Geometry

Question Analysis

This question involves the properties of angles formed by intersecting lines. The main focus is to find the measure of ∠4 using the concepts of vertical angles and linear pairs.

 

Key Concept Explanation

Vertical Angles: When two lines intersect, vertical angles are opposite each other and have equal measures.

Linear Pairs: Adjacent angles that share a common vertex and a common side, with non - common sides forming a straight line. The sum of angles in a linear pair is 180°.

 

Step - by - Step Solution

1. Observe the relationship between ∠4 and the 105° angle:

∠4 and the 105° angle form a linear pair. They are adjacent, and their non - common sides form a straight line.

According to the property of linear pairs, the sum of the measures of angles in a linear pair is 180°. So, we can set up the equation m∠4 + 105°= 180°.

To solve for m∠4, we subtract 105° from both sides of the equation: m∠4 = 180° - 105°.

Performing the subtraction, 180° - 105° = 75°. So, m∠4 = 75°.

 

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