Question

given the diagram below, hannah writes \\(m\angle 1 + m\angle 4 = 180^{\circ}\\), \\(m\angle 2 + m\angle 5 = 180^{\circ}\\), and \\(m\angle 3 + m\angle 6 = 180^{\circ}\\). which of the following reasons allows her to write these sums?

a. addition property of equality
b. corresponding angles theorem
c. triangle sum theorem
d. definition of linear pair

Explanation:

Analyze the given angle pairs

We examine the pairs of angles in the diagram:

• \(\angle 1\) and \(\angle 4\)
• \(\angle 2\) and \(\angle 5\)
• \(\angle 3\) and \(\angle 6\)

Each pair consists of adjacent angles that form a straight line.

Apply geometric definitions

Two adjacent angles forming a straight line are a linear pair.
By definition, a linear pair of angles is supplementary.
Therefore, their measures sum to \(180^\circ\).

Match with the given options

The reason Hannah can write these sums is the definition of a linear pair.
This directly matches option D.

Answer:

• A. Addition property of equality
• B. Corresponding angles theorem
• C. Triangle sum theorem
• D. Definition of linear pair (Correct answer)