Question

△ghk ≅ △ihj. complete the proof that ∠gjk ≅ ∠ikj.

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

1. △ghk ≅ △ihj (given)
2. ( overline{gk} cong overline{ij} ) (cpctc)
3. ( overline{gh} cong overline{hi} ) (cpctc)
4. ( overline{hk} cong overline{hj} ) (cpctc)
5. ( gj = gh + hj ) (additive property of length)
6. ( ik = hi + hk ) (additive property of length)
7. ( gj = hi + hk ) (reason to be filled)
8. ( gj = ik ) (transitive property of equality)
9. ( overline{jk} cong overline{jk} ) (reflexive property of congruence)
10. ( \triangle gjk cong \triangle ikj ) (sss)
11. ( angle gjk cong angle ikj ) (cpctc))

Explanation:

Step1: Identify Substituted Values

From step 3, \( \overline{GH} \cong \overline{HI} \) (so \( GH = HI \)) and from step 4, \( \overline{HK} \cong \overline{HJ} \) (so \( HK = HJ \)).

Step2: Substitute into \( GJ \)

In step 5, \( GJ = GH + HJ \). Substitute \( GH = HI \) and \( HJ = HK \) (from congruent segments). So \( GJ = HI + HK \).

Step3: Reason for Step 7

The reason is the Substitution Property of Equality (substituting \( GH \) with \( HI \) and \( HJ \) with \( HK \) in the expression for \( GJ \)).

Answer:

Substitution Property of Equality