Definition formula properties and solved examples of 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
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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°.
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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
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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
1. What are complementary and supplementary angles?
Complementary angles are two angles whose measures add up to 90°, while supplementary angles are two angles whose measures add up to 180°.
- If ∠A + ∠B = 90°, they are complementary angles.
- If ∠A + ∠B = 180°, they are supplementary angles.
- Complementary angles form a right angle, and supplementary angles form a straight line.
2. What is the formula for complementary angles?
The formula for complementary angles is Angle 1 + Angle 2 = 90°.
- If one angle is x, the other angle is 90° − x.
- Example: If one angle is 35°, the other is 90° − 35° = 55°.
3. What is the formula for supplementary angles?
The formula for supplementary angles is Angle 1 + Angle 2 = 180°.
- If one angle is x, the other angle is 180° − x.
- Example: If one angle is 120°, the other is 180° − 120° = 60°.
4. How do you find a missing complementary angle?
To find a missing complementary angle, subtract the given angle from 90°.
- Step 1: Identify the given angle.
- Step 2: Use the formula 90° − given angle.
- Example: If one angle is 48°, the other is 90° − 48° = 42°.
5. How do you find a missing supplementary angle?
To find a missing supplementary angle, subtract the given angle from 180°.
- Step 1: Note the given angle.
- Step 2: Apply 180° − given angle.
- Example: If one angle is 135°, the other is 180° − 135° = 45°.
6. What is the difference between complementary and supplementary angles?
The main difference is that complementary angles add to 90°, while supplementary angles add to 180°.
- Complementary angles form a right angle.
- Supplementary angles form a straight angle.
- Complementary angles are always acute, but supplementary angles can include one acute and one obtuse angle.
7. Can two obtuse angles be complementary or supplementary?
Two obtuse angles cannot be complementary or supplementary.
- An obtuse angle is greater than 90°.
- Two obtuse angles would sum to more than 180°.
- Complementary angles must total 90°, and supplementary angles must total 180°.
8. Are vertical angles complementary or supplementary?
Vertical angles are equal in measure but are not necessarily complementary or supplementary.
- Vertical angles are formed when two lines intersect.
- They are always equal.
- They are complementary only if each angle is 45°.
- They are supplementary only if each angle is 90°.
9. Can complementary or supplementary angles be adjacent?
Yes, complementary and supplementary angles can be adjacent, but they do not have to be.
- Adjacent angles share a common vertex and side.
- Two adjacent angles that sum to 90° are complementary.
- Two adjacent angles that form a straight line (sum 180°) are supplementary.
10. What are some real-life examples of complementary and supplementary angles?
Complementary and supplementary angles appear in everyday objects and designs involving right and straight angles.
- A corner of a square room shows 90°, where two angles can be complementary.
- A straight road represents a 180° straight angle, showing supplementary angles.
- Carpentry, architecture, and engineering use these angle relationships for accurate measurements.
