Question

(a) adjust the figure on the right to explore the different triangles that can be formed. then answer the question. note that the figure will not be graded.
image of two triangles
which of the following is true?
○ the two triangles must be congruent.
reason: select
○ the two triangles do not have to be congruent.
(b) adjust the figure on the right to explore the different triangles that can be formed. then answer the question. note that the figure will not be graded.
image of two triangles
which of the following is true?
○ the two triangles must be congruent.
reason: select
○ the two triangles do not have to be congruent.

Explanation:

Part (a)

To determine if the two triangles are congruent, we analyze the given information. The triangles have two sides marked as equal (indicating two pairs of congruent sides) and a pair of congruent angles (the right angles, or the marked angles). However, we need to check the congruence criterion. If the angle is not the included angle between the two sides, the triangles may not be congruent. But in the first figure, if we consider the marked angles and sides, let's assume the angle is the included angle? Wait, no—wait, the first triangle: let's see, the two sides with marks and the angle. Wait, actually, in the first case, if we have two sides and a non - included angle, the triangles may not be congruent. But wait, maybe the angle is the included angle? Wait, no, let's re - examine. The first set of triangles: if we have two sides and a angle, but if the angle is not between the two sides, then by SSA (Side - Side - Angle), which is not a valid congruence criterion (it can lead to non - congruent triangles). But wait, in the first figure, maybe the angle is the included angle? Wait, no, the first triangle: let's look at the marks. The two sides with the same number of marks, and the angle. Wait, actually, in the first problem, the correct answer is "The two triangles do not have to be congruent" because SSA does not guarantee congruence. Wait, no, maybe I made a mistake. Wait, let's think again. If we have two sides and the included angle (SAS), then they are congruent. But if the angle is not included, then SSA. In the first figure, maybe the angle is not included. So the two triangles do not have to be congruent.

Part (b)

In the second figure, we have two sides and the included angle (because the angle is between the two sides with marks). By the SAS (Side - Angle - Side) congruence criterion, 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. So the two triangles must be congruent.

Answer:

Part (a)

The two triangles do not have to be congruent.

Part (b)

The two triangles must be congruent.