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On the circle above, construct an angle with a vertex at the center of the circle. The angle being constructed should also cut off the same arc. In other words, construct the corresponding central angle.
Observe the measures of the angle and the arc intercepted by the angle. Start by moving B so that B, O, and A are collinear. Then, move it once again so that B, O, and C are collinear.
As can be seen, when B, O, and C are collinear, BC becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of BC is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.
In the diagram, the vertex of ∠UVT is on the circle O, and the sides of the angle are chords of the circle. Given the measure of UT, find the measure of the angle.
Write the answer without the degree symbol.
The Inscribed Angle Theorem can be used to find the measure of the angle.
The angle shown in the diagram fits the definition of an inscribed angle. For this reason, the measure of the angle can be found using the Inscribed Angle Theorem. The theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
The arc intercepted by ∠UVT measures 40∘. Therefore, ∠UVT measures 20∘.Find the measure of the inscribed angle in the circle.
Similarly, given the measure of an inscribed angle, the measure of its corresponding central angle can be found using the Inscribed Angle Theorem. This can be done because the measure of the central angle is the same as the measure of the arc that the central angle cuts off.
In the circle, ∠KLM measures 67∘.
Find the measure of the corresponding central angle.
Start by drawing the corresponding central angle.
Recall that a central angle is an angle whose vertex lies at the center of the circle. Additionally, the inscribed angle and its corresponding central angle intercept the same arc KM for this example. Therefore, the corresponding central angle is ∠KOM.
The arc KM is included in ∠KOM. Therefore, the measure of ∠KOM is equal to the measure of KM. Using the Inscribed Angle Theorem, the measure of KM can be found. Therefore, ∠KOM also measures 134∘.Given the measure of an inscribed angle, find the measure of its corresponding central angle.
Up to now, the relationship between inscribed angles and their corresponding central angles has been discussed. Now the relationship between two inscribed angles that intercept the same arc will be investigated.
As can be observed, the angles are congruent, so long as they intercept the same arc.
Find the measure of the central angle.
The following example involving circumscribed angles and inscribed angles could require the use of the previously learned theorems.
Two tangents from P to ⊙M are drawn. The measure of ∠KPL is 40∘.
Find the measure of the inscribed angle that intercepts the same arc as ∠KPL?
Use the Circumscribed Angle Theorem and Inscribed Angle Theorem.
This lesson defined three angles related to circles as well as the relationships between these angles. The diagram below shows the definitions and the main theorems of this lesson.