Question 38

Triangle BCD is congruent to triangle RST by Angle-Angle-Side (AAS) theorem.

Triangle BCD can be proven congruent to triangle RST using the Angle-Angle-Side (AAS) theorem, which states that if two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A) Side-Side-Angle (SSA)

The Side-Side-Angle (SSA) condition does not guarantee triangle congruence, as it can result in ambiguous cases where two different triangles may be formed. Therefore, SSA cannot be used to prove the congruence of triangles BCD and RST.

B) Hypotenuse-Leg (HL)

The Hypotenuse-Leg (HL) theorem applies specifically to right triangles, stating that if the hypotenuse and one leg of a right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. Without confirmation that triangles BCD and RST are right triangles, HL cannot be applied here.

C) Angle-Angle-Side (AAS)

The Angle-Angle-Side (AAS) theorem is applicable in this scenario, as it allows for the congruence of triangles if two angles and the side opposite one of these angles are congruent. This theorem provides a valid method to establish that triangle BCD is congruent to triangle RST.

D) Angle-Angle-Angle (AAA)

The Angle-Angle-Angle (AAA) condition indicates that if all three angles of one triangle are congruent to all three angles of another triangle, the triangles are similar, but not necessarily congruent. Therefore, AAA cannot be used to prove triangle BCD is congruent to triangle RST.

Conclusion

The AAS theorem is the only correct choice for proving the congruence of triangles BCD and RST, as it meets the necessary criteria of having two angles and the non-included side congruent. The other options, SSA, HL, and AAA, either do not provide sufficient criteria for congruence or are not applicable to the triangles in question. Thus, AAS is definitively the appropriate theorem to use in this context.