Complementary Angles
Introduction
In geometry, angles are formed when two rays share a common endpoint, called the vertex. When the combined measurement of two angles equals `90` degrees, they are called complementary angles.
If you add up the measurements of these two angles, you'll always get `90` degrees. For example, let's say we have one angle that measures `30°`. Its complementary angle would then be `60°` because when you add `30°` and `60°` together, you get `90°`. So, in this case, `30°` and `60°` are complementary angles.
In real-life scenarios, complementary angles can be found in various scenarios. For instance, consider a corner of your room. The two walls meet at `90` degrees in the corner. One way to think about it is that they fit together perfectly to make a right angle.
What are Complementary Angles?
Complementary angles are a pair of angles that, when added together, equal `90°`. In simpler terms, if you have `∠1` and `∠2`, and their measures sum up to `90°`, then `∠1` and `∠2` are considered complementary. We refer to `∠1` and `∠2` as each other's complements based on this relationship.
For example, consider two angles, one measuring \(55^\circ\) and the other \(35^\circ\). When you add \(55^\circ\) to \(35^\circ\), the total measure becomes \(90^\circ\). Thus, these two angles are complementary angles.
These angles can be part of the same shape or different shapes altogether. They don't have to be next to each other or facing the same way.
Types of Complementary Angles
Complementary angles can be categorized into two types:
- Adjacent Complementary Angles
- Non-Adjacent Complementary Angles
Each type has its characteristics, explained below.
Adjacent Complementary Angles
Adjacent complementary angles are two angles that share a common vertex and a common arm. In simpler terms, they are angles that are next to each other and share one side.
Consider the illustration below, where we have two angles, `∠XYZ` and `∠YZA`, with a common vertex at point `Y` and a common arm `YZ`. `∠XYZ` measures `30°` and `∠YZA` measures `60°`. Their measurements add up to `90` degrees. Hence, they are considered adjacent complementary angles. They are adjacent complementary angles because `30° + 60°` equals `90°`.
Non-adjacent Complementary Angles
Non-adjacent complementary angles are two angles that do not share a common vertex or a common arm. In other words, they are not next to each other and don't have any sides in common.
In the figure given below, `∠PQR` and `∠STU` are non-adjacent angles. They do not share a common vertex or a common side—however, their measures, when added, result in `90°`. For instance, `∠PQR` measures `40°` and `∠STU` measures `50°`, leading to a total of `90°`. Thus, these angles, `∠PQR` and `∠STU`, are termed as non-adjacent complementary angles. When placed together, they form a right angle.
It is interesting to note that in a right-angled triangle, the two acute angles are complementary. This means that their sum equals `90` degrees, making them a perfect example of non-adjacent complementary angles.
Understanding these types of complementary angles helps us classify and solve problems involving angles more efficiently.
How to Find the Complement of an Angle?
To find the complement of an angle, you subtract the angle's measurement from `90°`. Since complementary angles add up to `90` degrees, each angle is like the missing piece that completes the other.
Example: Find the complement of `65°`.
Solution:
Subtract `65°` from `90°`:
`90° - 65° = 25°`
Thus, the complement of `65°` is `25°`.
An interesting scenario is when you have an angle that is equal to its complement. For instance, let's find such an angle.
Example: Find the angle which is equal to its complement.
Solution:
Let the required angle be \( y \). Then, its complement is \( (90 - y) \).
If the angle is equal to its complement, we can set up the equation as \( y = 90 - y \).
Solving this equation,
\( 2y = 90 \)
\( y = 45 \)
Hence, the angle which is equal to its complement is \( 45 \)°.
Complementary Angles vs Supplementary Angles
Complementary angles and supplementary angles both involve pairs of angles that have specific relationships with each other.
Complementary Angles
Complementary angles are two angles that when added together, result in `90°`. They can either be adjacent or non-adjacent. These angles essentially complete each other to form a right angle.
Supplementary Angles
Supplementary angles, on the other hand, are two angles whose sum equals `180°`. They can also be adjacent or non-adjacent. When combined, they form a straight line or a straight angle.
Trick to Remember
To differentiate between complementary and supplementary angles, remember:
"`S`" for "Supplementary" equals "Straight" angle
"`C`" for "Complementary" equals "Corner" (right) angle.
Complementary Angles Theorem
The complementary angles theorem states that "Complements of the same angle are congruent." In simpler terms, if two angles are complementary to the same angle, they are congruent to each other. This means that if one angle is complementary to a given angle, then any other angle that is also complementary to that given angle must be equal in measure to the first complementary angle.
Let’s say we have `3` angles `∠1, ∠2,` and `∠3`.
`∠1` has a measure of `62°` and is complementary to `∠2` and `∠3`.
`∠2` has a measure of `28°` and is complementary to `∠1`.
`∠3` has an unknown measure and is complementary to `∠1`.
We don’t know the measure of `∠1`, but we know that `∠2` and `∠3` are both complementary to `∠1`. As per the complementary angles theorem, `∠3` is congruent to `∠2` and must also have a measure of `28°`.
