Basic Geometry • Topic 3 of 3

Pairs of Angles: Adjacent & Vertically Opposite

What are adjacent and vertically opposite angles? Angles can also be paired based on their positions relative to one another when lines cross or meet.

Adjacent Angles: Two angles are adjacent if they are next-door neighbors. To be adjacent, they must meet three strict rules:

They share a common vertex (corner point).
They share a common arm (side).
They do not overlap (their interiors are completely separate).

Think of two adjacent rooms in a house that share a single common wall.

Vertically Opposite Angles: When two straight lines cross each other like an 'X', they create four angles. The angles that sit directly across from each other are called vertically opposite angles.

Crucial Property: Vertically opposite angles are always equal in measure.
Think of an open pair of scissors; as you open the handles wider, the blades open wider by the exact same amount.

Solved Examples

Worked Example

In the given figure, two lines intersect. If one of the angles formed is $55^\circ$, find the measure of its vertically opposite angle and the remaining two angles.

Solution*Step 1: Let the given angle be $\angle 1 = 55^\circ$. *Step 2: The vertically opposite angle to $\angle 1$ is $\angle 3$. Since vertically opposite angles are equal, $\angle 3 = 55^\circ$. *Step 3: The angle adjacent to $\angle 1$ on a straight line is $\angle 2$. They form a linear pair, so they add up to $180^\circ$. *Step 4: Calculate $\angle 2 = 180^\circ - 55^\circ = 125^\circ$. *Step 5: The vertically opposite angle to $\angle 2$ is $\angle 4$. Therefore, $\angle 4 = 125^\circ$.

Key Points

Adjacent angles share a common vertex, a common arm, and do not overlap.
Vertically opposite angles are formed across from each other when two lines cross.
Vertically opposite angles are always completely equal in measurement.
When adjacent angles sit together on a straight line, they form a linear pair and add up to $180^\circ$.
---

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Adjacent angles share a common vertex and a common:

Explanation: Common arm.

Q2.Vertically opposite angles are:

Explanation: Equal.

Q3.A linear pair of angles sums to:

Explanation: $180^\circ$.

Q4.If one angle of a linear pair is $70^\circ$, the other is:

Explanation: $180-70$.

Practice more MCQs → 📝 Assignment · 35 marks