Question

which two triangles are congruent by the sss theorem? complete the congruence statement.

Explanation:

Step1: Analyze triangle \( \triangle ABC \)

In \( \triangle ABC \), the sides: \( AB \) has two marks, \( BC \) has three marks, \( AC \) has two marks. So the side lengths (by congruency marks) are: \( AC = AB \) (two marks), \( BC \) (three marks).

Step2: Analyze triangle \( \triangle STU \)

In \( \triangle STU \), the sides: \( ST \) has three marks, \( SU \) has two marks, \( TU \) has two marks. So \( ST \) (three marks), \( SU = TU \) (two marks).

Step3: Analyze triangle \( \triangle JHI \)

In \( \triangle JHI \), the sides: \( JI \) has two marks, \( JH \) has three marks, \( HI \) has one mark. Wait, no, re - check: Wait, \( \triangle ABC \): \( AB \) (two), \( BC \) (three), \( AC \) (two). \( \triangle STU \): \( ST \) (three), \( SU \) (two), \( TU \) (two). \( \triangle JHI \): \( JI \) (two), \( JH \) (three), \( HI \) (one)? No, maybe I misread. Wait, the SSS theorem says three sides equal. Let's check the mark counts:

For \( \triangle ABC \): Sides with two marks: \( AC \) and \( AB \); side with three marks: \( BC \).

For \( \triangle STU \): Sides with two marks: \( SU \) and \( TU \); side with three marks: \( ST \).

For \( \triangle JHI \): Side \( JI \) (two), \( JH \) (three), \( HI \) (one). Wait, no, maybe the triangle \( \triangle ABC \) and \( \triangle STU \): Let's match the mark counts. In \( \triangle ABC \), two sides with two marks, one with three. In \( \triangle STU \), two sides with two marks, one with three. So the correspondence: \( AC = SU \) (two marks), \( AB = TU \) (two marks), \( BC = ST \) (three marks). So by SSS, \( \triangle ABC \cong \triangle TUS \) (or \( \triangle ABC \cong \triangle STU \) with correct vertex correspondence). Wait, maybe the correct pair is \( \triangle ABC \) and \( \triangle STU \) because their side - mark patterns match: two sides with two congruency marks and one side with three congruency marks.

Answer:

\( \triangle ABC \cong \triangle STU \) (by SSS, as their corresponding sides have the same congruency mark counts, implying equal lengths)