Angles Opposed By The Vertex (with Solved Exercise)

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.

Article index

  • one

    Important concepts about angles

    • 1.1

      Sides and vertices of an angle

    • 1.2

      Angles formed by two lines that intersect

  • two

    Perpendicular lines and right angle

    • 2.1

      Rays on the same line and plane angle

    • 2.2

      Null angle and full angle

  • 3

    Angle measurement 

    • 3.1

      Sexagesimal system

  • 4

    Vertex Angles Theorem

    • 4.1

      Demonstration

  • 5

    Exercise resolved

    • 5.1

      Solution

  • 6

    References

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.

Angles formed by two lines that intersect

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:

-SOQ

,

QOR

,

ROP

Y

POS

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

Angle measurement 

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.

Sexagesimal system

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 one-fourth 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

”).

Vertex Angles Theorem

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.

Demonstration

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.

Exercise resolved

Referring to Figure 4: Suppose that β = 2 α. 
Find the measure of the angles

SOQ

,

QOR

Y

ROP

in sexagesimal degrees.

Solution

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. 

References

  1. Baldor, JA 1973. Plane and Space Geometry. Central American Cultural. 
  2. Mathematical laws and formulas. Angle measurement systems. Recovered from: ingemecanica.com.
  3. Wikipedia. Opposite angles by the vertex. Recovered from: es.wikipedia.com
  4. Wikipedia. Conveyor. Recovered from: es.wikipedia.com
  5. Zapata F. Goniómetro: history, parts, operation. Recovered from: lifeder.com

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