Question

1. given: \\( \triangle fgh \\) and \\( \triangle fjh \\) are right triangles, \\( \overline{gh} \cong \overline{hj} \\) prove: \\( \triangle fgh \cong \triangle fjh \\)

statements	reasons
2. \\( \angle fgh \cong \angle fjh \\)	2. definition of right angle
3. \\( \overline{fh} \cong \overline{fh} \\)	3. reflexive property
4. \\( \overline{hg} \cong \overline{hj} \\)	4. given
5. \\( \triangle fgh \cong \triangle fjh \\)	5. sas

Explanation:

Step1: Identify the given information

Given that \(\triangle FGH\) and \(\triangle FJH\) are right - triangles and \(\overline{GH}\cong\overline{HJ}\).

Step2: Analyze the right - angle property

Since \(\triangle FGH\) and \(\triangle FJH\) are right - triangles, \(\angle FGH=\angle FJH = 90^{\circ}\) (by the definition of a right - triangle).

Step3: Consider the common side

\(\overline{FH}\cong\overline{FH}\) (by the reflexive property of congruence, which states that any segment is congruent to itself).

Step4: Use the given congruent segment

We are given \(\overline{GH}\cong\overline{HJ}\).

Step5: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle FGH\) and \(\triangle FJH\), we have \(\overline{GH}\cong\overline{HJ}\), \(\angle FGH=\angle FJH\), and \(\overline{FH}\cong\overline{FH}\). So, by the SAS congruence criterion (\(SAS\): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle FGH\cong\triangle FJH\).

Answer:

The proof is correct as it uses the \(SAS\) (Side - Angle - Side) congruence criterion. The right angles (\(\angle FGH\) and \(\angle FJH\)) are the included angles between the pairs of congruent sides (\(\overline{GH}\cong\overline{HJ}\) and \(\overline{FH}\cong\overline{FH}\)). So, \(\triangle FGH\cong\triangle FJH\) by \(SAS\).