Linear pair and vertically opposite angles — with proof and applications

Question

What is a linear pair of angles? What are vertically opposite angles? Prove that vertically opposite angles are equal. If two lines intersect and one angle is 65°, find all other angles.

Solution — Step by Step

A linear pair consists of two adjacent angles whose non-common arms form a straight line. The sum of a linear pair is always 180°.

If two angles $\angle AOB$ and $\angle BOC$ form a linear pair, then:

\angle AOB + \angle BOC = 180°

When two straight lines intersect, they form two pairs of vertically opposite angles. These angles are across from each other at the intersection point and are always equal.

Let two lines intersect forming angles $a$ , $b$ , $c$ , $d$ (going clockwise).

$a$ and $b$ form a linear pair: $a + b = 180°$ … (i) $b$ and $c$ form a linear pair: $b + c = 180°$ … (ii)

From (i) and (ii): $a + b = b + c$

Therefore: $a = c$ (vertically opposite angles are equal).

Similarly, $b = d$ .

If one angle is 65°, its vertically opposite angle is also 65°.

The adjacent angle forms a linear pair: $180° - 65° = \mathbf{115°}$ .

Its vertically opposite angle is also 115°.

The four angles are: 65°, 115°, 65°, 115°.

flowchart TD
    A[Two lines intersect] --> B{Identify angle pairs}
    B --> C[Adjacent angles: Linear pair = 180°]
    B --> D[Opposite angles: Vertically opposite = equal]
    C --> E[If one angle = x, adjacent = 180 - x]
    D --> F[Opposite angle also = x]

Why This Works

The proof relies on a simple property: angles on a straight line add to 180°. Since both angles in a linear pair equal 180° minus the shared angle, the vertically opposite angles must be equal. No additional axioms needed.

Common Mistake

Students confuse adjacent angles with vertically opposite angles. Adjacent angles share a common arm and are next to each other (they form a linear pair if on a line). Vertically opposite angles are across from each other with NO common arm.

In any intersection of two lines, you get exactly 4 angles. You only need to find ONE — then all others follow: opposite = same, adjacent = 180° minus it. Always check that your four angles add to 360°.

Question

Solution — Step by Step

Why This Works

Common Mistake

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