Question

triangle $\triangle abc$ is the result of reflecting $\triangle abc$ across the line $l$.
select all of the correct statements about the unchanged properties of $\triangle abc$ and $\triangle abc$.
choose all answers that apply:
$square$ a $overline{bc}$ and $overline{bc}$ are both perpendicular to the $y$-axis.
$square$ b $angle b$ and $angle b$ have the same measures.
$square$ c the perimeters of $\triangle abc$ and $\triangle abc$ are the same.
$square$ d none of the above

Explanation:

Step1: Analyze property of reflection

Reflection is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures.

Step2: Check option A

\(\overline{BC}\) is parallel to the \(x\) - axis (perpendicular to the \(y\) - axis). After reflection, \(\overline{B'C'}\) is also parallel to the \(x\) - axis (perpendicular to the \(y\) - axis) since reflection preserves the orientation of line segments parallel or perpendicular to the axes in this case.

Step3: Check option B

Since reflection is a rigid transformation, \(\angle B\cong\angle B'\), so \(m\angle B = m\angle B'\)

Step4: Check option C

Since reflection is a rigid transformation, \(AB = A'B'\), \(BC=B'C'\), \(AC = A'C'\). The perimeter of \(\triangle ABC=AB + BC+AC\) and the perimeter of \(\triangle A'B'C'=A'B'+B'C'+A'C'\). So the perimeters are equal.

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.