CONJUGATE INTERNAL AND EXTERNAL ANGLES: EXAMPLES, EXERCISES - MATHS - 2024

The angles conjugates are those added to the results to be 360, regardless of said angles are adjacent or not. Two conjugate angles are shown in Figure 1, denoted α and β.

In this case, the angles α and β in the figure have a common vertex and their sides are common, therefore they are adjacent. The relationship between them is expressed as follows:

α + β = 360º

Figure 1. Two conjugated central angles, sum. Source: Wikimedia Commons. No machine-readable author provided. Thiago R Ramos assumed (based on copyright claims). It is a classification of the angles by their sum. Other important definitions include complementary angles, whose sum is 90º, and supplementary angles, which total 180º.

On the other hand, let us now consider two parallel lines cut by a secant, whose arrangement is shown below:

Figure 2. Parallel lines cut by a secant. Source: F. Zapata.

The lines MN and PQ are parallel, while the line RS is secant, intersecting the parallels at two points. As can be seen, this configuration determines the formation of 8 angles, which have been denoted with lowercase letters.

Well, according to the definition given at the beginning, the angles a, b, c and d are conjugated. And in the same way e, f, g and h are, since both cases are true:

a + b + c + d = 360º

AND

e + f + g + h = 360º

For this configuration, two angles are conjugated if they are on the same side with respect to the secant line RS and both are internal or external. In the first case we speak of internal conjugate angles, while in the second, they are external conjugate angles.

Examples

In figure 2, the external angles are those that are outside the region delimited by the lines MN and PQ, they are the angles A, B, G and H. While the angles that lie between the two lines are C, D, E and F.

Now it is necessary to analyze which angles are to the left and which to the right of the secant.

To the left of RS are angles A, C, E, and G. And to the right are angles B, D, F, and H.

We immediately proceed to determine the conjugate angle pairs, according to the definition given in the previous section:

-A and G, external and to the left of RS.

-D and F, internal and to the right of RS.

-B and H, external and to the right of RS.

-C and E, internal and to the left of RS.

Property of conjugated angles between parallel lines

The conjugated angles between parallel lines are supplementary, that is, their sum is equal to 180º. In this way, for figure 2 the following is true:

A + G = 180º

D + F = 180º

B + H = 180º

C + E = 180º

The pairs of corresponding angles for parallel lines

They are those that are on the same side of the secant line, they are not adjacent and one of them is internal and the other is external. It is important to visualize them, since their measure is the same, because they are opposite angles by the vertex.

Returning to figure 2, the corresponding pairs of angles are identified as:

-A and E

-C and G

-B and F

-D and H

Internal angles of a quadrilateral

Quadrilaterals are 4-sided polygons, among them the square, the rectangle, the trapezoid, the parallelogram and the rhombus, for example. Regardless of their shape, in any of them it is true that the sum of their internal angles is 360º, therefore they meet the definition given at the beginning.

Let's see some examples of quadrilaterals and how to calculate the value of their internal angles according to the information in the preceding sections:

Examples

a) Three of the angles of a quadrilateral measure 75º, 110º and 70º. How much should the remaining angle measure?

b) Find the value of the angle ∠Q in figure 3 i.

c) Calculate the measure of the angle ∠A in figure 3 ii.

Solution to

Let α be the missing angle, it is satisfied that:

α + 75º + 110º + 70º = 360º → α = 105º

Solution b

Figure 3i shown is a trapezoid and two of its internal angles are right, which have been marked with a colored square at the corners. For this quadrilateral the following is verified:

∠R + ∠S + ∠P + ∠Q = 360º; ∠S = ∠R = 90 °; ∠P = 60º

Thus:

∠ Q = 2 x 90º + 60º = 240º

Solution c

The quadrilateral in figure 3 ii is also a trapezoid, for which the following is true:

∠A + ∠B + ∠C + ∠D = 360º

Thus:

4x -5 + 3x + 10 +180 = 360

7x + 5 = 180

x = (180 - 5) / 7

x = 25

To determine the angle requested in the statement, we use that ∠A = 4x - 5. Substituting the previously calculated value of x it follows that ∠A = (4 × 25) -5 = 95º

Exercises

- Exercise 1

Knowing that one of the angles shown is 125º, find the measures of the 7 remaining angles in the following figure and justify the answers.

Figure 4. The lines and angles of exercise 1. Source: F. Zapata.

Solution

Angle 6 and angle 125º are internal conjugates, whose sum is 180º, according to the property of conjugated angles, therefore:

∠6 + 125º = 180º → ∠6 = 180º - 125º = 55º

On the other hand ∠6 and ∠8 are opposite angles by the vertex, whose measure is the same. Therefore ∠8 measures 55º.

The angle ∠1 is also opposite by the vertex at 125º, then we can affirm that ∠1 = 125º. We can also appeal to the fact that the corresponding pairs of angles have the same measure. In the figure these angles are:

∠7 = 125 º

∠2 = ∠6 = 55 º

∠1 = ∠5 = 125º

∠4 = ∠8 = 55 º

- Exercise 2

Find the value of x in the following figure and the values ​​of all the angles:

Figure 5. Lines and angles for exercise 2. Source: F. Zapata.

Solution

Since they are corresponding pairs, it follows that F = 73º. And on the other hand the sum of the conjugated pairs is 180º, therefore:

3x + 20º + 73º = 180º

3x = 180º - 73º -20º = 87

Finally the value of x is:

x = 87/3 = 29

As for all the angles, they are listed in the following figure:

Figure 6. Angles resulting from exercise 2. Source: F. Zapata.

References

1. Angle Groups. Complementary, Supplementary and Explementary Angles Explanation. Recovered from: thisiget.com/
2. Baldor, A. 1983. Plane and Space Geometry and Trigonometry. Patria Cultural Group.
3. Corral, M. Mathematics LibreTexts: Angles. Recovered from: math.libretexts.org.
4. Mathmania. Classifying and constructing angles by their measurement. Recovered from: mathemania.com/
5. Wentworth, G. Plane Geometry. Recovered from: gutenberg.org.
6. Wikipedia. Conjugate angles. Recovered from: es.wikipedia.org.