Supplementary angles are those angles that measure up to 180 degrees. For example, angle 130° and angle 50° are supplementary because on adding 130° and 50° we get 180°. Similarly, complementary angles add up to 90 degrees. The two supplementary angles, if joined together, form a straight line and a straight angle.
But it should be noted that the two angles that are supplementary to each other, do not have to be next to each other. Hence, any two angles can be supplementary, if their sum equal to 180°.
Geometry is one of the important branches of mathematics that deals with the study of different shapes. It initiates the study of lines and angles. A straight line is a line without curves and it is defined as the shortest distance between two points. An angle is formed when the line segment meets at a point.
The classification of a pair of angles is given as:
- Complementary angles
- Supplementary angles
- Corresponding Angles
- Linear pairs of angles
- Alternate angles
- Vertically opposite angles
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Definition
Two angles are said to be supplementary angles when they add up to 180 degrees. Two angles are supplementary, if
- One of its angles is an acute angle and another angle is an obtuse angle.
- Both of the angles are right angles.
This means that ∠A + ∠B = 180°.
See the figure below for a better understanding of the pair of angles that are supplementary.
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Properties
The important properties of supplementary angles are:
- The two angles are said to be supplementary angles when they add up to 180°.
- The two angles together make a straight line, but the angles need not be together.
- “S” of supplementary angles stands for the “Straight” line. This means they form 180°.
How to Find Supplementary Angles?
For example, if you had given that two angles form supplementary angles and you are provided with one angle and then asked to find the other angle, you can easily find the other angle using the formula.
∠A = 180° – ∠B (or)
∠B = 180° – ∠A
Adjacent and Non-Adjacent Supplementary Angles
The supplementary angles may be classified as either adjacent or non-adjacent. The adjacent supplementary angles share the common line segment or arm with each other, whereas the non-adjacent supplementary angles do not share the line segment or arm.
Supplementary Angles Theorem
The supplementary angle theorem states that if two angles are supplementary to the same angle, then the two angles are said to be congruent.
Proof:
If ∠x and ∠y are two different angles that are supplementary to a third angle ∠z, such that,
∠x + ∠z = 180 ……. (1)
∠y + ∠z = 180 ……. (2)
Then, from the above two equations we can say,
∠x = ∠y
Examples
Some of the examples of supplementary angles are:
- 120° + 60° = 180°
- 90° + 90° = 180°
- 140° + 40° = 180°
- 96° + 84° = 180°
Difference between Complementary and Supplementary Angles
Complementary Angles | Supplementary Angles |
Sum of two angles is 90° | Sum of two angles is 180° |
Ex: ∠A + ∠B = 90°. | Ex: ∠A + ∠B = 180°. |
Complementary angles form a right-angled triangle. | Supplementary angles form a straight line. |
Problems and Solutions
Question 1: Find the measure of an unknown angle from the given figure.
Solution:
We know that the supplementary angles add up to 180°.
X + 55° + 40° = 180°
X + 95° = 180°
X = 180°- 95°
X = 85°
Therefore, the unknown angle, X = 85°
Question.2: If ∠x and ∠y are supplementary angles and ∠x = 67, then find ∠y.
Solution: Given, ∠x and ∠y are supplementary angles
And
∠x = 67°
Since, ∠x + ∠y = 180°
∠y = 180 – ∠x
∠y = 180 – 67
∠y = 113°
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Frequently Asked Questions – FAQs
Can 2 Acute Angles be Supplementary Angles?
By definition, acute angles are the angles that measure the angle greater than 0° and less than 90°. If you add two acute angles in which each angle is large as possible, its sum will be less than 180°. By the definition of supplementary angles, it is impossible to get the supplementary angle when we add two acute angles.
Example: 80° +60° = 140° which is not a supplementary angle.
But, in the case, if we add more than two acute angles, we can get supplementary angles.
Can 2 Obtuse Angles be Supplementary Angles?
By the definition of obtuse angles, the angles that measure greater than 90° are obtuse. If you add two obtuse angles, the sum will be greater than 180°. It will not satisfy the property of the supplementary angles when we add obtuse angles.
Example: 110° + 95° = 205° which is not a supplementary angle. [205° > 180° ]
Can 2 Right Angles be Supplementary Angles?
When two right angles are added, it is possible to get the supplementary angle. because 90° + 90° = 180°, as it satisfies the condition of supplementary angles.