Question

$overline{ij} cong overline{gj}$ and $overline{hi} cong overline{gh}$. complete the proof that $overline{gk} cong overline{ik}$.

(image of a quadrilateral with points j, g, h, i and intersection k)

	statement	reason
2	$overline{hi} cong overline{gh}$
3	$overline{hj} cong overline{hj}$
4	$\triangle ghj cong \triangle ihj$
5	$angle ghj cong angle ihj$
6	$overline{hk} cong overline{hk}$
7	$\triangle ghk cong \triangle ihk$
8	$overline{gk} cong overline{ik}$

Explanation:

Step1: Identify Given Information

The problem states \(\overline{IJ} \cong \overline{GJ}\) (wait, original is \(\overline{IJ} \cong \overline{GJ}\)? Wait, original first statement: \(\overline{IJ} \cong \overline{GJ}\) (maybe typo, but given as \(\overline{IJ} \cong \overline{GJ}\)) and \(\overline{HI} \cong \overline{GH}\). Also, \(\overline{HJ}\) is common.

Step2: Reason for Statement 1

Statement 1: \(\overline{IJ} \cong \overline{GJ}\) (Given, as per problem statement)

Step3: Reason for Statement 2

Statement 2: \(\overline{HI} \cong \overline{GH}\) (Given, as per problem statement)

Step4: Reason for Statement 3

Statement 3: \(\overline{HJ} \cong \overline{HJ}\) (Reflexive Property of Congruence, a segment is congruent to itself)

Step5: Reason for Statement 4

Statement 4: \(\triangle GHJ \cong \triangle IHJ\) (SSS Congruence Postulate: since \(\overline{IJ} \cong \overline{GJ}\), \(\overline{HI} \cong \overline{GH}\), and \(\overline{HJ} \cong \overline{HJ}\), so three sides congruent)

Step6: Reason for Statement 5

Statement 5: \(\angle GHJ \cong \angle IHJ\) (Corresponding Parts of Congruent Triangles are Congruent (CPCTC), from \(\triangle GHJ \cong \triangle IHJ\))

Step7: Reason for Statement 6

Statement 6: \(\overline{HK} \cong \overline{HK}\) (Reflexive Property of Congruence, a segment is congruent to itself)

Step8: Reason for Statement 7

Statement 7: \(\triangle GHK \cong \triangle IHK\) (SAS Congruence Postulate: \(\overline{HI} \cong \overline{GH}\) (given), \(\angle GHJ \cong \angle IHJ\) (from CPCTC), \(\overline{HK} \cong \overline{HK}\) (reflexive), so two sides and included angle congruent)

Step9: Reason for Statement 8

Statement 8: \(\overline{GK} \cong \overline{IK}\) (CPCTC, from \(\triangle GHK \cong \triangle IHK\), corresponding sides are congruent)

Answer:

Statement	Reason
2. \(\overline{HI} \cong \overline{GH}\)	Given
3. \(\overline{HJ} \cong \overline{HJ}\)	Reflexive Property of Congruence
4. \(\triangle GHJ \cong \triangle IHJ\)	SSS Congruence Postulate
5. \(\angle GHJ \cong \angle IHJ\)	CPCTC (from \(\triangle GHJ \cong \triangle IHJ\))
6. \(\overline{HK} \cong \overline{HK}\)	Reflexive Property of Congruence
7. \(\triangle GHK \cong \triangle IHK\)	SAS Congruence Postulate ( \(\overline{HI} \cong \overline{GH}\), \(\angle GHJ \cong \angle IHJ\), \(\overline{HK} \cong \overline{HK}\))
8. \(\overline{GK} \cong \overline{IK}\)	CPCTC (from \(\triangle GHK \cong \triangle IHK\))

(Note: The first statement's segment might have a typo, but following the given, we use the provided labels. The key is to apply congruence postulates and CPCTC.)