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Which of the following conditions is sufficient proof that triangle EFG is congruent to triangle LMN?

A) Different sequences of rigid motions can be used to map \overline{EF} onto \overline{LM} and \overline{FG} onto \overline{MN}.

B) Different sequences of rigid motions can be used to map \angle E onto \angle L, \angle F onto \angle M, and \angle G onto \angle N.

C) Different sequences of rigid motions can be used to map point E onto point L, \overline{FG} onto \overline{MN}, and \angle G onto \angle N.

D) Different sequences of rigid motions can be used to map \overline{EF} onto \overline{LM}, \overline{FG} onto \overline{MN}, \overline{GE} onto \overline{NL}, \angle E onto \angle L, \angle F onto \angle M, and \angle G onto \angle N.

Answer :

To determine which condition is a sufficient proof that triangle EFG is congruent to triangle LMN, we need to understand the concept of triangle congruence using rigid motions. Rigid motions include transformations such as translations, rotations, and reflections, which preserve the size and shape of geometric figures.

Triangles can be proven congruent using a few well-known postulates and theorems, which are:

  1. Side-Side-Side (SSS): If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

  2. Side-Angle-Side (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

  3. Angle-Side-Angle (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

  4. Angle-Angle-Side (AAS): If two angles and the non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.

  5. Hypotenuse-Leg (HL) for Right Triangles: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Let's evaluate the options provided:

  • Option A: Mapping only the line segments [tex]\overline{EF}[/tex] to [tex]\overline{LM}[/tex] and [tex]\overline{FG}[/tex] to [tex]\overline{MN}[/tex] without information about the third side or angles does not ensure congruence.

  • Option B: Mapping only the angles [tex]\angle E[/tex] to [tex]\angle L[/tex], [tex]\angle F[/tex] to [tex]\angle M[/tex], and [tex]\angle G[/tex] to [tex]\angle N[/tex] doesn’t satisfy any of the congruence postulates since the side lengths are needed.

  • Option C: Mapping [tex]E[/tex] to [tex]L[/tex], [tex]\overline{FG}[/tex] to [tex]\overline{MN}[/tex], and [tex]\angle G[/tex] to [tex]\angle N[/tex] does not provide enough information about the correspondence of all sides and angles.

  • Option D: Mapping all sides [tex]\overline{EF}[/tex] to [tex]\overline{LM}[/tex], [tex]\overline{FG}[/tex] to [tex]\overline{MN}[/tex], [tex]\overline{GE}[/tex] to [tex]\overline{NL}[/tex], and angles [tex]\angle E[/tex] to [tex]\angle L[/tex], [tex]\angle F[/tex] to [tex]\angle M[/tex], [tex]\angle G[/tex] to [tex]\angle N[/tex] provides complete congruence statements for all corresponding parts, effectively utilizing the SSS or SAS postulates.

Therefore, Option D is the correct answer as it ensures that triangle EFG is congruent to triangle LMN by appropriately mapping all corresponding sides and angles, which is sufficient proof of congruence.

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