Adjacent Angles
When you open a book and look at its pages, the two angles formed by the pages on either side of the spine share the spine as a common arm and the binding as the common vertex. These are adjacent angles.
Adjacent angles are a pair of angles that are placed next to each other, sharing a common vertex and a common arm, without overlapping. They are one of the fundamental angle pairs studied in geometry.
Recognising adjacent angles is important because many other angle relationships — linear pair, vertically opposite angles, and angles formed by parallel lines and transversals — are built on the idea of adjacency.
In NCERT Class 7 Mathematics, adjacent angles are introduced in the chapter Lines and Angles. The concept is simple but forms the foundation for more advanced angle theorems.
What is Adjacent Angles?
Definition: Two angles are called adjacent angles if they satisfy ALL three conditions:
- They have a common vertex (same corner point).
- They have a common arm (one side is shared between them).
- Their non-common arms are on opposite sides of the common arm (they do not overlap).
Key terms:
- Common vertex: The point where the arms of both angles meet.
- Common arm: The ray (side) that is shared by both angles.
- Non-common arms: The other two rays, one belonging to each angle.
Important:
- Adjacent angles share a vertex and an arm but do NOT overlap.
- Adjacent angles are NOT necessarily equal.
- Adjacent angles do NOT necessarily add up to any fixed value (unless they form a special pair like a linear pair).
- If two angles have the same vertex but no common arm, they are not adjacent.
Adjacent Angles Formula
Properties of adjacent angles:
General rule:
- There is no fixed formula relating adjacent angles in general. They can have any measure.
- Adjacent angles simply share a vertex and an arm without overlapping.
Special cases:
If adjacent angles form a linear pair: Angle 1 + Angle 2 = 180°
If adjacent angles are complementary: Angle 1 + Angle 2 = 90°
Note:
- Every linear pair is a pair of adjacent angles, but not every pair of adjacent angles is a linear pair.
- A linear pair is formed when the non-common arms of adjacent angles form a straight line (opposite rays).
Types and Properties
Types and special cases of adjacent angles:
1. Adjacent complementary angles:
- Two adjacent angles whose measures add up to 90°.
- Example: 35° and 55° placed next to each other with a common arm.
2. Adjacent supplementary angles (Linear pair):
- Two adjacent angles whose measures add up to 180°.
- The non-common arms form a straight line.
- Example: 110° and 70° on a straight line.
3. Adjacent angles in shapes:
- In polygons, angles at the same vertex formed by a diagonal and a side are adjacent.
- In a rectangle, angles formed by a diagonal at a corner are adjacent and add up to 90°.
4. Adjacent angles that are equal:
- When a ray bisects an angle, it creates two equal adjacent angles.
- Example: An angle bisector divides a 60° angle into two adjacent 30° angles.
5. Non-adjacent angles (for contrast):
- Vertically opposite angles share a vertex but NOT a common arm — they are NOT adjacent.
- Angles at different vertices are NOT adjacent.
Solved Examples
Example 1: Example 1: Identifying adjacent angles
Problem: Two rays OB and OC are drawn from point O. ∠AOB = 40° and ∠BOC = 60°, where A, O, C are not on the same line. Are ∠AOB and ∠BOC adjacent angles?
Solution:
Check the three conditions:
- Common vertex? Yes — both angles have vertex O.
- Common arm? Yes — ray OB is the common arm.
- Non-common arms on opposite sides? Yes — ray OA and ray OC are on opposite sides of ray OB (they do not overlap).
Answer: Yes, ∠AOB and ∠BOC are adjacent angles.
Example 2: Example 2: Non-adjacent angles
Problem: Two lines intersect at point O, forming four angles. ∠1 and ∠3 are opposite each other. Are ∠1 and ∠3 adjacent?
Solution:
Check the conditions:
- Common vertex? Yes — both have vertex O.
- Common arm? No — ∠1 and ∠3 are across from each other. They do not share a common arm.
Since they fail condition 2, they are NOT adjacent.
Answer: No. ∠1 and ∠3 are vertically opposite angles, not adjacent angles.
Example 3: Example 3: Finding the measure of an adjacent angle (linear pair)
Problem: Two adjacent angles form a linear pair. One angle is 125°. Find the other angle.
Solution:
Given:
- Adjacent angles form a linear pair.
