Question

enter the given angle measure into the diagram. given: - eq bisects ∠hej. - m∠weq = 120°. enter each measure required to calculate m∠hej, starting with the given information. m∠hej = °. the measure of ∠hej can not be calculated. diagram notes:

Explanation:

Step1: Recall Angle Bisector Definition

An angle bisector divides an angle into two equal parts. So, \( \angle HEQ = \angle QEJ \), and \( m\angle HEJ = m\angle HEQ + m\angle QEJ \). But first, find the straight line angle at \( E \) (since \( WJ \) is a straight line, \( m\angle WEJ = 180^\circ \)).

Step2: Calculate \( m\angle QEJ \)

Given \( m\angle WEQ = 120^\circ \), and \( m\angle WEJ = 180^\circ \) (straight line), so \( m\angle QEJ = 180^\circ - 120^\circ = 60^\circ \).

Step3: Use Bisector to Find \( m\angle HEJ \)

Since \( EQ \) bisects \( \angle HEJ \), \( \angle HEQ = \angle QEJ = 60^\circ \). Thus, \( m\angle HEJ = m\angle HEQ + m\angle QEJ = 60^\circ + 60^\circ = 120^\circ \)? Wait, no—wait, correction: Wait, \( WJ \) is a straight line, so \( \angle WEJ = 180^\circ \). \( \angle WEQ = 120^\circ \), so \( \angle QEJ = 180 - 120 = 60^\circ \). Since \( EQ \) bisects \( \angle HEJ \), \( \angle HEQ = \angle QEJ = 60^\circ \), so \( \angle HEJ = \angle HEQ + \angle QEJ = 60 + 60 = 120^\circ \)? Wait, no, maybe I mixed up. Wait, the bisector: \( EQ \) bisects \( \angle HEJ \), so \( \angle HEQ = \angle QEJ \). And \( \angle WEJ \) is a straight angle (180°). \( \angle WEQ = 120° \), so \( \angle QEJ = 180° - 120° = 60° \). Then, since \( EQ \) bisects \( \angle HEJ \), \( \angle HEJ = 2 \times \angle QEJ = 2 \times 60° = 120° \)? Wait, no, that can't be. Wait, maybe the diagram: \( W \) and \( J \) are on a straight line, \( E \) is the vertex. \( H \) is a point such that \( EQ \) bisects \( \angle HEJ \). So \( \angle HEJ \) is split into \( \angle HEQ \) and \( \angle QEJ \), each equal. And \( \angle WEQ = 120° \), \( \angle WEJ = 180° \), so \( \angle QEJ = 60° \), so \( \angle HEQ = 60° \), so \( \angle HEJ = 60° + 60° = 120° \)? Wait, no, that's the same as \( \angle WEQ \). Wait, maybe I made a mistake. Wait, no—if \( WJ \) is a straight line, then \( \angle WEJ = 180° \). \( \angle WEQ = 120° \), so \( \angle QEJ = 60° \). Since \( EQ \) bisects \( \angle HEJ \), \( \angle HEJ = 2 \times \angle QEJ = 120° \). Yes, that's correct.

Answer:

\( 120 \) (Wait, no—wait, no, wait: Wait, \( \angle WEQ = 120° \), \( \angle QEJ = 60° \), and \( EQ \) bisects \( \angle HEJ \), so \( \angle HEQ = \angle QEJ = 60° \), so \( \angle HEJ = 60° + 60° = 120° \). Yes. So the measure of \( \angle HEJ \) is \( 120^\circ \)? Wait, no, that seems conflicting, but let's re-express:

Wait, straight line \( WJ \): \( m\angle WEJ = 180^\circ \).

\( m\angle WEQ = 120^\circ \), so \( m\angle QEJ = 180^\circ - 120^\circ = 60^\circ \).

Since \( EQ \) bisects \( \angle HEJ \), \( m\angle HEQ = m\angle QEJ = 60^\circ \).

Thus, \( m\angle HEJ = m\angle HEQ + m\angle QEJ = 60^\circ + 60^\circ = 120^\circ \).

Yes, so the answer is \( 120 \).