Vertically Opposite Angles
When two straight lines cross each other (intersect) at a point, they form four angles. The angles that are directly across from each other at the point of intersection are called vertically opposite angles.
Vertically opposite angles are always equal. This is one of the most important properties in the Lines and Angles chapter of Class 7 NCERT Mathematics.
Understanding vertically opposite angles helps in solving problems involving intersecting lines, parallel lines cut by a transversal, and many geometry proofs in higher classes.
What is Vertically Opposite Angles?
Definition: When two lines intersect at a point, the pairs of angles that are opposite to each other are called vertically opposite angles. They are also called vertical angles.
Key Terms:
- Intersecting Lines: Two lines that cross at exactly one point.
- Point of Intersection: The point where two lines cross each other.
- Vertically Opposite Angles: The pair of non-adjacent angles formed at the point of intersection. They are across from each other.
- Adjacent Angles: Angles that share a common arm and a common vertex.
- Linear Pair: Two adjacent angles that together form a straight line (sum = 180°).
How to identify vertically opposite angles:
- Two lines intersect at a point, forming 4 angles.
- Label them ∠1, ∠2, ∠3, ∠4 going around the point.
- ∠1 and ∠3 are vertically opposite (they face each other).
- ∠2 and ∠4 are vertically opposite (they face each other).
Vertically Opposite Angles Formula
Property of Vertically Opposite Angles:
Vertically opposite angles are always equal.
If two lines intersect and form angles a°, b°, c°, and d° (going around), then:
- a = c (vertically opposite)
- b = d (vertically opposite)
- a + b = 180° (linear pair)
- b + c = 180° (linear pair)
- a + b + c + d = 360° (angles around a point)
Finding unknown angles:
If one angle at the intersection is known, all four angles can be found:
- The vertically opposite angle = same value.
- Each adjacent angle = 180° - (given angle).
Derivation and Proof
Proof that vertically opposite angles are equal:
Let two lines AB and CD intersect at point O, forming four angles: ∠AOC, ∠COB, ∠BOD, and ∠DOA.
To prove: ∠AOC = ∠BOD (vertically opposite angles)
- ∠AOC and ∠COB form a linear pair (they are on a straight line AB).
- Therefore: ∠AOC + ∠COB = 180° ... (i)
- ∠COB and ∠BOD also form a linear pair (they are on a straight line CD).
- Therefore: ∠COB + ∠BOD = 180° ... (ii)
- From (i): ∠AOC = 180° - ∠COB
- From (ii): ∠BOD = 180° - ∠COB
- Since both equal (180° - ∠COB):
- ∠AOC = ∠BOD
Similarly, ∠COB = ∠DOA can be proved.
Numerical verification:
- Let ∠AOC = 70°
- ∠COB = 180° - 70° = 110° (linear pair)
- ∠BOD = 180° - 110° = 70° (linear pair with ∠COB)
- ∠AOC = ∠BOD = 70°. Verified.
Types and Properties
Problems on vertically opposite angles:
1. Finding the vertically opposite angle:
- If one angle is given, the vertically opposite angle equals it.
2. Finding all four angles at an intersection:
- Given one angle, find the other three using vertically opposite angles and linear pairs.
3. Finding unknown variables:
- Angles expressed as algebraic expressions. Set vertically opposite angles equal and solve.
4. Multiple intersecting lines:
- Three or more lines intersecting at a point. Use vertically opposite angles and linear pairs together.
5. Problems combining vertically opposite angles with parallel lines:
- A transversal cutting parallel lines creates vertically opposite angles at each intersection point.
Solved Examples
Example 1: Example 1: Finding the vertically opposite angle
Problem: Two lines intersect at a point. One of the angles formed is 65°. Find the vertically opposite angle.
Solution:
- Vertically opposite angles are equal.
- Vertically opposite angle = 65°
Answer: The vertically opposite angle is 65°.
Example 2: Example 2: Finding all four angles
Problem: Two lines intersect. One angle is 120°. Find all four angles.