- One angle = 125°
Since they form a linear pair:
- ∠1 + ∠2 = 180°
- 125° + ∠2 = 180°
- ∠2 = 180° − 125° = 55°
Answer: The other angle is 55°.
Example 4: Example 4: Adjacent complementary angles
Problem: Two adjacent angles are complementary. If one angle is 37°, find the other.
Solution:
Given:
- Adjacent angles are complementary (add up to 90°).
- One angle = 37°
Calculation:
- ∠1 + ∠2 = 90°
- 37° + ∠2 = 90°
- ∠2 = 90° − 37° = 53°
Answer: The other angle is 53°.
Example 5: Example 5: Angle bisector creating adjacent angles
Problem: Ray OC bisects ∠AOB = 84°. Find the measure of each adjacent angle formed.
Solution:
Given:
- ∠AOB = 84°
- OC bisects ∠AOB
Since OC bisects the angle:
- ∠AOC = ∠COB = 84° ÷ 2 = 42°
Check: ∠AOC and ∠COB are adjacent angles (common vertex O, common arm OC, non-common arms OA and OB on opposite sides).
Answer: Each adjacent angle is 42°.
Example 6: Example 6: Three adjacent angles on a straight line
Problem: Three adjacent angles are formed on a straight line. The angles are x°, 2x°, and 3x°. Find the value of x and each angle.
Solution:
Given:
- Three adjacent angles on a straight line.
- Angles: x°, 2x°, 3x°
Since they are on a straight line, their sum = 180°:
- x + 2x + 3x = 180
- 6x = 180
- x = 30
The three angles are:
- x = 30°
- 2x = 60°
- 3x = 90°
Verification: 30° + 60° + 90° = 180° ✓
Answer: x = 30; angles are 30°, 60°, and 90°.
Example 7: Example 7: Adjacent angles at a point
Problem: Four adjacent angles are formed around a point O. They are 90°, 85°, 95°, and x°. Find x.
Solution:
Given:
- Four adjacent angles around a point.
- Angles: 90°, 85°, 95°, x°
Since angles at a point add up to 360°:
- 90 + 85 + 95 + x = 360
- 270 + x = 360
- x = 360 − 270 = 90°
Answer: x = 90°.
Example 8: Example 8: Identifying from a figure description
Problem: In the figure, rays OA, OB, OC, and OD are drawn from point O. Name all pairs of adjacent angles.
Solution:
The pairs of adjacent angles are:
- ∠AOB and ∠BOC (common arm OB)
- ∠BOC and ∠COD (common arm OC)
- ∠COD and ∠DOA (common arm OD)
- ∠DOA and ∠AOB (common arm OA)
Non-adjacent pairs:
- ∠AOB and ∠COD (no common arm — these are vertically opposite if the lines cross).
- ∠BOC and ∠DOA (no common arm).
Answer: There are 4 pairs of adjacent angles.
Example 9: Example 9: Algebraic problem
Problem: Two adjacent angles are (3x + 10)° and (2x + 20)°. If they form a linear pair, find x and both angles.
Solution:
Given:
- Adjacent angles form a linear pair → sum = 180°
Setting up the equation:
- (3x + 10) + (2x + 20) = 180
- 5x + 30 = 180
- 5x = 150
- x = 30
Finding the angles:
- First angle = 3(30) + 10 = 100°
- Second angle = 2(30) + 20 = 80°
Verification: 100° + 80° = 180° ✓
Answer: x = 30; angles are 100° and 80°.
Example 10: Example 10: Real-life example — clock hands
Problem: At 3 o'clock, the minute hand points at 12 and the hour hand at 3, forming a 90° angle. If a fly sits on the clock face at the 1 mark, it divides this angle into two adjacent angles. The angle from 12 to 1 is 30°. Find the angle from 1 to 3.
Solution:
Given:
- Total angle (12 to 3) = 90°
- Angle (12 to 1) = 30°
The two adjacent angles:
- Angle from 12 to 1 = 30°
- Angle from 1 to 3 = 90° − 30° = 60°
These are adjacent angles: They share the common vertex (centre of clock) and common arm (direction of mark 1), with non-common arms at 12 and 3.
Answer: The angle from 1 to 3 is 60°.
Real-World Applications
Real-world examples of adjacent angles:
- Book pages: When a book is open, the two pages form adjacent angles at the spine. The spine is the common arm.
