Prove that vertically opposite angles are equal

Question

Prove that vertically opposite angles are equal.

Solution — Step by Step

When two straight lines intersect, they form 4 angles at the point of intersection. Let lines AB and CD intersect at point O.

Label the four angles:

$\angle AOC = \angle 1$ (top)
$\angle COB = \angle 2$ (right)
$\angle BOD = \angle 3$ (bottom)
$\angle DOA = \angle 4$ (left)

We need to prove: $\angle 1 = \angle 3$ and $\angle 2 = \angle 4$ .

Angles on a straight line add up to 180° (Linear Pair Axiom).

Since AOB is a straight line:

\angle 1 + \angle 2 = 180° \quad \text{...(i)}

Since COD is a straight line:

\angle 2 + \angle 3 = 180° \quad \text{...(ii)}

From equations (i) and (ii):

\angle 1 + \angle 2 = \angle 2 + \angle 3

Subtracting $\angle 2$ from both sides:

\boxed{\angle 1 = \angle 3}

Similarly, using lines AOB and COD in another pair of linear pairs:

\angle 1 + \angle 4 = 180° \quad \text{...(iii)}

From (i) and (iii): $\angle 1 + \angle 2 = \angle 1 + \angle 4$

Therefore $\boxed{\angle 2 = \angle 4}$

We have proved that vertically opposite angles are equal:

$\angle AOC = \angle BOD$ ( $\angle 1 = \angle 3$ )
$\angle COB = \angle DOA$ ( $\angle 2 = \angle 4$ )

This result is the Vertically Opposite Angles Theorem (VOA Theorem).

Why This Works

The proof relies on two facts:

A straight line makes an angle of 180°
If two quantities are each equal to the same third quantity, they are equal to each other (axiom of equality)

The key insight: both $\angle 1$ and $\angle 3$ are each supplementary to $\angle 2$ . Since both sum to the same value (180°), they must be equal to each other. This is a classic “same supplement” argument.

Common Mistake

Some students write “vertically opposite angles are equal” as a reason in other proofs without citing the theorem or proving it when asked. In CBSE board exams, if the question explicitly says “prove,” you must start from axioms (linear pair) and work up — you cannot cite the VOA theorem as a reason in its own proof. Always set up the angle names clearly first, then use the linear pair property.

For CBSE Class 9, this theorem is a 3-mark question. The structured proof (statement → diagram → given/to prove → proof) earns full marks. Write: Given: Lines AB and CD intersect at O. To Prove: $\angle AOC = \angle BOD$ . Proof: (steps as above). This format is exactly what CBSE expects.

Question

Solution — Step by Step

Why This Works

Common Mistake

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