Question

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

Explanation:

Step1: Identify the hypotenuse - leg (HL) congruence criterion

For right - angled triangles, if the hypotenuse and one leg of a right - triangle are equal to the hypotenuse and one leg of another right - triangle, then the two right - triangles are congruent.
In \(\triangle FGH\) and \(\triangle FJH\), \(\angle G=\angle J = 90^{\circ}\) (since they are right - triangles).
The hypotenuse is \(FH\) for both \(\triangle FGH\) and \(\triangle FJH\), so \(FH\cong FH\) (reflexive property of congruence: \(a\cong a\) for any geometric figure \(a\)).
It is given that \(GH\cong HJ\).

Step2: Apply the HL congruence criterion

By the Hypotenuse - Leg (HL) congruence criterion for right - triangles, since the hypotenuse \(FH\) is common (\(FH\cong FH\)) and \(GH\cong HJ\) (given), we can conclude that \(\triangle FGH\cong\triangle FJH\).

Answer:

\(\triangle FGH\cong\triangle FJH\) by the Hypotenuse - Leg (HL) congruence criterion for right - triangles.