- Clock hands: At any time, if a third mark divides the angle between two hands, the resulting two angles are adjacent.
- Door and wall: When a door is partially open, the angle between the door and the wall, and the angle between the door and the doorframe on the other side, are adjacent angles.
- Pizza slices: Two consecutive slices of pizza form adjacent angles at the centre of the pizza.
- Road intersections: At a T-junction, the angles formed on either side of the joining road are adjacent.
- Corner of a room: If you stand at a corner and look along two walls, the angles formed by any object in between create adjacent angles.
Key Points to Remember
- Adjacent angles share a common vertex and a common arm, and their non-common arms are on opposite sides of the common arm.
- Three conditions must ALL be satisfied: common vertex, common arm, no overlap.
- Adjacent angles are NOT necessarily equal.
- Adjacent angles do NOT necessarily add to a fixed value — unless they form a special pair.
- If adjacent angles form a linear pair, they add up to 180°.
- If adjacent angles are complementary, they add up to 90°.
- Every linear pair consists of adjacent angles, but not all adjacent angles are a linear pair.
- Vertically opposite angles are NOT adjacent — they share a vertex but not a common arm.
- An angle bisector divides an angle into two equal adjacent angles.
- Adjacent angles around a point add up to 360°.
Practice Problems
- Two adjacent angles have measures 45° and 75°. Can they form a linear pair? Explain.
- Two adjacent angles form a linear pair. One is three times the other. Find both angles.
- Ray OP bisects ∠MON = 124°. What are the two adjacent angles formed?
- Three adjacent angles on a straight line measure (x + 30)°, (2x)°, and (x − 10)°. Find x and all three angles.
- Name three pairs of adjacent angles in a rectangle ABCD where diagonal AC is drawn.
- At point O, five rays are drawn creating five adjacent angles that together complete a full rotation. Four of the angles are 60°, 80°, 90°, and 50°. Find the fifth angle.
- Give an example of two angles that share a common vertex but are NOT adjacent.
- Two adjacent angles are in the ratio 2:3. If they form a linear pair, find each angle.
Frequently Asked Questions
Q1. What are adjacent angles?
Adjacent angles are two angles that share a common vertex and a common arm, and their non-common arms lie on opposite sides of the common arm. In simple terms, they are angles placed next to each other without overlapping.
Q2. Do adjacent angles always add up to 180°?
No. Adjacent angles add up to 180° only when they form a linear pair (their non-common arms form a straight line). In general, adjacent angles can add up to any value.
Q3. What is the difference between adjacent angles and a linear pair?
All linear pairs are adjacent angles, but not all adjacent angles are linear pairs. A linear pair is a special case where the non-common arms of the two adjacent angles form a straight line (opposite rays), so they add up to 180°.
Q4. Are vertically opposite angles adjacent?
No. Vertically opposite angles share a common vertex but do NOT share a common arm. They are across from each other, not next to each other. So they fail the second condition for adjacency.
Q5. Can two adjacent angles be equal?
Yes. When a ray bisects an angle, it creates two equal adjacent angles. For example, if ∠AOB = 80° and ray OC bisects it, then ∠AOC = ∠COB = 40°. These are equal adjacent angles.
Q6. Can there be more than two adjacent angles?
Adjacent angles are always defined as a pair (two angles). However, multiple pairs of adjacent angles can exist at the same vertex. For example, if three rays OA, OB, OC emerge from O, then ∠AOB and ∠BOC are one pair, and ∠AOB and ∠AOC can be considered as well, depending on arrangement.
Q7. How many pairs of adjacent angles are formed by two intersecting lines?
When two lines intersect, they form 4 angles. There are 4 pairs of adjacent angles (each angle is adjacent to the two angles next to it). The non-adjacent pairs are vertically opposite.
Q8. Can adjacent angles be complementary?
Yes. If two adjacent angles add up to 90°, they are adjacent complementary angles. For example, in a right angle, if a ray divides it into 35° and 55°, these two adjacent angles are complementary.
Related Topics
- Linear Pair of Angles
- Vertically Opposite Angles
- Complementary Angles
- Supplementary Angles
- Transversal and Parallel Lines
- Corresponding Angles
- Alternate Interior Angles
- Co-Interior Angles
- Angles on a Straight Line
- Angles at a Point
- Alternate Exterior Angles
- Proving Lines are Parallel
- Word Problems on Lines and Angles
