Question

complete the proof that ( mangle f + mangle j = mangle hgj ) without using the triangle angle sum theorem.

(image of a geometric figure with points f, j, g, i, h and a table with statements and reasons)

1. ( overline{fj} parallel overline{gi} ) - given
2. ( angle f cong angle hgi ) - corresponding angles theorem
3. ( angle j cong angle igj ) - alternate interior angles theorem
4. ( mangle hgi + mangle igj = mangle hgj ) - blank for reason
5. ( mangle f + mangle j = mangle hgj ) - additive property of angle measure (and other options: additive property of length, all right angles are congruent, angles forming a linear pair sum to ( 180^circ ), definition of angle bisector, definition of complementary angles)

Explanation:

Step1: Analyze Statement 4

Statement 4 says \( m\angle HGI + m\angle IGJ = m\angle HGJ \). We need to find the reason for this. Looking at the diagram, \( \angle HGI \) and \( \angle IGJ \) are adjacent angles that together form \( \angle HGJ \). The Additive Property of Angle Measure states that if two angles are adjacent (share a common side and vertex, and their non - common sides form a larger angle), then the sum of their measures is equal to the measure of the larger angle they form.

Step2: Confirm the Reason

So the reason for \( m\angle HGI + m\angle IGJ = m\angle HGJ \) is the Additive Property of Angle Measure. Then, for statement 5, since \( \angle F\cong\angle HGI \) (so \( m\angle F = m\angle HGI \)) and \( \angle J\cong\angle IGJ \) (so \( m\angle J=m\angle IGJ \)), by substitution (which is part of the Additive Property of Angle Measure application here), we get \( m\angle F + m\angle J = m\angle HGJ \). But for statement 4, the key is recognizing the angle addition postulate (Additive Property of Angle Measure) for adjacent angles forming a larger angle.

Answer:

For statement 4, the reason is "Additive Property of Angle Measure" (specifically the Angle Addition Postulate which is part of the Additive Property of Angle Measure, stating that the measure of an angle formed by two adjacent angles is the sum of the measures of the two adjacent angles). For the overall proof, the missing reason for statement 4 is the Additive Property of Angle Measure.