Supplementary Angles
- Introduction to Supplementary Angles
- Definition of Supplementary Angles
- Types of Supplementary Angles
- Adjacent Supplementary angles
- Non-adjacent Supplementary Angles
- Complementary and Supplementary Angles
- How to Find the Supplement of an Angle?
- Solved Examples
- Practice Problems
- Frequently Asked Question
Introduction to Supplementary Angles
In geometry, angles are formed when two rays share a common endpoint, called the vertex. When the combined measurement of two angles equals `180` degrees, they are called supplementary angles.
Supplementary angles are pairs of angles whose measures add up to `180` degrees. For example, if one angle measures `70` degrees, then the other angle must measure `110` degrees to make a total of `180` degrees when added together.
In real-life scenarios, supplementary angles can be found in various configurations. For instance, consider a straight line. The angles on either side of the line are supplementary, totaling `180` degrees.
Definition of Supplementary Angles
Supplementary angles are a pair of angles that, when combined, result in a total of `180°`. This means that if you have two angles and when you add their measures together and the sum equals `180°`, these angles are termed as supplementary.
For example, consider two angles, one measuring \(110^\circ\) and the other \(70^\circ\). When you add \(110^\circ\) to \(70^\circ\), the total measure becomes \(180^\circ\). Thus, these two angles are supplementary angles.
In simpler terms, if you have `∠1` and `∠2`, and their measures sum up to `180°`, then `∠1` and `∠2` are considered supplementary. We refer to `∠1` and `∠2` as each other's supplements based on this relationship.
Types of Supplementary Angles
Supplementary angles can be categorized into two types:
- Adjacent Supplementary Angles
- Non-Adjacent Supplementary Angles
Each type has its characteristics, explained below.
Adjacent Supplementary Angles
Adjacent supplementary angles are two angles that share a common vertex and a common side. In simpler terms, they are angles that "sit next to each other" with one side in common.
Consider the illustration below, where `∠DEF` and `∠DEG` are adjacent angles. They share the vertex `D` and the side `DE`. Moreover, when their measures are added, they sum up to `180°`. For instance, `∠DEF` measures `40°` and `∠DEG` measures `140°`, resulting in a total of `180°`. Thus, these angles, `∠DEF` and `∠DEG`, are considered adjacent supplementary angles.
Non-Adjacent Supplementary Angles
Non-adjacent supplementary angles are two angles that do not share a common vertex or a common side. Despite not being adjacent, their measures still combine to form a total of `180°`.
In the given diagram, `∠LMN` and `∠PQR` are non-adjacent angles. They do not share a common vertex or a common side. However, their measures, when added, result in `180°`. For instance, `∠LMN` measures `85°` and `∠PQR` measures `95°`, leading to a total of `180°`. Thus, these angles, `∠LMN` and `∠PQR`, are termed as non-adjacent supplementary angles. When placed together, they form a straight angle.
Complementary Angles and Supplementary Angles
In geometry, angles often come in pairs, and two important types are complementary angles and supplementary angles. While supplementary angles combine to give a total of `180°`, complementary angles add up to `90°`. These concepts find practical use in various real-world scenarios, with one common example being crossroads.
Complementary Angles
If the sum of two angles equals `90°`, they are termed complementary angles. For instance, if one angle measures \(30^\circ\), its complementary angle would be `90° - 30° = 60°`.
Supplementary Angles
On the other hand, supplementary angles are those whose measures add up to `180°`. For example, if one angle measures \(120^\circ\), its supplementary angle would be `180° - 120° = 60°`.
Tips on Supplementary Angles vs Complementary Angles
- Remember that 'S' stands for both 'Supplementary' and 'Straight.' This association helps you recall that two supplementary angles form a straight angle when added together meaning `180°`.
- Similarly, 'C' is for both 'Complementary' and 'Corner.' This connection aids in remembering that two complementary angles create a corner (right) angle when combined meaning `90°`.
How to Find the Supplement of an Angle?
When two angles add up to `180°`, we call them supplementary angles. Each angle in such a pair is considered the supplement of the other. To find the supplement of a given angle, we subtract its measurement from `180°`.
Example: Find the supplement of `65°`.
Solution:
Subtract `65°` from `180°`:
`180° - 65° = 115°`
Thus, the supplement of `65°` is `115°`.
Solved Examples
Example `1`. If the ratio of two supplementary angles is `1:5`, what is the measure of each angle?
