Use the image to answer the question.

Question

An illustration shows four congruent triangles, two in different orientations. Congruent sides and angles are marked. A dotted arrow extends beyond the side of the triangle on the right.

Cailyn is using rigid transformations to justify the SAS Congruence Theorem. She started with △ABC
and △DFE
, as shown. She translated and then rotated △ABC
to create △A"B"C"
such that △A"C"
coincides with DE¯¯¯¯¯¯¯¯
. She then reflected △A"B"C"
across △A"C"
and drew EF−→−
.

What is the significance of EF−→−
with respect to Cailyn’s justification of the SAS Congruence Theorem?

Option #1: EF¯¯¯¯¯¯¯¯
must lie on EF−→−
.

Option #2: Both points C′′′
and E
must lie on EF−→−
.

Option #3: Both points B′′′
and F
must lie the same distance away from point E
on EF−→−
.

(1 point)
Option #
best describes the significance of EF−→−
with respect to Cailyn’s justification of the SAS Congruence Theorem.

Answer 1

To determine the correct option regarding the significance of \( \overrightarrow{EF} \) in Cailyn's justification of the SAS (Side-Angle-Side) Congruence Theorem, we need to consider the steps she took and what they imply about the congruence of the triangles.

Cailyn translated and then rotated \(\triangle ABC\) to align \(\triangle A''B''C''\) such that side \(\overline{A''C''}\) coincides with \(\overline{DE}\). She then reflected \(\triangle A''B''C''\) across \(\overline{A''C''}\) to create line \(EF\).

The SAS Congruence Theorem states that if two triangles have two sides of equal length and the angle between those sides is equal, then the triangles are congruent. In this scenario, the sides \(\overline{AB}, \overline{DE}\) and the angle \(\angle A''C''B''\) must correspond in the congruent triangles.

Considering the options given:

1. Option #1: \( \overline{EF} \) must lie on \( \overrightarrow{EF} \).
2. Option #2: Both points \( C''' \) and \( E \) must lie on \( \overrightarrow{EF} \).
3. Option #3: Both points \( B''' \) and \( F \) must lie the same distance away from point \( E \) on \( \overrightarrow{EF} \).

The correct interpretation seems to align with Option #3. Since she reflected \(\triangle A''B''C''\) across the line \( \overline{A''C''}\) to create point \( F\) from point \( B''\), both points \( B'''\) and \( F\) being the same distance from \( E\) on line \( \overrightarrow{EF} \) reinforces the congruence of triangles by showing that reflected points must maintain equal distance from the line, preserving the metric properties that support the SAS theorem.

Thus, the best option that describes the significance of \( \overrightarrow{EF} \) with respect to Cailyn's justification of the SAS Congruence Theorem is:

Option #3: Both points \( B''' \) and \( F \) must lie the same distance away from point \( E \) on \( \overrightarrow{EF} \).