Question

which theorem shows that $\triangle abc \cong \triangle def$?
a. the triangles are not congruent.
b. sas triangle congruence theorem
c. isosceles triangle theorem
d. sss triangle congruence theorem

Explanation:

To determine which theorem shows \(\triangle ABC \cong \triangle DEF\), we analyze the marked sides (congruent segments). The SSS (Side - Side - Side) Triangle Congruence Theorem states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Looking at the triangles, we can see that the three pairs of corresponding sides are marked as congruent (the tick marks indicate congruent segments). Option A is incorrect as the triangles are congruent. Option B (SAS) is incorrect because we are not given information about included angles, and we have three sides. Option C (Isosceles Triangle Theorem) deals with angles in isosceles triangles, not triangle congruence. So the SSS Triangle Congruence Theorem (Option D) applies.

Answer:

D. SSS Triangle Congruence Theorem