Complementary and Supplementary Angles
In Class 5, students learn about two special relationships between pairs of angles: complementary angles and supplementary angles. These concepts are fundamental in geometry and help in solving angle problems in triangles, quadrilaterals, and real-life situations.
Two angles are complementary if they add up to 90°. Two angles are supplementary if they add up to 180°. Knowing one angle in a complementary or supplementary pair lets you calculate the other.
What is Complementary and Supplementary Angles - Class 5 Maths (Geometry)?
Complementary Angles: Two angles whose sum is 90° (a right angle).
Example: 30° and 60° are complementary because 30° + 60° = 90°.
Supplementary Angles: Two angles whose sum is 180° (a straight angle).
Example: 110° and 70° are supplementary because 110° + 70° = 180°.
Key facts:
- Each angle in a complementary pair is called the complement of the other.
- Each angle in a supplementary pair is called the supplement of the other.
- Complementary angles are always acute (both less than 90°).
- In a supplementary pair, at least one angle can be obtuse.
Complementary and Supplementary Angles Formula
Complement of an angle = 90° − angle
Supplement of an angle = 180° − angle
Solved Examples
Example 1: Example 1: Finding the complement
Problem: Find the complement of 35°.
Solution:
Complement = 90° − 35° = 55°
Answer: The complement of 35° is 55°.
Example 2: Example 2: Finding the supplement
Problem: Find the supplement of 120°.
Solution:
Supplement = 180° − 120° = 60°
Answer: The supplement of 120° is 60°.
Example 3: Example 3: Checking if angles are complementary
Problem: Are 47° and 43° complementary?
Solution:
Sum = 47° + 43° = 90° ✓
Answer: Yes, 47° and 43° are complementary.
Example 4: Example 4: Checking if angles are supplementary
Problem: Are 95° and 75° supplementary?
Solution:
Sum = 95° + 75° = 170° ≠ 180°
Answer: No, they are not supplementary.
Example 5: Example 5: Finding a missing angle
Problem: Two complementary angles are in the ratio 2:3. Find both angles.
Solution:
Step 1: Let the angles be 2x and 3x.
Step 2: 2x + 3x = 90° → 5x = 90° → x = 18°
Step 3: Angles = 2 × 18° = 36° and 3 × 18° = 54°
Answer: The angles are 36° and 54°.
Example 6: Example 6: Supplementary angles in a ratio
Problem: Two supplementary angles are in the ratio 5:4. Find both angles.
Solution:
Step 1: Let the angles be 5x and 4x.
Step 2: 5x + 4x = 180° → 9x = 180° → x = 20°
Step 3: Angles = 100° and 80°
Answer: The angles are 100° and 80°.
Example 7: Example 7: Complement of the complement
Problem: Find the complement of the complement of 25°.
Solution:
Complement of 25° = 90° − 25° = 65°
Complement of 65° = 90° − 65° = 25°
Answer: 25° (the complement of the complement is the angle itself).
Example 8: Example 8: Can a right angle have a complement?
Problem: Does 90° have a complement?
Solution:
Complement = 90° − 90° = 0°
While mathematically 0° is the complement, practically a 0° angle does not form a visible angle.
Answer: The complement of 90° is 0°.
Example 9: Example 9: Word problem — Clock
Problem: At 3 o'clock, the angle between the hour and minute hands is 90°. The minute hand moves and creates a 55° angle with the hour hand. What is the supplement of this angle?
Solution:
Supplement of 55° = 180° − 55° = 125°
Answer: The supplement is 125°.
Key Points to Remember
- Complementary angles add up to 90°.
- Supplementary angles add up to 180°.
- Complement of x° = 90° − x°.
- Supplement of x° = 180° − x°.
- Both angles in a complementary pair must be acute (less than 90°).
- In a supplementary pair, one angle can be acute and the other obtuse, or both can be right angles (90° + 90° = 180°).
- An angle greater than 90° does not have a complement (the result would be negative).
- The complement of the complement of an angle is the angle itself.
Practice Problems
- Find the complement of 48°.
- Find the supplement of 135°.
- Are 65° and 25° complementary?
- Are 88° and 92° supplementary?
- Two supplementary angles are in the ratio 1:3. Find both angles.
- The complement of an angle is 15°. Find the angle.
- Find the supplement of the complement of 50°.
- An angle is 20° more than its complement. Find both angles.
Frequently Asked Questions
Q1. What are complementary angles?
Two angles are complementary when they add up to 90°. For example, 40° and 50° are complementary. Each angle is called the complement of the other.
Q2. What are supplementary angles?
Two angles are supplementary when they add up to 180°. For example, 110° and 70° are supplementary. Each angle is called the supplement of the other.
Q3. Can an obtuse angle have a complement?
No. Since complementary angles must sum to 90°, and an obtuse angle is already greater than 90°, subtracting it from 90° would give a negative result. Only acute angles have complements.
Q4. Can two obtuse angles be supplementary?
No. Two obtuse angles (each greater than 90°) would add up to more than 180°, so they cannot be supplementary.
Q5. What is the complement of 45°?
The complement of 45° is 90° − 45° = 45°. This means 45° is its own complement — it is the only angle with this property.
Q6. What is the supplement of 90°?
The supplement of 90° is 180° − 90° = 90°. Two right angles (90° each) are supplementary.
Q7. Do complementary and supplementary angles have to be next to each other?
No. The angles do not need to be adjacent (next to each other). Any two angles that sum to 90° are complementary, and any two that sum to 180° are supplementary, regardless of their position.
Q8. Where do we see complementary and supplementary angles in real life?
Complementary angles appear in right-angled triangles (the two non-right angles are complementary). Supplementary angles appear on a straight line (linear pair). They are used in architecture, construction, and navigation.
