Question

what are the necessary conditions to apply the sas triangle congruence theorem?
a. one angle and two sides of one triangle are congruent to the corresponding parts of another triangle.
b. two angles and the included side of one triangle are congruent to the corresponding parts of another triangle.
c. an angle and the two sides collinear with the angles rays are congruent to the corresponding parts of another triangle.
d. two sides and any angle of one triangle are congruent to the corresponding parts of another triangle.

Explanation:

The SAS (Side - Angle - Side) Triangle Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. Let's analyze each option:

• Option A: Just saying one angle and two sides, but it doesn't specify that the angle is included between the two sides. So this is incorrect.
• Option B: This describes the ASA (Angle - Side - Angle) Theorem, not SAS. So this is incorrect.
• Option C: The two sides collinear with the angle's rays mean that the angle is between the two sides (the included angle). This matches the SAS Theorem's requirements.
• Option D: "Any angle" is incorrect. The angle must be the included angle between the two sides.

Answer:

C. An angle and the two sides collinear with the angle's rays are congruent to the corresponding parts of another triangle.