Angles
🎯 What will you learn on this page?
👉 Meaning of an angle, its arms and vertex.
👉 Acute, right, obtuse, straight and reflex angles.
👉 Adjacent, complementary and supplementary angles.
👉 Linear pairs and vertically opposite angles.
👉 How to find unknown angles using simple equations.
👉 What is an Angle?
When two rays originate from the same endpoint and move in different directions, an angle is formed.
The angle formed by two rays lies on the same plane.
In a geometric figure, an angle is represented by three points, such as ∠AOB or ∠O.
Angle
∠AOB or ∠O
🔹 Types of Angles
Based on their measure, angles are divided into five types:
Angle
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
Reflex Angle
Acute Angle
0° < θ < 90°
An angle smaller than 90° is called an acute angle.
Right Angle
θ = 90°
An angle equal to 90° is called a right angle.
Obtuse Angle
90° < θ < 180°
An angle greater than 90° but smaller than 180° is called an obtuse angle.
Straight Angle
θ = 180°
An angle equal to 180° is called a straight angle.
Reflex Angle
180° < θ < 360°
An angle greater than 180° but smaller than 360° is called a reflex angle.
🔹 Relation Between Angles
Based on the relationship between two angles, they can be divided into four types:
Angle
Adjacent Angles
Complementary Angles
Supplementary Angles
Vertically Opposite Angles
Adjacent Angles
Adjacent Angles
∠AOC is adjacent to ∠BOC.
If two angles have a common arm and a common vertex, they are called adjacent angles.
Here, OC is the common arm and O is the vertex.
Complementary Angles
Complementary Angles
∠AOC + ∠BOC = 90°
If the sum of two angles is 90°, they are called complementary angles. They do not have to be adjacent.
Here, ∠AOC and ∠BOC are complementary to each other.
Supplementary Angles
Supplementary Angles
∠AOC + ∠BOC = 180°
If the sum of two angles is 180°, they are called supplementary angles. They do not have to be adjacent.
Here, ∠AOC and ∠BOC are supplementary to each other.
| Given | A, O and B lie on the same straight line, and OC is another ray. |
| ∵ | ∠AOB = 180° |
| ∴ | ∠AOC + ∠COB = 180° |
| Therefore, ∠AOC and ∠COB form a linear pair. |
Vertically Opposite Angles
Vertically Opposite Angles
∠AOC = ∠BOD
∠AOD = ∠COB
Each pair of vertically opposite angles is equal.
| ∵ | ∠AOC + ∠AOD = 180° |
| ∵ | ∠AOC + ∠BOC = 180° |
| ∴ | ∠AOD = ∠BOC |
| Similarly, ∠AOC = ∠BOD |
✏️ Solved Examples
Example 1: Identify the type of angle
Classify the following angles: 38°, 90°, 126°, 180° and 245°.
| Angle | Type |
|---|---|
| 38° | Acute angle |
| 90° | Right angle |
| 126° | Obtuse angle |
| 180° | Straight angle |
| 245° | Reflex angle |
Example 2: Find the complementary angle
If one angle is 37°, find its complement.
Required angle = 90° - 37° = 53°Therefore, the complementary angle is 53°.
Example 3: Find the supplementary angle
If one angle is 112°, find its supplement.
Required angle = 180° - 112° = 68°Therefore, the supplementary angle is 68°.
Example 4: Angles in a linear pair
Two angles in a linear pair are 3x and 2x. Find both angles.
3x + 2x = 180° 5x = 180° x = 36°So, the angles are 108° and 72°.
Example 5: Vertically opposite angles
Two straight lines intersect. One of the angles is 64°. Find the other three angles.
The vertically opposite angle is also 64°.
Each adjacent angle is supplementary to 64°:
180° - 64° = 116°So, the four angles are 64°, 116°, 64°, 116°.
🧩 Practice Problems
🔹 1. Classify these angles: 25°, 95°, 180°, 300°.
🔹 2. Find the complement of 28°.
🔹 3. Find the supplement of 74°.
🔹 4. Two complementary angles are in the ratio 2 : 3. Find the angles.
🔹 5. Two supplementary angles are in the ratio 4 : 5. Find the angles.
🔹 6. A linear pair is (x + 20)° and (2x + 10)°. Find x and both angles.
🔹 7. Two straight lines intersect. If one angle is 127°, find the other three angles.
🔹 8. Are two angles of 40° and 50° complementary even if they are not adjacent? Explain.
1. Acute, obtuse, straight, reflex.
2. 62°
3. 106°
4. 36° and 54°
5. 80° and 100°
6. x = 50; angles = 70° and 110°
7. 127°, 53°, 127°, 53°
8. Yes. Complementary angles only need a total measure of 90°; adjacency is not required.
⚠️ Common Misconceptions
🔹 Longer arms make a larger angle.
False. The size of an angle depends on the amount of turn between the two rays, not on the lengths of the rays.
🔹 Complementary and supplementary angles must be adjacent.
False. They may be separate. Only their total measures matter.
🔹 A rotated right angle is no longer a right angle.
False. Its orientation may change, but its measure remains 90°.
🔹 The wider-looking region is always the angle we measure.
Not necessarily. The intended angle is identified by its arms, vertex and notation.
✅ Quick Revision
🔹 Acute angle: 0° < θ < 90°
🔹 Right angle: θ = 90°
🔹 Obtuse angle: 90° < θ < 180°
🔹 Straight angle: θ = 180°
🔹 Reflex angle: 180° < θ < 360°
🔹 Complementary angles have a sum of 90°.
🔹 Supplementary angles have a sum of 180°.
🔹 Angles in a linear pair are adjacent and have a sum of 180°.
🔹 Vertically opposite angles are equal.
Further Learning
In the next chapters, we will study other types of angles such as Alternate Angles, Corresponding Angles, Internal Angles, and External Angles.