Solution:
Given: One angle = 120°
- Vertically opposite angle = 120°
- Adjacent angle = 180° - 120° = 60° (linear pair)
- Vertically opposite to 60° = 60°
The four angles are: 120°, 60°, 120°, 60°
Verification: 120 + 60 + 120 + 60 = 360°
Example 3: Example 3: Finding unknown variable
Problem: Two lines intersect forming angles (3x + 10)° and (5x - 30)° which are vertically opposite. Find x and the angles.
Solution:
Since vertically opposite angles are equal:
- 3x + 10 = 5x - 30
- 10 + 30 = 5x - 3x
- 40 = 2x
- x = 20
The angles:
- 3(20) + 10 = 70°
- 5(20) - 30 = 70°
All four angles: 70°, 110°, 70°, 110°
Answer: x = 20, the vertically opposite angles are 70° each.
Example 4: Example 4: More than two lines
Problem: Three lines pass through the same point, forming 6 angles. One angle is 40°, and the angle next to it is 70°. Find all six angles.
Solution:
Let the six angles be a, b, c, d, e, f going around the point.
- a = 40°, b = 70°
- c = 180° - 40° - 70° = 70° (since a + b + c = 180° on a straight line)
Wait, three lines create 6 angles. Adjacent angles on a straight line sum to 180°.
- a = 40°, b = 70°
- a + b + c forms a straight angle (if all three are on one side): c = 180° - 40° - 70° = 70°
- Vertically opposite: d = 40°, e = 70°, f = 70°
The six angles are: 40°, 70°, 70°, 40°, 70°, 70°
Verification: 40 + 70 + 70 + 40 + 70 + 70 = 360°
Example 5: Example 5: Identifying vertically opposite angles
Problem: Lines PQ and RS intersect at O. Name the pairs of vertically opposite angles.
Solution:
The four angles formed are: ∠POR, ∠ROS (wait — let me be precise).
Going around point O:
- ∠POR and ∠QOS are vertically opposite (they face each other).
- ∠POS and ∠QOR are vertically opposite (they face each other).
Answer: The two pairs of vertically opposite angles are: ∠POR = ∠QOS and ∠POS = ∠QOR.
Example 6: Example 6: Using vertically opposite angles and linear pair together
Problem: Two lines intersect. The ratio of two adjacent angles is 2:3. Find all four angles.
Solution:
Given: Adjacent angles are in ratio 2:3.
- Let the angles be 2x and 3x.
- Adjacent angles form a linear pair: 2x + 3x = 180°
- 5x = 180°
- x = 36°
The four angles:
- 2x = 72°
- 3x = 108°
- Vertically opposite to 72° = 72°
- Vertically opposite to 108° = 108°
Answer: The four angles are 72°, 108°, 72°, 108°.
Example 7: Example 7: Checking if given angles can be vertically opposite
Problem: Can 75° and 105° be vertically opposite angles?
Solution:
- Vertically opposite angles must be equal.
- 75° ≠ 105°
- Therefore, they cannot be vertically opposite angles.
Note: 75° and 105° add up to 180°, so they could be a linear pair (adjacent angles), but NOT vertically opposite.
Answer: No, they cannot be vertically opposite angles.
Example 8: Example 8: Finding angle with two intersecting lines
Problem: Two lines AB and CD intersect at O. If ∠AOC = 4y and ∠BOD = 6y - 40, find y and ∠AOC.
Solution:
Since ∠AOC and ∠BOD are vertically opposite:
- 4y = 6y - 40
- 40 = 6y - 4y
- 40 = 2y
- y = 20
∠AOC = 4(20) = 80°
All four angles: 80°, 100°, 80°, 100°
Answer: y = 20, ∠AOC = 80°.
Example 9: Example 9: Clock hands and vertically opposite angles
Problem: At what time do the hands of a clock form vertically opposite angles? Is this possible?
Solution:
- A clock has only TWO hands (hour and minute). Two lines form vertically opposite angles only when they cross each other.
- Clock hands share a common endpoint (the centre) but extend only in one direction from that point.
- They are rays, not full lines. So they form only ONE angle between them (or the reflex angle).
- Since clock hands do not extend in both directions from the centre, they do NOT form vertically opposite angles.
Answer: Clock hands cannot form vertically opposite angles because they are rays, not lines.
