Question

select the correct answer. which triangles in the diagram are congruent? triangle 1 triangle 2 triangle 3 triangle 4 a. triangle 1 and triangle 2 b. triangle 1, triangle 2, and triangle 3 c. triangle 1 and triangle 3 d. triangle 1, triangle 3, and triangle 4

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.

Step2: Analyze triangle 1 and triangle 3

• For triangle 1: Let the two equal - length sides be \(a\) and \(b\), and the angle of \(70^{\circ}\) is between them.
• For triangle 3: The two equal - length sides (marked with the same tick marks) are the same as in triangle 1, and the angle of \(70^{\circ}\) (although it looks like a vertex angle in triangle 3) is also between the two equal - length sides.

Step3: Analyze triangle 2

In triangle 2, the side - side - angle (SSA) condition is not a valid congruence criterion (except in the right - triangle case, which is not applicable here). The side - side - angle combination does not guarantee congruence.

Step4: Analyze triangle 4

In triangle 4, the side - side - angle (SSA) condition is not a valid congruence criterion. The side - side - angle combination does not guarantee congruence.

Answer:

C. triangle 1 and triangle 3