Angle Addition Postulate - Geometry | Luna Learn 24

Angle Addition Postulate

In this lesson, we will explore the Angle Addition Postulate. This postulate states that if a point lies on the interior of an angle, the measure of the larger angle is the sum of the two smaller angles formed. This concept is essential in Geometry, and we will provide clear examples, exercises, and a homework section to reinforce your understanding.

Outline

Explanation of the Angle Addition Postulate
Diagram examples illustrating the concept
Step-by-step examples
Exercises with solutions
Homework and revision section

1. Explanation of the Angle Addition Postulate

The Angle Addition Postulate states that if point \( P \) lies on the interior of \( \angle ABC \), then:

\[ \angle ABP + \angle PBC = \angle ABC \]

Example 1: Using the Angle Addition Postulate

Given that \( \angle ABP = 30^\circ \) and \( \angle PBC = 45^\circ \), find the measure of \( \angle ABC \).

Using the Angle Addition Postulate:

\[ \angle ABP + \angle PBC = \angle ABC \] \[ 30^\circ + 45^\circ = 75^\circ \]

So, \( \angle ABC = 75^\circ \).

2. Diagram Examples

Example 2: Visualizing the Angle Addition Postulate

In the diagram below, find the measure of \( \angle XYZ \) if \( \angle XYW = 40^\circ \) and \( \angle WYZ = 50^\circ \).

By applying the Angle Addition Postulate:

\[ \angle XYW + \angle WYZ = \angle XYZ \] \[ 40^\circ + 50^\circ = 90^\circ \]

So, \( \angle XYZ = 90^\circ \).

Example 3: Solving with Unknown Angles

If \( \angle ABC = 120^\circ \) and \( \angle ABP = 80^\circ \), find \( \angle PBC \).

Using the Angle Addition Postulate:

\[ \angle ABP + \angle PBC = \angle ABC \] \[ 80^\circ + \angle PBC = 120^\circ \] \[ \angle PBC = 120^\circ - 80^\circ = 40^\circ \]

Exercises

Exercise 1: Find \( \angle LMN \) if \( \angle LMP = 25^\circ \) and \( \angle PMN = 55^\circ \).

\[ \angle LMP + \angle PMN = \angle LMN \] \[ 25^\circ + 55^\circ = 80^\circ \]

The measure of \( \angle LMN \) is \( 80^\circ \).

Exercise 2: If \( \angle ABC = 150^\circ \) and \( \angle ABP = 90^\circ \), find \( \angle PBC \).

\[ \angle ABP + \angle PBC = \angle ABC \] \[ 90^\circ + \angle PBC = 150^\circ \] \[ \angle PBC = 150^\circ - 90^\circ = 60^\circ \]

Homework

Complete the following exercises:

Find \( \angle DEF \) if \( \angle DEP = 35^\circ \) and \( \angle PEF = 60^\circ \).
If \( \angle XYZ = 140^\circ \) and \( \angle XYW = 70^\circ \), find \( \angle WYZ \).