SUPPLEMENTARY ANGLES: WHAT ARE THEY, CALCULATION, EXAMPLES, EXERCISES - MATHS - 2024

Two or more are supplementary angles if the sum of their measures corresponds to the measure of a straight angle. The measure of a straight angle, also called a plane angle, in degrees is 180º and in radians it is π.

For example, we find that the three interior angles of a triangle are supplementary, since the sum of their measures is 180º. Three angles are shown in Figure 1. From the above it follows that α and β are supplementary, since they are adjacent and their sum completes a straight angle.

Figure 1: α and β are supplementary. α and γ are supplementary. Source: F. Zapata.

Also in the same figure, we have the angles α and γ that are also supplementary, because the sum of their measures is equal to the measure of a plane angle, that is, 180º. It cannot be said that the angles β and γ are supplementary because, as both angles are obtuse, their measures are greater than 90º and therefore their sum exceeds 180º.

Source: lifeder.com

Instead it can be stated that the measure of angle β is equal to the measure of angle γ, since if β is supplementary to α and γ is supplementary to α, then β = γ = 135º.

Examples

In the following examples it is asked to find the unknown angles, indicated with question marks in figure 2. They range from the simplest examples to some a little more elaborate that the reader should be more careful.

Figure 2. Several worked-out examples of supplementary angles. Source: F. Zapata.

Example A

In the figure we have that the adjacent angles α and 35º add up to a plane angle. That is, α + 35º = 180º and therefore it is true that: α = 180º- 35º = 145º.

Example B

Since β is supplementary with the angle of 50º, then it follows that β = 180º - 50º = 130º.

Example C

From figure 2C the following sum can be observed: γ + 90º + 15º = 180º. That is, γ is supplementary with the angle 105º = 90º + 15º. It is concluded then that:

γ = 180º- 105º = 75º

Example D

Since X is supplementary to 72º, it follows that X = 180º - 72º = 108º. Furthermore Y is supplementary with X, so Y = 180º - 108º = 72º.

And finally Z is supplementary with 72º, therefore Z = 180º - 72º = 108º.

Example E

The angles δ and 2δ are supplementary, therefore δ + 2δ = 180º. Which means that 3δ = 180º, and this in turn allows us to write: δ = 180º / 3 = 60º.

Example F

If we call the angle between 100º and 50º U, then U is supplementary to both of them, because it is observed that their sum completes a plane angle.

It immediately follows that U = 150º. Since U is opposite by the vertex to W, then W = U = 150º.

Exercises

Three exercises are proposed below, in all of them the value of angles A and B must be found in degrees, so that the relationships shown in figure 3 are fulfilled. The concept of supplementary angles is used in solving all of them.

Figure 3. Figure to solve exercises I, II and III on supplementary angles. All angles are in degrees. Source: F. Zapata.

- Exercise I

Determine the values ​​of angles A and B from part I) of Figure 3.

Solution

A and B are supplementary, from which we have that A + B = 180 degrees, then the expression of A and B is substituted as a function of x, as it appears in the image:

(x + 15) + (5x + 45) = 180

A first order linear equation is obtained. To solve it, the terms are grouped below:

6 x + 60 = 180

Dividing both members by 6 we have:

x + 10 = 30

And finally solving, it follows that x is worth 20º.

Now we must plug in the value of x to find the requested angles. Hence, the angle A is: A = 20 +15 = 35º.

And for its part, angle B is B = 5 * 20 + 45 = 145º.

- Exercise II

Find the values ​​of angles A and B from Part II) of Figure 3.

Solution

Since A and B are supplementary angles, we have that A + B = 180 degrees. Substituting the expression for A and B as a function of x given in part II) of figure 3, we have:

(-2x + 90) + (8x - 30) = 180

Again a first degree equation is obtained, for which the terms must be conveniently grouped:

6 x + 60 = 180

Dividing both members by 6 we have:

x + 10 = 30

From which it follows that x is worth 20º.

In other words, the angle A = -2 * 20 + 90 = 50º. While angle B = 8 * 20 - 30 = 130º.

- Exercise III

Determine the values ​​of angles A and B from part III) of Figure 3 (in green).

Solution

Since A and B are supplementary angles, we have that A + B = 180 degrees. We must substitute the expression for A and B as a function of x given in figure 3, from which we have:

(5x - 20) + (7x + 80) = 180

12 x + 60 = 180

Dividing both members by 12 to solve for the value of x, we have:

x + 5 = 15

Finally it is found that x is worth 10 degrees.

Now we proceed to substitute to find the angle A: A = 5 * 10 -20 = 30º. And for angle B: B = 7 * 10 + 80 = 150º

Supplementary angles in two parallels cut by a secant

Figure 4. Angles between two parallels cut by a secant. Source: F. Zapata.

Two parallel lines cut by a secant is a common geometric construction in some problems. Between such lines, 8 angles are formed as shown in figure 4.

Of those 8 angles, some pairs of angles are supplementary, which we list below:

1. The exterior angles A and B, and the exterior angles G and H
2. The interior angles D and C, and the interior angles E and F
3. The exterior angles A and G, and the exterior angles B and H
4. The interior angles D and E, and the interiors C and F

For completeness, the angles equal to each other are also named:

1. The internal alternates: D = F and C = E
2. The external alternates: A = H and B = G
3. The corresponding ones: A = E and C = H
4. The opposites by the vertex A = C and E = H
5. The corresponding ones: B = F and D = G
6. Vertex opposites B = D and F = G

- Exercise IV

Referring to Figure 4, which shows the angles between two parallel lines cut by a secant, determine the value of all angles in radians, knowing that the angle A = π / 6 radians.

Solution

A and B are supplementary external angles so B = π - A = π - π / 6 = 5π / 6

A = E = C = H = π / 6

B = F = D = G = 5π / 6

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. Wentworth, G. Plane Geometry. Recovered from: gutenberg.org.
4. Wikipedia. Supplementary angles. Recovered from: es.wikipedia.com
5. Wikipedia. Conveyor. Recovered from: es.wikipedia.com
6. Zapata F. Goniómetro: history, parts, operation. Recovered from: lifeder.com