The SSS Congruence Theorem Practice

Question

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Question
Use the image to answer the question.

Three congruent triangles marked with congruent corresponding sides are arranged on two overlapping circles. The top circle is larger. All triangles overlap the top circle.

Charlie is using rigid transformations to justify the SSS Congruence Theorem. He started with △ABC
and △DEF
that have three pairs of congruent sides. He translated and then rotated △ABC
to create △A"B"C"
such that A"B"¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
coincides with DE¯¯¯¯¯¯¯¯
. He then constructed the circles shown in the image and labeled Points 1 and 2. Why are Points 1 and 2 significant?

Option #1: They are the only points in the plane that are both a distance EF
away from point E
and a distance DF
away from point D
.

Option #2: They are the only points in the plane that are both a distance ED
away from point E
and a distance FE
away from point F
.

Option #3: They are the only points in the plane that are both a distance EF
away from point E
and a distance FE
away from point F
.

(1 point)
Option #
best describes the significance of Points 1 and 2.

Answer 1

To determine the correct option regarding the significance of Points 1 and 2 in Charlie's explanation of the SSS (Side-Side-Side) Congruence Theorem, let's analyze the statements:

For triangles △ABC and △DEF, Charlie has identified three pairs of congruent sides: AB ≅ DE, BC ≅ EF, and AC ≅ DF. When he translated and rotated △ABC to create △A"B"C", he ensured that certain segments coincide and created circles centered at points D and E with the respective radii.

Now, let's examine each option:

• Option #1: It states that Points 1 and 2 are both a distance EF away from point E and a distance DF away from point D. This implies that Points 1 and 2 lie along the arc of a circle drawn around D with radius DF and a circle around E with radius EF. This option suggests that these points are determined by distances related to the specific sides that are congruent to each other.

• Option #2: Similarly, this option indicates that Points 1 and 2 are both a distance ED away from point E and a distance FE away from point F. This doesn't fit, because ED is not one of the congruent side lengths; ED should correspond to side wraps of the triangles.

• Option #3: This states that Points 1 and 2 are both a distance EF away from point E and a distance FE away from point F. Since EF and FE are the same length (as they are congruent sides), this is a valid statement. However, FE refers to the same length as EF, which doesn’t contribute to distinguishing the distances correctly.

Given the descriptions, the best option is:

Option #1: They are the only points in the plane that are both a distance EF away from point E and a distance DF away from point D.

This accurately captures the significance of those points relative to the congruent sides discussed in the SSS theorem.