Vertically opposite angles are equal. If one is 60°, the opposite is also 60°.

Find Vertically Opposite Angles

The Question

Two straight lines intersect at a point. One of the angles formed is 60°. Find all four angles at the point of intersection.

What Are Vertically Opposite Angles?

When two lines cross each other at a point, they form four angles.

The angles that are directly across from each other (diagonally opposite, sharing only the vertex) are called vertically opposite angles.

They are called “vertically opposite” not because they are vertical (up-down), but because they share the same vertex and are opposite each other.

         A
         |
    C----O----D
         |
         B

Angles formed at O: ∠AOC, ∠COB, ∠BOD, ∠DOA

Vertically opposite pairs:
∠AOC and ∠BOD (across from each other)
∠COB and ∠AOD (across from each other)

Key property: Vertically opposite angles are always equal.

When two lines intersect: ∠1 = ∠3 (vertically opposite) ∠2 = ∠4 (vertically opposite) ∠1 + ∠2 = 180° (linear pair) ∠1 + ∠2 + ∠3 + ∠4 = 360° (complete angle at a point)

Why Are They Always Equal? (Proof)

Let’s say the four angles at intersection point O are ∠1, ∠2, ∠3, ∠4 going around.

∠1 and ∠2 form a linear pair → ∠1 + ∠2 = 180° … (i) ∠2 and ∠3 form a linear pair → ∠2 + ∠3 = 180° … (ii)

From (i) and (ii): ∠1 + ∠2 = ∠2 + ∠3

Subtract ∠2 from both sides: ∠1 = ∠3 ✓

Similarly, we can show ∠2 = ∠4.

Solution

Given: One angle at the intersection = 60°

       60°
    --------
       ?

Step 1: The angle opposite to 60° is vertically opposite → it is also 60°.

Step 2: The angle adjacent to 60° forms a linear pair with it:

Adjacent angle = 180° - 60° = 120°

Step 3: The angle opposite to the 120° is also vertically opposite → also 120°.

All four angles: 60°, 120°, 60°, 120°

Check: 60° + 120° + 60° + 120° = 360° ✓ (angles at a point sum to 360°)

Whenever two lines intersect, you always get two pairs of equal angles. The four angles always come in two matching pairs that together add to 360°.

Common mistake: Saying vertically opposite angles are supplementary (add to 180°).

Vertically opposite angles are EQUAL — not supplementary (unless both are 90°).

60° and 60° are NOT 180°. They are just equal.

They are supplementary to the other pair: 60° + 120° = 180° (that’s the linear pair).

Try These Similar Problems

Problem 1: Two lines intersect. One angle is 45°. Find all four angles.

One angle = 45°. Vertically opposite = 45°. Linear pair = 180° - 45° = 135°. Vertically opposite of 135° = 135°.

Four angles: 45°, 135°, 45°, 135° Check: 45 + 135 + 45 + 135 = 360° ✓

Problem 2: Two lines intersect. Two of the angles are equal. What type of angles are they?

If two adjacent angles are equal and they form a linear pair, then each = 90°. The lines are perpendicular to each other. All four angles would be 90°.

But more generally, when two of the four angles are equal, they must be the vertically opposite pair.

Problem 3: When two lines intersect, one angle is (3x + 10)° and its vertically opposite angle is (5x - 20)°. Find x and the measure of the angles.

Vertically opposite angles are equal: 3x + 10 = 5x - 20 10 + 20 = 5x - 3x 30 = 2x x = 15

Angles = 3(15) + 10 = 45 + 10 = 55° Check: 5(15) - 20 = 75 - 20 = 55° ✓

The vertically opposite angles are both 55°. The other pair = 180° - 55° = 125° each.

Exam tip: Vertically opposite angle problems in CBSE exams often involve setting up equations (like Problem 3 above) to find x. Always state: “Vertically opposite angles are equal” as your reason before writing the equation. This shows the examiner you know the theorem.

Find Vertically Opposite Angles — When Two Lines Cross