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 it goes 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.
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. Opposite angles 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
[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
colored blue, while the angle
hOi
It has been colored red.
The blue and red angles in Figure 1 are opposite by 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.
Important concepts about angles
Sides and vertices of an angle
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:
Figure 2. The POQ angle defines two angular sectors. Prepared by: F. Zapata.
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. The lines (PQ) and (RS) intersect at O and form 4 angles. Prepared by: F. Zapata.
Figure 3 shows the two lines
(PQ)
Y
(RS)
that are intercepted in
OR
. There it can be seen that four angles are determined:
The angles
SOQ
Y
QOR
,
QOR
Y
ROP, ROP
Y
POS
,
POS
Y
SOQ
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
.
Perpendicular lines and right angle
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.
Rays on the same line and plane angle
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.
Null angle and full angle
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:
the measure 360º (three hundred and sixty degrees) is assigned to it.
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 onefourth of the full angle, that is, 90º (ninety degrees).
Protractor or goniometer
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 intercept 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.
Figure 4. α, β and γ are the measures of the angles SOQ, QOR and ROP. Prepared by: F. Zapata.
The angle
SOQ
has measure α; and
l angle
QOR
has measure β and e
l 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 look at the two previous equations, the only way that both are fulfilled is for α to be 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
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
it is α = 60º.
The measure of
QOR
is β = 2 α = 2 * 60º = 120º.
Finally like
ROP
is opposite by 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.

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