Vertical Angles and Linear Pairs
In this lesson, we will explore Vertical Angles and Linear Pairs. Understanding these concepts is crucial in Geometry. Vertical angles are opposite angles formed by two intersecting lines, and linear pairs are adjacent angles that form a straight line. We will cover definitions, provide examples, and offer exercises to help reinforce your knowledge.
Outline
- Definition of Vertical Angles
- Definition of Linear Pairs
- Examples and Diagrams
- Exercises with Solutions
- Homework and Revision Section
1. Definition of Vertical Angles
Vertical angles are the angles opposite each other when two lines intersect. These angles are always equal.

Example 1: Identifying Vertical Angles
In the diagram below, identify the vertical angles and state their measures if one of the vertical angles is \( 70^\circ \).

Vertical angles are equal, so the measure of the other angle is also \( 70^\circ \).
2. Definition of Linear Pairs
A linear pair consists of two adjacent angles that form a straight line. The sum of the angles in a linear pair is always \( 180^\circ \).

Example 2: Identifying Linear Pairs
In the diagram, if one angle of the linear pair is \( 120^\circ \), find the measure of the adjacent angle.

The sum of angles in a linear pair is \( 180^\circ \):
\[ 120^\circ + \text{Adjacent Angle} = 180^\circ \] \[ \text{Adjacent Angle} = 180^\circ - 120^\circ = 60^\circ \]Examples and Practice
Example 3: Finding Vertical Angles
If two intersecting lines create one pair of vertical angles measuring \( 45^\circ \), what are the measures of the other three vertical angles?
Vertical angles are equal:
\[ \text{Other Three Angles} = 45^\circ \]Example 4: Solving Linear Pair Problems
Find the measure of the missing angle in a linear pair if one angle is \( 85^\circ \).
The sum of angles in a linear pair is \( 180^\circ \):
\[ 85^\circ + \text{Missing Angle} = 180^\circ \] \[ \text{Missing Angle} = 180^\circ - 85^\circ = 95^\circ \]Exercises
Exercise 1: In the diagram, one angle of a linear pair is \( 110^\circ \). Find the measure of the other angle.
Exercise 2: If two vertical angles are \( x \) and \( 3x \), find the value of \( x \).
Since vertical angles are equal:
\[ x = 3x \] \[ x = 3x \text{ (This equation is not true, which indicates a mistake in the problem or an incorrect assumption.)} \]Homework
Complete the following exercises:
- Find the measures of both angles in a linear pair where one angle is \( 75^\circ \).
- Determine the measures of all four vertical angles if two intersecting lines form angles of \( 60^\circ \) and \( 120^\circ \).
Revision Section
Review the concepts of Vertical Angles and Linear Pairs:
- Understand that vertical angles are always equal.
- Know that linear pairs add up to \( 180^\circ \).
Watch this Video for More Explanation:
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