Question

problem 11: (first taught in lesson 18) from this given statement, select the definition, property, postulate, or theorem that justifies the prove statement. given: $overleftrightarrow{dj}$ and $overleftrightarrow{gf}$ intersect at e. prove: $angle def$ and $angle gej$ are congruent. diagram of intersecting lines at e (points d, j, g, f) multiple choice options a, b, c, d, e; “reflexive property” shown; “go” button

Explanation:

1. Identify the Angles: \( \angle DEF \) and \( \angle GEJ \) are formed by intersecting lines \( \overleftrightarrow{DJ} \) and \( \overleftrightarrow{GF} \) at \( E \), so they are vertical angles.
2. Recall the Theorem: The Vertical Angles Theorem states that vertical angles are congruent. The Reflexive Property (e.g., \( x = x \)) does not apply here, as it relates to a figure being congruent to itself, not to vertical angles. Thus, the Vertical Angles Theorem justifies \( \angle DEF \cong \angle GEJ \).

Answer:

The correct justification should be the "Vertical Angles Theorem" (not the Reflexive Property). The Reflexive Property is for a quantity being equal to itself (e.g., \( \angle A \cong \angle A \) or \( AB = AB \)), while vertical angles (formed by intersecting lines) are congruent by the Vertical Angles Theorem. Since \( \overleftrightarrow{DJ} \) and \( \overleftrightarrow{GF} \) intersect at \( E \), \( \angle DEF \) and \( \angle GEJ \) are vertical angles, so they are congruent by the Vertical Angles Theorem. If we assume the options had the Vertical Angles Theorem (even though the shown option is Reflexive Property which is incorrect), the correct concept is Vertical Angles Theorem. But based on the problem's context of intersecting lines and proving vertical angles congruent, the right principle is Vertical Angles Theorem.