Question

complete the proof that \\(\overline{gj} \cong \overline{ij}\\).
1 \\(\overline{gh} \cong \overline{hi}\\)
2 \\(\overline{gi} \perp \overline{hj}\\)
3 \\(\angle hgi \cong \angle gih\\)
4 \\(\angle gkh \cong \angle hki\\)
5 \\(\triangle ghk \cong \triangle ihk\\)
6 \\(\angle ghj \cong \angle ihj\\)
7 \\(\overline{hj} \cong \overline{hj}\\)
8 \\(\triangle ghj \cong \triangle ihj\\)
9 \\(\overline{gj} \cong \overline{ij}\\)

Explanation:

Step1: Analyze Statement 1

The segments \(\overline{GH}\) and \(\overline{HI}\) are marked as congruent in the diagram (the red tick marks). So the reason for \(\overline{GH} \cong \overline{HI}\) is "Given (marked congruent segments)".

Step2: Analyze Statement 2

\(\overline{GI} \perp \overline{HJ}\) is shown by the right angle symbol at \(K\), so this is "Given (perpendicular lines)".

Step3: Analyze Statement 3

In \(\triangle GHI\), since \(\overline{GH} \cong \overline{HI}\), \(\triangle GHI\) is isosceles with base \(\overline{GI}\). By the Isosceles Triangle Theorem, the base angles are congruent, so \(\angle HGI \cong \angle GIH\) (Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent).

Step4: Analyze Statement 4

\(\angle GKH\) and \(\angle HKI\) are right angles (because \(\overline{GI} \perp \overline{HJ}\)), so all right angles are congruent. Thus, \(\angle GKH \cong \angle HKI\) (All right angles are congruent).

Step5: Analyze Statement 5

We have \(\overline{GH} \cong \overline{HI}\) (Statement 1), \(\angle HGI \cong \angle GIH\) (Statement 3), and \(\overline{HK}\) is common to both \(\triangle GHK\) and \(\triangle IHK\) (or we can use the angles and side). Wait, actually, with \(\overline{GH} \cong \overline{HI}\), \(\angle GKH \cong \angle HKI\) (right angles), and \(\angle HGI \cong \angle GIH\), we can use ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side). Let's check: \(\angle HGI \cong \angle GIH\), \(\overline{GH} \cong \overline{HI}\), \(\angle GKH \cong \angle HKI\) (right angles), so by AAS, \(\triangle GHK \cong \triangle IHK\) (AAS Congruence Criterion).

Step6: Analyze Statement 6

Since \(\triangle GHK \cong \triangle IHK\) (Statement 5), their corresponding angles \(\angle GHJ\) and \(\angle IHJ\) are congruent (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Step7: Analyze Statement 7

\(\overline{HJ} \cong \overline{HJ}\) is the Reflexive Property of Congruence (A segment is congruent to itself).

Step8: Analyze Statement 8

We have \(\overline{GH} \cong \overline{HI}\) (Statement 1), \(\angle GHJ \cong \angle IHJ\) (Statement 6), and \(\overline{HJ} \cong \overline{HJ}\) (Statement 7). By SAS (Side - Angle - Side) Congruence Criterion, \(\triangle GHJ \cong \triangle IHJ\) (SAS: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent).

Step9: Analyze Statement 9

Since \(\triangle GHJ \cong \triangle IHJ\), their corresponding sides \(\overline{GJ}\) and \(\overline{IJ}\) are congruent (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. Given (marked congruent segments)
2. Given (perpendicular lines)
3. Isosceles Triangle Theorem
4. All right angles are congruent
5. AAS Congruence Criterion
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
7. Reflexive Property of Congruence
8. SAS Congruence Criterion
9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)