Vertical Angles and Linear Pairs - Geometry | Luna Learn 24

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.

\[ 110^\circ + \text{Other Angle} = 180^\circ \] \[ \text{Other Angle} = 180^\circ - 110^\circ = 70^\circ \]

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 \( 120^\circ \).