Question

fj || gi. complete the proof that m∠hgj = m∠f + m∠j without using the triangle angle sum theorem.

image of a diagram with points j, i, h, f, g and a table with statements and reasons

1. fj || gi reason: ?
2. ∠f ≅ ∠hgi reason: ?
3. ∠j ≅ ∠igj reason: ?
4. m∠hgj = m∠hgi + m∠igj reason: ?
5. m∠hgj = m∠f + m∠j reason: ?

Explanation:

Step1: Identify Given

The problem states \(\overline{FJ} \parallel \overline{GI}\) as given. So for statement 1, the reason is "Given".

Step2: Corresponding Angles (∠F and ∠HGI)

Since \(\overline{FJ} \parallel \overline{GI}\) and \(FG\) is a transversal, \(\angle F \cong \angle HGI\) by the Corresponding Angles Postulate (when two parallel lines are cut by a transversal, corresponding angles are congruent).

Step3: Alternate Interior Angles (∠J and ∠IGJ)

Since \(\overline{FJ} \parallel \overline{GI}\) and \(JG\) is a transversal, \(\angle J \cong \angle IGJ\) by the Alternate Interior Angles Theorem (when two parallel lines are cut by a transversal, alternate interior angles are congruent).

Step4: Angle Addition Postulate

The angle addition postulate states that if a point lies in the interior of an angle, the sum of the two smaller angles equals the larger angle. Here, \(I\) is in the interior of \(\angle HGJ\), so \(m\angle HGJ = m\angle HGI + m\angle IGJ\) (Angle Addition Postulate).

Step5: Substitution Property

From steps 2 and 3, we know \(\angle F \cong \angle HGI\) (so \(m\angle F = m\angle HGI\)) and \(\angle J \cong \angle IGJ\) (so \(m\angle J = m\angle IGJ\)). Substituting these into the equation from step 4 gives \(m\angle HGJ = m\angle F + m\angle J\) (Substitution Property of Equality).

Answer:

1. Reason for statement 1: Given
2. Reason for statement 2: Corresponding Angles Postulate
3. Reason for statement 3: Alternate Interior Angles Theorem
4. Reason for statement 4: Angle Addition Postulate
5. Reason for statement 5: Substitution Property of Equality