Solution:
Let's denote the smaller angle as \( x \) and the larger angle as \( 5x \) since their ratio is `1:5`.
Since the angles are supplementary, their sum is equal to \( 180^\circ \):
\( x + 5x = 180^\circ \)
Combine like terms:
\( 6x = 180^\circ \)
Divide both sides by `6` to find the value of \( x \):
\( x = 30^\circ \)
Now that you know \( x \) is \( 30^\circ \), find the larger angle by multiplying \( x \) by `5`:
\( 5x = 5 \times 30^\circ = 150^\circ \)
Therefore, the smaller angle measures `30°`, and the larger angle measures `150°`.
Example `2`. If one angle in a pair of supplementary angles measures `120°`, what is the measurement of the other angle?
Solution:
Since supplementary angles add up to `180°`, subtract the given angle from `180°`.
Measure of the other angle `= 180° - 120° = 60°`
Example `3`. If \(2x + 13\) and \(x + 17\) are two supplementary angles, find the measures of each angle.
Solution:
Given that the angles are supplementary, their measures add up to \(180^\circ\).
So, we can write the equation:
\( (2x + 13) + (x + 17) = 180 \)
Now, let's solve for \(x\):
\( 2x + 13 + x + 17 = 180 \)
\( 3x + 30 = 180 \)
Subtract \(30\) from both sides:
\( 3x = 150 \)
Divide both sides by \(3\):
\( x = 50 \)
Now that we have found \(x\), we can find the measures of the angles:
\( \text{Measure of first angle} = 2x + 13 \)
\( \text{Measure of first angle} = 2(50) + 13 = 100 + 13 = 113^\circ \)
\( \text{Measure of second angle} = x + 17 \)
\( \text{Measure of second angle} = 50 + 17 = 67^\circ \)
So, the measure of the two angles are \(113^\circ\) and \(67^\circ\).
Example `4`. What is the supplement of `110°`?
Solution:
Subtract the given angle from `180` degrees to find its supplement.
Supplement `= 180` degrees `- 110` degrees `= 70` degrees.
Example `5`. If the supplement of an angle is `85°`, what is the measurement of the angle?
Solution:
Since the supplement is given, subtract it from `180` degrees to find the angle.
Measure of the angle `= 180` degrees `- 85` degrees `= 95` degrees.
Practice Problems
Q`1`. If \(2x + 9\) and \(115^\circ\) are two supplementary angles, find the measures of the angle `(2x + 9)°`.
- `30°`
- `40°`
- `65°`
- `90°`
Answer: c.
Q`2`. Find the supplement of an angle measuring `120` degrees.
- `60°`
- `80°`
- `100°`
- `140°`
Answer: a.
Q`3`. Two angles are supplementary. One is double the size of the other. What is the size of the bigger angle?
- `45°`
- `90°`
- `120°`
- `180°`
Answer: c.
Q`4`. If `(2x - 29)°` and `(3x+14)°` are two supplementary angles, find both angles.
- `49°` and `131°`
- `59°` and `121°`
- `65°` and `115°`
- `69°` and `111°`
Answer: a
Q`5`. `PQS` is a right-angled triangle. Which of the following pairs of angles are adjacent supplementary angles?
- `∠QPR` and `∠RPS`
- `∠PRQ` and `∠PRS`
- `∠PQS` and `∠PSQ`
- `∠RPS` and `∠RSP`
Answer: b.
Q`6`. `PQS` is a right-angled triangle. Which of the following pairs of angles are non-adjacent supplementary angles?
- `∠QPR` and `∠RPS`
- `∠PRQ` and `∠PRS`
- `∠PQS` and `∠PSQ`
- `∠RPS` and `∠RSP`
Answer: c.
Frequently Asked Questions
Q`1`. What are supplementary angles?
Answer: Supplementary angles are two angles whose measures add up to `180°`.
Q`2`. How do you find the supplement of an angle?
Answer: The supplement of an angle is found by subtracting its measure from `180°`.
Q`3`. What is the difference between supplementary and complementary angles?
Answer: Supplementary angles add up to `180°`, while complementary angles add up to `90°`.
Q`4`. Can two acute angles be supplementary?
Answer: No, two acute angles cannot be supplementary because their sum would be less than `180°`.
Q`5`. Can two obtuse angles be supplementary?
Answer: No, two obtuse angles cannot be supplementary because their sum would exceed `180°`.