Question

hj ≅ fg, gh ≅ jk, and hi ≅ fk. complete the proof that ∠h ≅ ∠k.

statement | reason
1 hj ≅ fg | given
2 gh ≅ jk | given
3 hi ≅ fk | given
4 gk = jk + gj | additive property of length
5 hj = gh + gj | additive property of length
6 gk = gh + gj | substitution
7 hj = gk | transitive property of equality
8 |

Explanation:

Step1: Analyze Triangle Congruence

We have sides: \(\overline{IJ} \cong \overline{FG}\), \(\overline{GH} \cong \overline{JK}\), \(\overline{HI} \cong \overline{FK}\). By SSS (Side - Side - Side) congruence criterion, \(\triangle HIJ \cong \triangle FKG\).

Step2: Corresponding Angles

In congruent triangles, corresponding angles are congruent. Since \(\triangle HIJ \cong \triangle FKG\), the corresponding angles \(\angle H\) and \(\angle K\) are congruent. So the statement for step 8 should be \(\angle H \cong \angle K\) and the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Answer:

Statement: \(\boldsymbol{\angle H \cong \angle K}\); Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)