Example 10: Example 10: Real-life example
Problem: Two roads cross at a junction. The angle between road A and road B (measured on one side) is 55°. Find all four angles at the junction.
Solution:
The junction is like two intersecting lines.
- Angle 1 = 55°
- Vertically opposite angle = 55°
- Adjacent angle = 180° - 55° = 125°
- Vertically opposite to 125° = 125°
Answer: The four angles at the junction are 55°, 125°, 55°, 125°.
Real-World Applications
Road Intersections: When two roads cross, they form vertically opposite angles. Traffic engineers use these angles to design safe junctions.
Scissors: When scissors are opened, the two blades and the handles form two pairs of vertically opposite angles at the pivot point.
Letter X: The letter X is formed by two crossing lines. The opposite angles of the X are vertically opposite and equal.
Geometry Proofs: Vertically opposite angles are used as a basic step in many geometry proofs, especially those involving parallel lines and triangles.
Engineering: Structural engineers use the property of vertically opposite angles when analysing forces at joints where beams cross.
Key Points to Remember
- When two lines intersect, they form 4 angles at the point of intersection.
- Vertically opposite angles are always equal.
- There are 2 pairs of vertically opposite angles at each intersection.
- Adjacent angles at an intersection form a linear pair (sum = 180°).
- The sum of all four angles at an intersection = 360°.
- If one angle is known, all four can be found.
- Vertically opposite angles are NOT adjacent — they are across from each other.
- This property applies only when two full lines (not rays or segments) intersect.
- The proof uses the linear pair axiom: adjacent angles on a straight line sum to 180°.
- Vertically opposite angles are used frequently in proofs involving parallel lines and transversals.
Practice Problems
- Two lines intersect. One angle is 47°. Find all four angles.
- The ratio of two adjacent angles formed by two intersecting lines is 5:4. Find all four angles.
- Two lines intersect forming angles (2x + 15)° and (3x - 25)° which are vertically opposite. Find x and the angles.
- Three lines pass through the same point. One of the six angles is 50° and the one next to it is 60°. Find all six angles.
- Can 90° and 90° be vertically opposite angles? If yes, what are the other two angles?
- Two streets cross at an angle of 35°. What are the other three angles at the crossing?
Frequently Asked Questions
Q1. What are vertically opposite angles?
When two lines intersect at a point, the angles that are directly opposite each other (across the intersection) are called vertically opposite angles. They are always equal.
Q2. Why are they called 'vertically opposite'?
The word 'vertical' here comes from 'vertex' (the point of intersection), not from 'up and down'. They are opposite each other at the vertex point.
Q3. Are vertically opposite angles supplementary?
Not necessarily. Vertically opposite angles are EQUAL. They are supplementary (sum to 180°) only when each angle is 90°. Adjacent angles at an intersection are supplementary (linear pair), not vertically opposite ones.
Q4. How many pairs of vertically opposite angles are formed when two lines intersect?
Two pairs. If the four angles are a, b, c, d (going around), then a = c and b = d. So there are 2 pairs of vertically opposite angles.
Q5. Can vertically opposite angles be complementary?
Only if each angle is 45° (since they must be equal and sum to 90°). In that case, each is 45° and the adjacent angles are 135° each.
Q6. Do three intersecting lines form vertically opposite angles?
Yes. When three lines pass through one point, they form 6 angles. There are 3 pairs of vertically opposite angles. Each pair consists of the angles directly across from each other.
Q7. What is the difference between vertically opposite angles and a linear pair?
Vertically opposite angles are OPPOSITE each other and are EQUAL. A linear pair consists of ADJACENT angles that sum to 180°. At an intersection, each angle is part of one vertically opposite pair AND two linear pairs.
Q8. If one angle at an intersection is 90°, what are the other angles?
If one angle is 90°, all four angles are 90°. The vertically opposite angle is 90°, and each adjacent angle is 180° - 90° = 90°. The lines are perpendicular.
Q9. Can vertically opposite angles be formed by parallel lines?
No. Parallel lines never intersect, so they cannot form vertically opposite angles. Vertically opposite angles are formed only at the point where two lines cross.
Q10. Are vertically opposite angles congruent?
Yes. 'Congruent' means equal in measure. Vertically opposite angles are always congruent (equal).
