Question

Which statement about the triangles is true?

a) Triangles ABC and DCB are congruent by the Angle-Angle Triangle Congruence Theorem. (True/False)
b) Triangles ABC and BCD are congruent by the Angle-Side-Angle Triangle Congruence Theorem. (True/False)
c) Triangles ABC and BCD are congruent by the Side-Side-Side Triangle Congruence Theorem. (True/False)
d) Triangles ABC and DCB are congruent by the Side-Angle-Side Triangle Congruence Theorem. (True/False)

Answer 1

Without specific details about triangles ABC, DCB, and BCD, we cannot confirm if the congruence statements are true. The congruence theorems require knowing specific congruent angles or sides. In vector context, the Pythagorean theorem applies, vectors can form right triangles, but knowing only angles is not enough to find resultant vector angles.

Step-by-step explanation:

The accuracy of the statements about the triangles depends on specific information about the triangles, which is not provided. Generally:

• Angle-Angle (AA) Congruence Theorem says that two triangles are congruent if two angles of one triangle are congruent to two angles of the other triangle.
• Angle-Side-Angle (ASA) Congruence Theorem states that 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.
• Side-Side-Side (SSS) Congruence Theorem indicates that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
• Side-Angle-Side (SAS) Congruence Theorem asserts that 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.

Without specific information about the triangles ABC, DCB, and BCD, we cannot definitively determine the truth value of each of the congruence statements (a to d). The congruence theorems require specific angles or sides to be known to be congruent between the triangles. In context of vectors:

1. The Pythagorean theorem can be used to calculate the length of the resultant vector when two vectors are at right angles to each other.
2. A vector can indeed form a right angle triangle with its x and y components, meaning it can have both magnitude and direction.
3. Knowing just the angles of two vectors is not sufficient to find the angle of their resultant addition vector; we would also need to know their magnitudes.