What is the Difference Between Complementary and Supplementary Angles?
The concept of complementary and supplementary angles is a vital building block in geometry and is frequently used in both exams and everyday problem-solving. Understanding the distinction between these angle pairs makes angle questions easier across school-level mathematics.
What Are Complementary and Supplementary Angles?
Complementary and supplementary angles are special pairs of angles based on their degree sum. Complementary angles are two angles whose measures add up to 90°. Supplementary angles are two angles whose measures add up to 180°. These concepts are widely applied in geometry, trigonometry, and even real-life examples like clocks and engineering designs.
Key Formula for Complementary and Supplementary Angles
Here are the standard formulas you’ll use:
- If angle A is complementary to angle B: A + B = 90°. To find one angle: Complement = 90° – given angle.
- If angle C is supplementary to angle D: C + D = 180°. To find one angle: Supplement = 180° – given angle.
Comparison Table: Complementary vs Supplementary Angles
| Feature | Complementary Angles | Supplementary Angles |
|---|---|---|
| Definition | Pair sums to 90° | Pair sums to 180° |
| Example Pair | 40° & 50° | 110° & 70° |
| Formula | 90° – given angle | 180° – given angle |
| Real-life Illustration | Angles in a right triangle (other than the right angle) | Angles on a straight line |
| Possible Angle Types | Both must be acute (< 90°) | One acute & one obtuse, or both 90° |
Step-by-Step Examples and Calculations
-
Find the complement of 38°.
1. Use the formula: Complement = 90° – 38°
2. Calculate: 90 – 38 = 52°
3. So, the complement of 38° is 52°.
-
What is the supplement of 86°?
1. Use the formula: Supplement = 180° – 86°
2. Calculate: 180 – 86 = 94°
3. So, the supplement of 86° is 94°.
-
If two angles are complementary and one angle is (2x + 10)°, find the other angle in terms of x.
1. Let the second angle be y.
2. By definition: (2x + 10) + y = 90
3. Rearranged: y = 90 – (2x + 10) = 80 – 2x
-
Two angles are supplementary. If one angle is twice the other, what are the angles?
1. Let first angle = x; second = 2x
2. x + 2x = 180
3. 3x = 180 ⇒ x = 60
4. Second angle = 2 × 60 = 120°.
5. So the angles are 60° and 120°.
Frequent Errors and Misunderstandings
- Confusing complementary (90°) with supplementary (180°) due to similar-sounding terms.
- Forgetting that complementary angles must both be less than 90° (both acute).
- Thinking angles must be adjacent (they don’t have to be; only their sum matters).
- Incorrectly using “sum” in algebraic problems (always use 90 or 180 appropriately).
Memory Tip: Remember: “C” (Complementary, Corner, 90° like a corner) comes before “S” (Supplementary, Straight, 180° like a straight line).
Try These Yourself
- Find the complement of 47°.
- If angle Y is 35°, what is its supplement?
- Check if angles 55° and 35° are complementary.
- If one angle is 5x° and the other is (85 – x)°, are they complementary or supplementary?
Relation to Other Concepts
Complementary and supplementary angles are related to other geometric concepts. To deepen your knowledge, explore topics such as types of angles, adjacent and vertical angles, and linear pair of angles. These links help you see how different angle relationships play a role in classifying and solving geometry problems.
Real-Life Applications
- Clock: The hands create complementary angles when they make a right angle, and supplementary angles when they make a straight line.
- Corners: The corner of a book or wall forms a right angle—complementary angles add up here.
- Engineering: Bridges and supports often use supplementary angles for strength.
- Triangles: In a right triangle, the two non-right angles are always complementary.
Classroom Tip
A quick way to distinguish them is: “Complementary for Corners, Supplementary for Straight lines.” Vedantu teachers recommend drawing a box for 90° and a line for 180° alongside your exercise to help visual memory.
Wrapping It All Up
We explored complementary and supplementary angles—their definitions, formulas, solved examples, and memory tricks. By connecting these to other geometry topics and practicing regularly (especially with Vedantu’s resources), you’ll become more confident in solving angle problems for school or competitive exams.
Continue learning with Vedantu for clear, exam-focused summaries and live support on all angle types, single angle measures, and properties related to angle sums in shapes and linear pairs. Practice makes perfect!
FAQs on Complementary and Supplementary Angles Explained with Examples
1. What are complementary and supplementary angles in maths?
In mathematics, complementary angles are two angles whose sum is 90 degrees (a right angle). Supplementary angles are two angles whose sum is 180 degrees (a straight angle).
2. What is the difference between complementary and supplementary angles?
The key difference lies in their sum: complementary angles add up to 90°, forming a right angle, while supplementary angles add up to 180°, forming a straight line. Complementary angles are always acute, whereas supplementary angles can include one acute and one obtuse angle, or two right angles.
3. Is 45° and 45° complementary or supplementary?
45° and 45° are complementary angles because their sum (45° + 45° = 90°) equals 90 degrees.
4. How do you find the supplementary angle of a given angle?
To find the supplementary angle, subtract the given angle's measure from 180°. For example, the supplement of 70° is 180° - 70° = 110°.
5. What is a real-life example of complementary angles?
The two legs of a right-angled triangle form complementary angles. The corner of a square or rectangle also shows complementary angles (each corner being 90°).
6. Can two obtuse angles be supplementary?
No, two obtuse angles cannot be supplementary. An obtuse angle is greater than 90°, and the sum of two obtuse angles would exceed 180°.
7. How are complementary and supplementary angles used in trigonometry proofs?
Complementary angles are used extensively in trigonometry to establish identities like sin(90° - x) = cos x and cos(90° - x) = sin x. Supplementary angles help in proving identities related to sine and tangent functions in different quadrants.
8. Does the order of angles affect whether they are complementary or supplementary?
No, the order does not matter. If angles A and B are complementary (A + B = 90°), then B and A are also complementary. The same applies to supplementary angles.
9. Can there be more than two angles that are supplementary at the same time?
Yes. For example, three angles measuring 60°, 60°, and 60° are supplementary since their sum is 180°.
10. Is there a memory trick to quickly distinguish between complementary and supplementary angles?
Think of "Complementary" as relating to a Corner (90°), and "Supplementary" as relating to a Straight line (180°).
11. What is the complement of 40°?
The complement of 40° is 50° (90° - 40° = 50°).
12. What is the supplement of 2x - 13°?
The supplement of 2x - 13° is 180° - (2x - 13°) = 193° - 2x.
