Question

The SAS Congruence Theorem Practice
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Question
Use the image to answer the 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

option 3 was the answer

Answer 2

To analyze the significance of \( \overrightarrow{EF} \) in the context of Cailyn’s justification of the SAS Congruence Theorem, we should consider what the theorem states and how Cailyn's transformations work.

The SAS (Side-Angle-Side) Congruence Theorem states that if two triangles have two sides and the included angle of one triangle congruent to two sides and the included angle of another triangle, then the triangles are congruent.

In this situation, Cailyn has proven that \( \triangle ABC \) is congruent to \( \triangle DFE \) by:

1. **Translating** \( \triangle ABC \).
2. **Rotating** it to coincide with side \( DE \).
3. **Reflecting** to construct segment \( \overrightarrow{EF} \).

Given these transformations and the context of the SAS theorem, the best option that describes the significance of \( \overrightarrow{EF} \) is:

**Option #1:** \( \overline{EF} \) must lie on \( \overrightarrow{EF} \).

This signifies that the length of segment \( EF \) (the segment corresponding to side \( AC \)) must correspond directly to the transformed segment. Thus, this option holds the most significance in demonstrating that the segments are congruent as required by the SAS theorem.

Answer 3

Thank you for your patience, and I appreciate the clarification. Let's analyze option 3 in the context of the SAS Congruence Theorem:

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

This choice makes sense because it emphasizes that the segments radiating from a point (in this case, from point \( E \) to points \( B''' \) and \( F \)) must be of equal length due to the congruence of the triangles. In other words, \( B''' \) in \( \triangle A''B''C'' \) must correspond to \( F \) in \( \triangle DFE \) along the \( \overrightarrow{EF} \), thus ensuring they are equidistant from point \( E \).

Cailyn's reflection step demonstrates that corresponding points in congruent triangles must be equidistant from a line (in this case, line \( EF \)), reinforcing their congruence as stated by the SAS theorem.

Thank you for highlighting the correct option! If you have any further questions or need clarification, feel free to ask.