The Opposite angles 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 theorem of the angles opposed by the vertex reads like this: two angles opposite by the vertex have the same measure.
Language is often abused by saying that the angles opposite the vertex are equal, which is not correct. Just because 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.
Recall 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 [Of) and the ray [Og) of common origin OR. Figure 1 also shows the angle hOi (red) composed of the ray [I heard) and the ray [Oh) both with origin OR.
Two angles opposed by the vertex are two different geometric figures. To highlight this, in figure 1 the angle has been colored fOg blue, while the angle hOi has been colored red.
The blue and red angles in Figure 1 are opposite at the vertex because: the ray [Of) of the blue angle is the prolongation of the ray [Oh) of the red angle and the ray [Og) of the blue angle is the prolongation of the ray [I heard) of the red angle.
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The geometric figure that consists of two rays with common origin is an angle. The following image shows the angle POQ formed by the two rays [OP) Y [OQ) of common origin OR:
The rays [OP) Y [OQ) are the angle sides POQ, while the common point O is called angle vertex.
Angular sector: An angle divides the plane that contains it into two angular sectors. One of them is the convex angular sector and the other is the concave angular sector. The union of the two sectors gives the complete plane.
Figure 2 shows the angle POQ and its two angular sectors. The convex angular sector is the one with a pointed shape, while the concave is the angular sector of the plane that lacks the convex sector.
Two intersecting lines of a plane form four angles and divide the plane into four angular sectors.
Figure 3 shows the two lines (PQ) Y (RS) that are intercepted in OR. There it can be seen that four angles are determined:
-SOQ, QOR, ROP Y POS
The angles SOQ Y QOR, QOR Y ROP, ROP Y POS, POS Y SOQ They are adjacent angles each other, while SOQ Y ROP they are opposite at the vertex. They are also Opposite angles by the vertex The angles QOR Y POS.
Two secant lines (intersecting lines) are Perpendicular straight lines if they determine four angular sectors of equal measure. If each of the four sectors are symmetric with the adjacent angular sector, then they have the same measure.
Each of the angles that determine the two perpendicular lines is called right angle. All right angles have the same measure.
Given a line and a point on it, two rays are defined. Those two rays define two plane angles.
In figure 3 you can see the line (RS) and the point OR which belongs to (RS). The angle SOR is a plane angle. It can also be stated that the angle ROS is a plane angle. All plane angles have the same measure.
A single ray defines two angles: one of them that of the convex angular sector is the null angle and the other, that of the concave angular sector is the full angle. In figure 3 you can see the null angle SOS and the full angle SOS.
There are two number systems that are frequently used to give the measure of an angle.
One of them is the sexagesimal system, that is, based on the number 60. It is an inheritance of the ancient Mesopotamian cultures. The other system of angle measurement is the radian system, based on the number π (pi) and is a legacy of the ancient Greek sages who developed geometry.
Null angle: in the sexagesimal system the null angle measures 0º (zero degrees).
Full angle: it is assigned the measure 360º (three hundred and sixty degrees).
Plane angle: in the sexagesimal system the plane angle measures 180º (one hundred and eighty degrees).
Right angle: two perpendicular lines divide the plane into four angles of equal measure called right angles. The measure of a right angle is the fourth part of the complete angle, that is to say 90º (ninety degrees).
The protractor is the instrument used to measure angles. It consists of a semicircle (usually clear plastic) divided into 180 angular sections. Since a semicircle forms a plane angle, then the measure between two consecutive sections is 1º.
The goniometer is similar to the protractor and consists of a circle divided into 360 angular sections.
An angle whose sides start from the center of the goniometer intersect two sectors and the measure of that angle in degrees is equal to the number n of sections between the two intercepted sectors, in this case the measure will be nº (it reads “Jan degrees”).
Formally, the theorem is stated this way:
If two angles are vertically opposite, then they have the same measure.
The angle SOQ has measure α; the angle QOR has measure β and angle ROP has measure γ. The sum of the angle SOQ more him QOR form the plane angle SOR measuring 180º.
That is:
α + β = 180º
On the other hand and using the same reasoning with the angles QOR Y ROP you have:
β + γ = 180º
If we observe the two previous equations, the only way that both are fulfilled is that α is equal to γ.
What 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 opposite the vertex have the same measure.
Referring to Figure 4: Suppose that β = 2 α. Find the measure of the angles SOQ, QOR Y ROP in sexagesimal degrees.
As the sum of the angle SOQ more him QOR form the plane angle SOR you have:
α + β = 180º
But they tell us that β = 2 α. Substituting this value of β we have:
α + 2 α = 180º
Namely:
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 like ROP is opposite by the vertex a SOQ then according to the already proven theorem they have the same measure. That is, the measure of ROP is γ = α = 60º.
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