The inscribed angle of a circle It is the one that has its vertex on the circumference and its rays are secant or tangent to it. As a consequence, the inscribed angle will always be convex or flat..
In figure 1 several angles inscribed in their respective circumferences are represented. The angle ∠EDF is inscribed by having its vertex D on the circumference and its two rays [DE) and [DF) intersecting the circumference.
Similarly, the angle ∠HGI is inscribed, as it has its vertex on the circumference and its sides secant to it.
The angles ∠KJR and ∠UST are also inscribed on the circumference. The first one has a secant side and the other tangent, while the second has its two sides tangent to the circumference, forming a plane inscribed angle (180º).
Some authors call the semi-inscribed angle that one of its sides has tangent to the circumference, but in this article it is considered inscribed..
Every inscribed angle defines or subtends an arc associated with it. For example in figure 2 the inscribed angle ∠ABC subtends the arc A⌒C of length d.
In the same figure the angle ∠DOE is shown, which is not inscribed in the circumference because its vertex does not have its circumference, but in the center O.
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In addition to the inscribed angle, in a circumference the central angle, which is the one whose vertex is in the center of the circumference and whose sides intersect the circumference.
The measure in radians of a central angle is the quotient between the subtending arc, that is, the arc of circumference between the sides of the angle, and the radius of the circumference.
If the circumference is unitary (of radius 1), then the length of the arc in the same units of radius is the measure of the angle in radians.
And when the angle measure in degrees is required, then the measure in radians is multiplied by the factor 180º / π.
Angle measuring instruments always use a central angle and the length of the arc subtended by it is directly calibrated in degrees. This means that whenever an angle is measured, in the background what is measured is the length of the arc subtended by the central angle.
The measure of an inscribed angle is half the measure of the central angle, if both angles subtend the same arc.
In figure 4 two angles ∠ABC and ∠AOC are shown, which intersect the same arc of circumference A⌒C.
If the measure of the inscribed angle is α, then the measure β of the central angle is twice the measure of the inscribed angle (β = 2 α) because both subtend the same arc of measure d.
To prove theorem 1, we will start by showing several particular cases, until we reach the general case.
Let us suppose an inscribed angle, in which one of its sides passes through the center of the circumference, as shown in figure 5.
In this case, the isosceles triangle COB is formed, since [OC] = [OB].
In an isosceles triangle, the angles adjacent to the base are equal, therefore ∠BCO = ∠ABC = α. On the other hand ∠COB = 180º - β.
Considering the sum of the internal angles of the triangle COB, we have:
α + α + (180º - β) = 180º
From which it follows that 2 α = β, or what is equivalent: α = β / 2. This coincides with what theorem 1 states: the measure of the inscribed angle is half the central angle, if both angles subtend the same chord [AC].
In this case we have an inscribed angle ∠ABC, in which the center O of the circumference is within the angle.
To prove Theorem 1 in this case, the auxiliary ray [BO) is drawn, so that we have two inscribed angles ∠ABO and ∠OBC adjacent to said ray.
Similarly we have the central angles β1 and βtwo adjacent to said ray. In this way we have the same situation as in proof 1a, so it can be stated that αtwo = βtwo / 2 and α1 = β1 /two. Since α = α1 + αtwo and β = β1 + βtwo therefore, α = α1 + αtwo = β1 / 2 + βtwo / 2 = (β1 + βtwo) / 2 = β / 2.
In conclusion α = β / 2, which fulfills theorem 1.
If two or more inscribed angles subtend the same arc, then they have the same measure.
The inscribed angles that subtend chords of the same measure are equal.
Show that the inscribed angle that subtends the diameter is a right angle.
The central angle ∠AOB associated with the diameter is a plane angle, whose measure is 180º.
According to theorem 1, every angle inscribed in the circumference that subtends the same chord (in this case the diameter), has as a measure half of the central angle that subtends the same chord, which for our example is 180º / 2 = 90º.
The line (BC) tangent at A to the circumference C, determines the inscribed angle ∠BAC (see figure 10).
Verify that the theorem 1 of the inscribed angles is fulfilled.
The angle ∠BAC is inscribed because its vertex is on the circumference, and its sides [AB) and [AC) are tangent to the circumference, so the definition of inscribed angle is satisfied.
On the other hand, the inscribed angle ∠BAC subtends the arc A ,A, which is the entire circumference. The central angle that subtends the arc A⌒A is a convex angle whose measure is the full angle (360º).
The inscribed angle that subtends the entire arc measures half the associated central angle, that is, ∠BAC = 360º / 2 = 180º.
With all of the above, it is verified that this particular case fulfills theorem 1.
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