Solved Examples
Example `1`: If the measure of angle `XYZ` is `40` degrees, what is the measure of its complement?
Solution:
The measure of angle `XYZ = 40` degrees
Complement angle `= 90° -` Measure of `∠XYZ`
`= 90° - 40°`
`= 50°`
The complement of angle `XYZ` is `50` degrees.
Example `2`: Are the angles measuring `30` degrees and `60` degrees complementary or supplementary?
Solution:
Measure of first angle `= 30°`, Measure of second angle `= 60°`
Since `30° + 60° = 90°`, the angles are complementary.
The angles measuring `30` degrees and `60` degrees are complementary.
Example `3`: If the ratio of two complementary angles is `3:6`, what is the measure of each angle?
Solution:
Let's denote the smaller angle as \( 3x \) and the larger angle as \( 6x \) since their ratio is `3:6`.
Since the angles are complementary, their sum is equal to \( 90^\circ \):
\( 3x + 6x = 90^\circ \)
Combine like terms:
\( 9x = 90^\circ \)
Divide both sides by `9` to find the value of \( x \):
`x = \frac{90^\circ}{9}`
\( x = 10^\circ \)
Now that you know \( x \) is \( 10^\circ \), find the larger angle by multiplying \( x \) by `6`:
\( 6x = 6 \times 10^\circ = 60^\circ \)
Next, find the smaller angle by multiplying \( x \) by `3`:
\( 3x = 3 \times 10^\circ = 30^\circ \)
Therefore, the smaller angle measures \( 30^\circ \), and the larger angle measures \( 60^\circ \).
Example `4`: \( (4x + 1)^\circ \) and \( (9 + x)^\circ \) are two complementary angles. Find the measures of each angle.
Solution:
Given that the angles are complementary, their measures add up to \(90^\circ\).
So, we can write the equation:
\( (4x + 1) + (9 + x) = 90 \)
Now, let's solve for \(x\):
\( 4x + 1 + 9 + x = 90 \)
\( 5x + 10 = 90 \)
Subtract \(10\) from both sides:
\( 5x = 80 \)
Divide both sides by \(5\):
\( x = 16 \)
Now that we have found \(x\), we can find the measures of the angles:
Let's denote the first angle as \( \text{Angle A} = 4x + 1 \) degrees:
\( \text{Angle A} = 4(16) + 1 = 64 + 1 = 65^\circ \)
Let's denote the second angle as \( \text{Angle B} = 9 + x \) degrees:
\( \text{Angle B} = 9 + 16 = 25^\circ \)
Therefore, the measure of the two angles are \(65^\circ\) and \(25^\circ\).
Example `5`: Jack is building a ramp for his skateboard. If the angle of inclination of the ramp measures `25` degrees, what is the measure of its complementary angle?
Solution:
Angle of inclination `= 25°`
Complement angle `= 90° -` Angle of inclination
`= 90° - 25°`
`= 65°`
The measure of the complementary angle is `65°`.
Practice Problems
Q`1`. Which of the following angles is complementary to a `50°` angle?
- `30°`
- `60°`
- `40°`
- `70°`
Answer: c
Q`2`. \((4x +7)^\circ\) and \( (3x - 22)^\circ \) are two complementary angles. Find the measure of the angle \((4x + 7)^\circ\).
- `38°`
- `67°`
- `62°`
- `85°`
Answer: b
Q`3`. Which pair of angles are complementary?
- `45°` and `45°`
- `90°` and `100°`
- `30°` and `60°`
- `80°` and `70°`
Answer: a
Q`4`. Two angles are complementary. One is triple the size of the other. What is the size of the bigger angle?
- `60°`
- `30.5°`
- `45°`
- `67.5°`
Answer: d
Q`5`. Given that \( 38^\circ \) and \( (5x + 2)^\circ \) form a pair of complementary angles, find the value of `x`.
- `15`
- `20`
- `10`
- `30`
Answer: c
Frequently Asked Questions
Q`1`. What are complementary angles?
Answer: Complementary angles are two angles whose measures add up to `90` degrees. In other words, when you add the measures of complementary angles together, the result is always `90°`.
Q`2`. How do I find the complement of an angle?
Answer: To find the complement of an angle, subtract its measure from `90°`. The resulting value is the measure of the complement angle.
Q`3`. Can three angles be complementary?
Answer: No, three angles cannot be complementary. Complementary angles are always a pair of two angles whose measures add up to `90` degrees.
Q`4`. What is the difference between complementary and supplementary angles?
Answer: Complementary angles add up to `90°`, while supplementary angles add up to `180°`. Complementary angles complete a right angle, whereas supplementary angles form a straight line.
Q`5`. Are complementary angles always acute?
Answer: Yes, complementary angles are always acute angles, meaning their measures are less than `90°`. This is because their sum must equal `90` degrees, which is the measure of a right angle.