The inscribed angle of a circle is one that has its vertex on the circle 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 =.
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 agrees with what theorem 1 states: the measure of the inscribed angle is half the central angle, if both angles subtend the same chord.
Demonstration 1b
Figure 6. Auxiliary construction to show that α = β / 2. Source: F. Zapata with Geogebra.
In this case we have an inscribed angle ∠ABC, in which the center O of the circle is within the angle.
To prove Theorem 1 in this case, draw the auxiliary ray).push ({});
Similarly, the central angles β 1 and β 2 are adjacent to said ray. Thus we have the same situation as show 1a, so can be said that α 2 = β 2 /2 and α 1 = β 1 /2. As α = α 1 + α 2 and β = β 1 + β 2 have therefore that α = α 1 + α 2 = β 1 /2 + β 2 /2 = (β 1 + β 2) / 2 = β / two.
In conclusion α = β / 2, which fulfills theorem 1.
- Theorem 2
Figure 7. Inscribed angles of equal measure α, because they subtend the same arc A⌒C. Source: F. Zapata with Geogebra.
- Theorem 3
The inscribed angles that subtend chords of the same measure are equal.
Figure 8. Inscribed angles that subtend chords of equal measure have equal measure β. Source: F. Zapata with Geogebra.
Examples
- Example 1
Show that the inscribed angle that subtends the diameter is a right angle.
Solution
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.
Figure 9. Every inscribed angle that subtends to diameter is a right angle. Source: F. Zapata with Geogebra.
- Example 2
The line (BC) tangent at A to the circumference C, determines the inscribed angle ∠BAC (see figure 10).
Verify that Theorem 1 of the inscribed angles is fulfilled.
Figure 10. Inscribed angle BAC and its central convex angle AOA. Source: F. Zapata with Geogebra.
Solution
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.
References
- Baldor. (1973). Geometry and trigonometry. Central American cultural publishing house.
- EA (2003). Geometry elements: with exercises and compass geometry. University of Medellin.
- Geometry 1st ESO. Angles on the circumference. Recovered from: edu.xunta.es/
- All Science. Proposed exercises of angles in the circumference. Recovered from: francesphysics.blogspot.com
- Wikipedia. Inscribed angle. Recovered from: es.wikipedia.com