Linear pair and vertically opposite angles — with proof and applications

Question

What are linear pairs and vertically opposite angles, how do we prove that vertically opposite angles are equal, and how do we apply these in problems?

Solution — Step by Step

A linear pair is formed when two adjacent angles share a common arm and their non-common arms form a straight line (opposite rays).

\angle AOB + \angle BOC = 180°

The two angles in a linear pair are supplementary — they always add up to 180 degrees. This follows directly from the fact that a straight line makes an angle of 180 degrees.

When two straight lines intersect, they form two pairs of vertically opposite angles (also called vertical angles). These are the angles that are across from each other at the intersection point.

If lines $AB$ and $CD$ intersect at $O$ :

$\angle AOC$ and $\angle BOD$ are vertically opposite
$\angle AOD$ and $\angle BOC$ are vertically opposite

Let $\angle AOC = x$ and $\angle AOD = y$ .

Since $\angle AOC$ and $\angle AOD$ form a linear pair:

x + y = 180° \quad \ldots (1)

Since $\angle AOD$ and $\angle DOB$ form a linear pair:

y + \angle DOB = 180° \quad \ldots (2)

From (1): $y = 180° - x$ Substituting in (2): $(180° - x) + \angle DOB = 180°$

\angle DOB = x = \angle AOC

Therefore, vertically opposite angles are equal. $\square$

Example: Two lines intersect. One of the angles is $65°$ . Find all four angles.

The vertically opposite angle = $65°$
Each adjacent angle = $180° - 65° = 115°$ (linear pair)

So the four angles are: $65°, 115°, 65°, 115°$ .

Rule: When two lines intersect, you only need ONE angle to find all four.

flowchart TD
    A["Two lines intersect at a point"] --> B["4 angles formed"]
    B --> C["Adjacent angles form linear pairs: sum = 180"]
    B --> D["Opposite angles are vertically opposite: equal"]
    C --> E["Know one angle? Adjacent = 180 minus that angle"]
    D --> F["Know one angle? Opposite angle is the same"]
    E --> G["All 4 angles determined from just 1 angle"]
    F --> G

Why This Works

The proof uses nothing more than the straight angle property (a straight line = 180 degrees) applied twice. Since two intersecting lines create two straight lines through the point, each pair of adjacent angles must sum to 180 degrees. This forces the opposite angles to be equal — it is a simple consequence of the supplementary property used twice.

Alternative Method

For a visual proof, fold the figure along the angle bisector of one angle. The two vertically opposite angles overlap perfectly, showing they are equal. While this is not a formal proof, it builds strong geometric intuition.

Common Mistake

Students confuse “adjacent angles” with “vertically opposite angles” when three or more lines intersect at a point. With three lines through a point, you get 6 angles — and not all opposite-looking pairs are vertically opposite. Vertically opposite angles are formed by the same two lines. Always identify which specific pair of lines creates the angle pair before claiming they are vertically opposite.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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