Think about the following questions.
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 so that and are collinear. Then, move it once again so that and are collinear.
As can be seen, when and are collinear, becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.
In this figure, the measure of is half the measure of
Let the measures of and be and respectively.
Therefore, by the Isosceles Triangle Theorem, the measures of and will also be and respectively.
By the Triangle Exterior Angle Theorem, it is recognized that and
The measure of an intercepted arc is the same as the measure of its central angle. Therefore, the measures of and are and respectively. From here, using the Angle Addition Postulate and the Arc Addition Postulate, and can be written in terms of and Consequentially, the measure of is half the measure of The proof of Case II has been completed.
By applying similar logic as the procedure above, Case I and Case III can be proven.
Write the answer without the degree symbol.
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.
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, measures
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 for this example. Therefore, the corresponding central angle is
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.
By this theorem, in the above diagram and are congruent angles.
Consider two inscribed angles that intercept the same arc in a circle.
By the Inscribed Angle Theorem, the measures of and are half the measure of
Therefore, by the Transitive Property of Equality, it can be said that these two angles have the same measure. Consequently, by the definition of congruent angles, and are congruent.
Mark and Jordan have been asked to find the measure of
The inscribed angles and intercept
By the Inscribed Angles of a Circle Theorem, is congruent to Accordingly, half the measure of can be found as Additionally, and intercept the same arc,
Again by the Inscribed Angles of a Circle Theorem, is congruent to Therefore, Since neither Mark's claim nor Jordan's claim is correct.
Inscribed angles, or the central angles, are not the only angles related to circles. In the next part, the angles constructed outside the circles will be examined. To construct an angle outside a circle, tangents can be used.
In the diagram, is tangent to the circle
Line is tangent to
Assume that line is tangent to the circle and is not perpendicular to By the Perpendicular Postulate, there is another segment from that is perpendicular to Let this segment be The aim is to show that must be this segment.
Since is perpendicular to then is a right triangle. In this right triangle, whose length is units, is the hypotenuse and therefore the longest side. Consequently, is less than In the diagram, it can be seen that contains a radius of Because part of the segment lies outside the circle, the unknown portion can be assigned another variable,
Line is tangent to
For the second part, it will be assumed that is perpendicular to the radius at and that line is not tangent to In this case, line intersects at a second point
Since is perpendicular to at is a right triangle and is the hypotenuse. Therefore, must be less than However, both and are radii of This means that they must have the same length. These two statements contradict each other. Therefore, both cannot be true. The contradiction came from supposing that line was not tangent to Consequently, is a tangent line to the circle. This completes the second part of the proof.
line is tangent to
Having proven both parts completes the proof for the theorem.
Line is tangent to
It has been given that By the Tangent to Circle Theorem, is perpendicular to In other words,
A circumscribed angle is supplementary to the central angle it cuts off.
Considering the above diagram, the following relation holds true.
By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since is a circumscribed angle, and are tangents to at points and respectively. By the Tangent to Circle Theorem, is perpendicular to and is perpendicular to
Find the measure of the central angle.
Two tangents from to are drawn. The measure of is
Find the measure of the inscribed angle that intercepts the same arc as
The inscribed angle that intercepts is By the Inscribed Angle Theorem, is half Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore, is equal to
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.