The angles opposed by the vertex are those that fulfill the following: the sides of one of them are the extensions of the sides of the other angle. The Fundamental Vertex Angles Theorem goes like this: Two vertex angles have the same measure.
Language is often abused by saying that the angles opposite the vertex are equal, which is not correct. The fact that two angles have the same measure does not mean that they are equal. It is like saying that two children who are the same height are equal.
Figure 1. Angles opposed by the vertex. Prepared by: Fanny Zapata.
Remember that an angle is defined as the geometric figure composed of two rays with the same origin.
Figure 1 shows the angle fOg (blue) composed of the ray ).push ({});
Vertex Angles Theorem
Original text
Formally, the theorem is stated this way:
Figure 4. α, β and γ are the measures of the angles SOQ, QOR and ROP. Prepared by: F. Zapata.
Demonstration
The angle SOQ has measure α; angle QOR has measure β and angle ROP has measure γ. The sum of the angle SOQ plus the QOR forms the plane angle SOR of measure 180º.
That is:
α + β = 180º
On the other hand and using the same reasoning with the angles QOR and ROP, we have:
β + γ = 180º
If we look at the two previous equations, the only way that they both hold is for α to be equal to γ.
Since SOQ has measure α and is opposite by the vertex to ROP of measure γ, and since α = γ, it is concluded that the angles opposed by the vertex have the same measure.
Exercise resolved
Referring to Figure 4: Suppose that β = 2 α. Find the measure of the angles SOQ, QOR, and ROP in sexagesimal degrees.
Solution
Since the sum of the angle SOQ plus the QOR forms the plane angle SOR, we have:
α + β = 180º
But they tell us that β = 2 α. Substituting this value of β we have:
α + 2 α = 180º
That is to say:
3 α = 180º
Which means that α is the third part of 180º:
α = (180º / 3) = 60º
Then the measure of SOQ is α = 60º. The measure of QOR is β = 2 α = 2 * 60º = 120º. Finally, as ROP is opposite by the vertex to SOQ then according to the theorem already proven they have the same measure. That is, the measure of ROP is γ = α = 60º.
References
- Baldor, JA 1973. Plane and Space Geometry. Central American Cultural.
- Mathematical laws and formulas. Angle measurement systems. Recovered from: ingemecanica.com.
- Wikipedia. Opposite angles by the vertex. Recovered from: es.wikipedia.com
- Wikipedia. Conveyor. Recovered from: es.wikipedia.com
- Zapata F. Goniómetro: history, parts, operation. Recovered from: lifeder.com