Question

which statement would be included in a proof of why ( mangle c + mangle g + mangle h = 180^circ )?

• angles f and h are congruent.
• angles c and e are congruent.
• angles a and g are congruent.
• angles f and c are congruent.

Explanation:

Step1: Analyze Angle Relationships

To prove \( m\angle C + m\angle G + m\angle H = 180^\circ \), we use alternate interior angles (from parallel lines) and triangle angle sum. Angles \( F \) and \( H \) are alternate interior angles (if lines are parallel), so they are congruent. This helps substitute angles to form a triangle's angle sum.

Step2: Evaluate Options

• "Angles \( C \) and \( E \)": Vertical angles? Not helpful here.
• "Angles \( A \) and \( G \)": No clear parallel - line relation.
• "Angles \( F \) and \( C \)": Not congruent (no reason).
• "Angles \( F \) and \( H \)": Alternate interior angles (congruent), useful for substituting \( \angle F \) with \( \angle H \) to form a triangle with \( \angle C \) and \( \angle G \), leading to \( 180^\circ \).

Answer:

Angles \( F \) and \( H \) are congruent.