connect the inscribed angle to the central angle.
Proof steps:
This construction proves the configuration shown, where lies on the arc opposite . Other positions can be split into the same isosceles-triangle cases. In every case, the two angles being compared must intercept the same arc.
The central-angle relationship gives two useful consequences. A diameter determines a straight central angle. Keeping the intercepted arc fixed also keeps its central angle fixed, even when the vertex of the inscribed angle moves.
Every inscribed angle that subtends a diameter of a circle measures (right angle).
A semicircle gives a straight central angle, so:
, both angles subtend the same arc . In other words, sliding to along the same arc does not change the measure.
Here the two vertices lie on the same major arc. Both intercept the minor arc whose central angle is .
Given central angle . Find the measure of inscribed angle that subtends the same arc!
Solution:
Given inscribed angle . Find the measure of central angle that subtends the same arc!
Solution:
Identify the vertex and intercepted arc first. Use half or twice the angle for central-inscribed pairs, then apply the diameter and equal-arc properties in the last two problems.
If the central angle of a circle is , what is the measure of the inscribed angle that subtends the same arc?
Inscribed angle . Find the measure of central angle that intercepts the same arc.
In a circle, inscribed angle subtends a diameter. What is the measure of ?
Two inscribed angles subtend the same arc. If one angle measures , find the measure of the other angle!
Solution 1
The inscribed angle subtends the same arc as the central angle, so its measure is half the central angle: .
Solution 2
The central angle and intercept arc . The central angle is twice the inscribed angle:
Solution 3
A diameter cuts off a semicircle, whose central angle is . The inscribed angle is half of that angle, so .
Solution 4
Inscribed angles that intercept the same arc have equal measures. The other angle is also .
Published: . Updated: .
