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 flat angle, in degrees is 180º and in radians 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.

Also in that same figure, there are angles α and γ, which 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, being both obtuse angles, their measures are greater than 90º and therefore their sum exceeds 180º.

On the other hand, it can be affirmed 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.

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## examples

In the following examples, you are asked to find the unknown angles, marked with a question mark in figure 2. They range from the simplest examples to some a little more elaborate that the reader should be more careful with.

### 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 to the 50º angle, then it follows that β = 180º – 50º = 130º.

### Example C

Figure 2C shows the following sum: γ + 90º + 15º = 180º. That is to say, that γ is supplementary with the angle 105º = 90º + 15º. It is then concluded that:

γ = 180º- 105º = 75º

### Example D

Since X is supplementary with 72º, it follows that X = 180º – 72º = 108º. Also Y is supplementary to 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 U the angle that is between 100º and 50º, then U is supplementary to both of them, because it is observed that their sum completes a plane angle.

It follows immediately that U = 150º. Since U is opposite W by the vertex, 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 the resolution of all of them.

### – Exercise I

Determine the values of angles A and B in part I) of figure 3.

**Solution**

A and B are supplementary, from where we have that A + B = 180 degrees, then se substitutes the expression of A and B 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, immediately group the terms:

6 x + 60 = 180

Dividing both members by 6 we have:

x + 10 = 30

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

Now we must substitute the value of x to find the requested angles. From there we have that 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 in part II) of figure 3.

**Solution**

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

(-2x + 90) + (8x – 30) = 180

Once 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 where it follows that x is worth 20º.

That is to say that the angle A = -2*20 + 90 = 50º. While angle B = 8*20 – 30 = 130º.

### – Exercise III

Determine the values of angles A and B in 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 of 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 isolate the value of x, we have:

x + 5 = 15

Finally it is found that x is equal to 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

Two parallel lines cut by a transversal 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 supplementarywhich we list below:

The exterior angles A and B, and the exterior angles G and H

Interior angles D and C, and interior angles E and F

The exterior angles A and G, and the exterior angles B and H

Interior angles D and E, and interior angles C and F

For completeness, the angles equal to each other:

Internal alternates: D = F and C = E

The outer alternates: A = H and B = G

The corresponding ones: A = E and C = H

Vertex opposites A = C and E = H

The corresponding ones: B = F and D = G

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 angle A = π/6 radians.

**Solution**

A and B are supplementary exterior angles therefore B = π – A = π – π/6 = 5π/6

A = E = C = H = π/6

B = F = D = G =5π/6

## References

Baldor, JA 1973. Plane and space geometry. Central American culture.

Laws and mathematical formulas. Angle measurement systems. Recovered from: ingemecanica.com.

Wentworth, G. Plane Geometry. Retrieved from: gutenberg.org.

Wikipedia. Supplementary Angles. Recovered from: en.wikipedia.com

Wikipedia. Conveyor. Recovered from: en.wikipedia.com

Zapata F. Goniometer: history, parts, operation. Retrieved from: .com