Question

fj || gi. complete the proof that m∠hgj = m∠f + m∠j 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. fj || gi (given)
2. ∠f ≅ ∠hgi (corresponding angles theorem)
3. ∠j ≅ ∠igj (alternate interior angles theorem)
4. m∠hgj = m∠hgi + m∠igj (dropdown for reason)
5. m∠hgj = m∠f + m∠j (dropdown for reason with options: aa similarity, aas, additive property of angle measure, additive property of length, all right angles are congruent, alternate exterior angles theorem)

Explanation:

Step1: Analyze Statement 4

Statement 4 says \( m\angle HGJ = m\angle HGI + m\angle IGJ \). We need to find the reason for this. The Additive Property of Angle Measure states that if a point lies in the interior of an angle, the sum of the measures of the two smaller angles formed is equal to the measure of the larger angle. Here, \( \angle HGI \) and \( \angle IGJ \) are adjacent angles forming \( \angle HGJ \), so their measures add up to \( \angle HGJ \)'s measure.

Step2: Confirm the Reason

Among the options, "Additive Property of Angle Measure" fits this description. Other options like AA Similarity, AAS, etc., are for triangle similarity or congruence, not angle addition. Additive Property of Length is for segments, not angles. All right angles congruent and Alternate Exterior Angles Theorem don't apply here.

Answer:

Additive Property of Angle Measure