Two congruent triangles are drawn on an 8 by 7 unit grid. Sides and angles are marked as congruent.

Question

Brice wants to use rigid transformations to justify the SAS Congruence Theorem. He starts with △ABC
and △DEF
, which have one pair of congruent angles and two pairs of congruent sides. Before he starts his transformations, he notices that AC¯¯¯¯¯¯¯¯≆DF¯¯¯¯¯¯¯¯
. He wonders if he has already made a mistake. What happened?

Option #1: Brice made a mistake matching up the corresponding segments. AC¯¯¯¯¯¯¯¯≆DF¯¯¯¯¯¯¯¯
because AC¯¯¯¯¯¯¯¯
does not correspond to DF¯¯¯¯¯¯¯¯
.

Option #2: Brice made a mistake choosing △ABC
and △DEF
. Even though these triangles have one pair of congruent angles and two pairs of congruent sides, because the angles are not between the sides, the SAS Congruence Theorem does not apply. The triangles are not congruent.

Option #3: Brice has not made a mistake. If he continues with the rigid transformations, he will see that △ABC
can be mapped to △DEF
such that they overlap perfectly; therefore, the triangles are congruent.

(1 point)
Option #
best describes what happened.

Answer 1

The best option that describes what happened is:

Option #2: Brice made a mistake choosing △ABC and △DEF. Even though these triangles have one pair of congruent angles and two pairs of congruent sides, because the angles are not between the sides, the SAS Congruence Theorem does not apply. The triangles are not congruent.

The SAS Congruence Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the angle between those sides is also congruent, then the triangles are congruent. In Brice's case, he has one pair of congruent angles but they are not situated between the congruent sides, which means the triangles cannot be confirmed as congruent using the SAS theorem.