Question

△efi ≅ △gfh. complete the proof that ∠ghi ≅ ∠eih.

statement
1 △efi ≅ △gfh
2 (overline{ei} cong overline{gh})
3 (overline{ef} cong overline{fg})
4 (overline{fi} cong overline{fh})
5 (eh = ef + fh)
6 (gi = fg + fi)
7 (eh = fg + fi)
8 (eh = gi)
9 (overline{hi} cong overline{hi})
10 △ehi ≅ △gih
11 ∠ghi ≅ ∠eih

reason
given
cpctc
cpctc
cpctc

additive property of length
substitution

reflexive property of congruence
sss
cpctc

Explanation:

Step1: Determine the reason for statement 5

The formula \(EH = EF+FH\) is based on the fact that if a point \(F\) lies on a line segment \(EH\), then the length of \(EH\) is the sum of the lengths of \(EF\) and \(FH\). This is the Additive Property of Length.

Step2: Determine the reason for statement 8

Since \(EH = FG + FI\) (from statement 7) and \(GI=FG + FI\) (from statement 6), by the transitive property of equality (if \(a=b\) and \(b = c\), then \(a=c\)), we can conclude that \(EH = GI\).

Answer:

1. Additive Property of Length
2. Transitive Property of Equality