Question

17). complete each proof. write the letter and reason of the correct answer from the word bank.
given: f is the midpoint of \\( \overline { e g } , \overline { e h } \cong \overline { f j } , \overline { h f } \cong \overline { j g } \\)
prove: \\( \angle e f h \cong \angle f g j \\)

word bank (there are extras and can be used more than once)
\\( a - \overline { h f } \cong \overline { j g } \\)
\\( b - \triangle e h f \cong \triangle f j g \\)
\\( c - s s s \\)
\\( d - \\) definition of midpoint
\\( e - \\) given
\\( f - \overline { e h } \cong \overline { f j } \\)
\\( g - c p c t c \\)
\\( h - \overline { e f } \cong \overline { f g } \\)
\\( k - s a s \\)

Explanation:

Step1: State the given information

1. Statements: F is the midpoint of \( \overline{EG} \)

Reasons: \( E \) (Given)

Step2: Use the mid - point definition

1. Statements: \( \overline{EF}\cong\overline{FG} \)

Reasons: \( D \) (Definition of Midpoint)

Step3: State another given side - congruence

1. Statements: \( \overline{EH}\cong\overline{FJ} \)

Reasons: \( F \) (Given \( \overline{EH}\cong\overline{FJ} \))

Step4: State the third given side - congruence

1. Statements: \( \overline{HF}\cong\overline{JG} \)

Reasons: \( A \) (Given \( \overline{HF}\cong\overline{JG} \))

Step5: Prove triangle congruence

1. Statements: \( \triangle EHF\cong\triangle FJG \)

Reasons: \( C \) (SSS - Side - Side - Side congruence criterion, since \( \overline{EF}\cong\overline{FG},\overline{EH}\cong\overline{FJ},\overline{HF}\cong\overline{JG} \))

Step6: Use CPCTC

1. Statements: \( \angle EFH\cong\angle FGJ \)

Reasons: \( G \) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. \( E \) (Given)
2. \( H \), \( D \) (\( \overline{EF}\cong\overline{FG} \), Definition of Midpoint)
3. \( F \) (Given \( \overline{EH}\cong\overline{FJ} \))
4. \( A \) (Given \( \overline{HF}\cong\overline{JG} \))
5. \( B \), \( C \) (\( \triangle EHF\cong\triangle FJG \), SSS)
6. \( \angle EFH\cong\angle FGJ \), \( G \) (CPCTC)