While dealing with Mathematics, a normal brain will be confused in its initial stages about what complementary and supplementary angles are. Therefore, it is important to learn in-depth about complementary supplementary angles. Complementary angles might be sounding like we are complementing an angle, right? But, no! The complementary angles definition is different when it is learned under Mathematics. So what exactly do complementary supplementary angles mean? Let us understand more about them.
Complementary Angles: More than complementing angles
We have arrived at the point where we will define complementary angles? One common question in everyone’s mind – what are complementary angles? In that case, complementary angles are those whose sum is equal to 90 degrees. Yes, exactly 90 degrees. In other terms, angles making a right angle can be termed complementary angles. The word complementary comes from a Latin word, completum, meaning completed. So, can we say a right angle is a complete angle? Think it over.
More fascinating though is, cutting a slice of bread into two triangles will yield two right angles having a pair each of complementary angles. So it is clear why the right angle is known as a complete angle.
Do you refer to Merriam Webster while looking for any definition of an unknown word or phrase? For example, while looking for complementary angles, one will see complementary air and complementary cell. That is not what we want, right? Yet, the look-up popularity in Merriam Webster for complementary angles is top 7% of words. How fascinating is that!
According to Merriam Webster, the definition of the complementary angle is two angles that add up to 90 degrees. Moreover, be careful while spelling the word. Many make mistakes and spell complementary instead of complementary. The former is incorrect, while the latter is correct.
Consider the figure as shown. On the left side, the angle AXB is a right angle, i.e., the value of angle AXB is equal to 90 degrees. On the right-hand side, angles BXC and CXD make a right angle when joined together. When these angles are considered separately, they are referred to as complementary angles.
As a result, angle BXC and angle CXD are complementary angles or complement each other. Speak like this: angle BXC complements angle CXD and vice versa. Mathematically, if angle BXC plus angle CXD is equal to 90 degrees, they are complementary.
Finding complementary angles is not difficult anymore!
Can we determine which one is a complementary angle and which one is not? Not so easily. After knowing the definition of the complementary angle, let us look at how to determine the complementary angles. We can measure the angles in a complementary angle with the help of a protractor. If the angle is less than 90 degrees, we can say that the angle is complementary. If it’s greater than 90, then? Need to wait for a little while when we jump to supplementary angles.
Are complementary angles always joined with each other?
Some people may try to figure out whether complementary angles are always joined with each other or not. What do you think? Are they always joined with each other? Two types of complementary angles are known in geometry: Non-adjacent and adjacent complementary angles. What are they? Continue reading further.
Adjacent complementary angles
As the term adjacent denotes, these complementary angles must be near each other. The angles joined with a common vertex/common arm or are a part of the same right angle are referred to as adjacent complementary angles.
Consider the figure demonstrated below. Since both the angles BOC and BOA have a common vertex O and both make a right angle, they are said to be adjacent complementary angles. Why a right angle? Because 70 + 20 = 90 degrees, a right angle!
Non-adjacent complementary angles
Since we know about complementary angles, we can determine what non-adjacent complementary angles mean. Those angles whose vertex is not common and make up a right angle are non-adjacent complementary angles. Simple to understand.
Let us understand non-adjacent complementary angles by a pictorial representation. As it can be seen, two angles, ABC and QPR, add up to 90 degrees and make a right angle. How? 40 + 50 = 90 degrees. However, these angles do not have the same vertex. Can we still refer to them as complementary? Well, yes! Non-adjacent complementary angles do not have a common vertex but add up, making a right angle.
Finding the complement of a complementary angle
Since now we know what are complementary angles, can we notice a common point in the definitions we studied above? Right angle and 90 degrees! The entire concept of complementary angles revolves around the right angle. So, to find the complement of a complementary angle, one has to subtract the known value of the angle from 90. The resulting value will be the complement of the complementary angle.
Complementary Angle Theorem: Special case of complementary angles
After learning what are complementary angles, time to know a special case of complementary angles. According to the complementary angle theorem, the angles will be congruent if they are complementary.
Consider the illustration below. From this, if we say angle POA and angle POQ are complementary angles, and angle POQ and angle QOR are complementary angles, then due to a common angle, we can say angle POA is equal to QOR. And that is the proof of the complementary angle theorem.
Mathematically, angle POA + angle POQ = angle POA + angle QOR.
Therefore, angle POQ = angle QOR, according to the complementary angle theorem.
Supplementary Angles: Straight and non-complementary
Suppose people were confused between complementary supplementary angles, and they knew one deal with 90 degrees and the other with 180 degrees. In that case, it might be clear by now that supplementary angles deal with 180 degrees. Therefore, if the sum of two angles equals 180 degrees, i.e., the angle is straight or formed as a line, it is a supplementary angle.
The term supplementary angles were first used in 1924. The look-up popularity in Merriam Webster is top 6%. One might have trouble while learning complementary and supplementary angles. Rule of thumb is complementary means corner, focus on the letter C, whereas supplementary is straight, focus on the word S. And that’s it! As easy as it is!
Let us understand it with the help of an illustration. In the figure below, angles AXD and CXD are supplementary because they form 180 degrees, i.e., a straight line. Similarly, angles BXC and CXD are supplementary. Can you figure out the other two pairs of supplementary angles?
One must notice the special case in this figure. Angles BXC and AXD will be equal to each other. And angels BXA and CXD will be equal to each other. How? That is the property of supplementary angles. If they intersect at one common point, the alternate angles will be equal. This is a special property used in Mathematics while solving supplementary angle questions. Check the angles by yourself using a protractor.
Supplementary angle types: Adjacent and non-adjacent
After going through the adjacent and non-adjacent complementary angles in complementary angles definition, one can determine what adjacent and non-adjacent supplementary angles will be. For example, if two angles have a common arm or vertex and add up to 180 degrees, they are adjacent supplementary angles. Whereas, if they do not have a common arm or vertex and still add up to 180 degrees, they are referred to as non-adjacent supplementary angles.
Check the demonstrated images below to figure out adjacent and non-adjacent supplementary angles.
Adjacent supplementary angle Non-adjacent supplementary angle
Finding the supplement of a supplementary angle
Have you figured out how to find the supplement of an angle? If yes, it’s quite simple, no? The way complementary angles deal with 90 degrees, the same way supplementary angles revolve around 180 degrees. Therefore, to find the supplement of a supplementary angle, subtract the known angle from 180, and the leftover value will be the supplement.
Time to practice complementary and supplementary angles
Example 1: What will be the value of complementary angles P and Q if they are (3x – 10)° and (3x – 50)°, respectively?
Solution: Since two angles, P and Q, are complementary; therefore, according to complementary angles definition their sum will be equal to 90 degrees.
angle P + angle Q = 90°
3x – 10 + 3x – 50 = 90
On solving the above equation, we get, x = 25.
Therefore, the value of angles P and Q = (3(25) – 10)° and (3(25) – 50)° = 65° and 25°, respectively.
Example 2: Find the value of angles X and Y if their value is (4x – 80) and (6x – 45), respectively. Given that X and Y are supplementary angles.
Solution: Since the given angles X and Y are supplementary; therefore, their sum will be equal to 180°.
angle X + angle Y = 180°
4x – 80 + 6x – 45 = 180
Solving the above equation, we get, x = 30.5.
Therefore, angle X = 4(30.5) – 80 = 42°, and angle Y = 6(30.5) – 45 = 138°.
Example 3: What is the value of the other angle as shown in the figure?
Solution: Since the given angle is a part of a right angle; therefore, the other angle must be 90 – 79 = 11 degrees.