Question

select all of the correct statements about the unchanged properties of \\( \triangle abc \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\).
choose all answers that apply:
\\( \square \\) a \\( \overline { b c } \\) and \\( \overline { b ^ { prime } c ^ { prime } } \\) are both perpendicular to the \\( y \\)-axis.
\\( \square \\) b \\( \angle b \\) and \\( \angle b ^ { prime } \\) have the same measures.
\\( \square \\) c the perimeters of \\( \triangle a b c \\) and \\( \triangle a ^ { prime } b ^ { prime } c ^ { prime } \\) are the same.
\\( \square \\) d none of the above

Explanation:

Step1: Analyze option A

Lines perpendicular to the \(y -\)axis are horizontal lines. If \(\triangle ABC\) is transformed (assuming a rigid transformation like translation, rotation, reflection) to \(\triangle A'B'C'\), the orientation of line segments can change. But if \(\overline{BC}\) is horizontal (perpendicular to \(y -\)axis), after a rigid transformation \(\overline{B'C'}\) will also be horizontal (since rigid transformations preserve shape and orientation of line - segments in terms of parallelism and perpendicularity).

Step2: Analyze option B

In a rigid transformation (translation, rotation, reflection), the measures of angles are preserved. So, \(m\angle B=m\angle B'\)

Step3: Analyze option C

In a rigid transformation, the lengths of line segments are preserved. If \(AB = A'B'\), \(BC=B'C'\) and \(AC = A'C'\), then the perimeter \(P=\ AB + BC+AC\) and \(P'=A'B'+B'C'+A'C'\) implies \(P = P'\)

Answer:

A. \(\overline{BC}\) and \(\overline{B'C'}\) are both perpendicular to the \(y -\)axis; B. \(\angle B\) and \(\angle B'\) have the same measures; C. The perimeters of \(\triangle ABC\) and \(\triangle A'B'C'\) are the same.