Which of the following statements is sufficient to conclude that two triangles are congruent?

The correct answer is: These have two equal sides and the same perimeter.

To determine which statement is sufficient to conclude that two triangles are congruent, let us analyze the conditions under which triangles are congruent. Two triangles are congruent if all three corresponding sides and all three corresponding angles are equal. This can be established using standard congruence criteria, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), etc.

Let us analyze the correct option step-by-step:

Suppose we have two triangles, Triangle 1 and Triangle 2.

Let the side lengths of Triangle 1 be $a$, $b$, and $c$.

Let the side lengths of Triangle 2 be $x$, $y$, and $z$.

According to the statement, the two triangles have two equal sides. Without loss of generality, let:

$a=x$

and

$b=y$

We are also given that the two triangles have the same perimeter, $P$.

For Triangle 1, the perimeter is given by:

$P=a+b+c$

We can solve for the third side $c$ as:

$c=P-a-b$

For Triangle 2, the perimeter is given by:

$P=x+y+z$

We can solve for the third side $z$ as:

$z=P-x-y$

Since $x=a$ and $y=b$, we can substitute these values into the equation for $z$:

$z=P-a-b$

Comparing the expressions for $c$ and $z$, we find that:

$c=z$

Since all three corresponding sides of the two triangles are equal ($a=x$, $b=y$, and $c=z$), the two triangles are congruent by the Side-Side-Side (SSS) congruence criterion. Therefore, this statement is sufficient.

Why the other statements are insufficient:
1. "One side and one angle of both triangles are equal": This is not enough information to establish congruence, as we need at least three corresponding parts (e.g., SSS, SAS, ASA) to match.
2. "These have the same area and the same base": Since Area = 0.5 × base × height, having the same area and base only guarantees they have the same height. The shape of the triangles can be completely different (for example, one could be a right triangle and the other could be an obtuse triangle), so they are not necessarily congruent.
3. "These have the same base and the same height": Similar to the area argument, having the same base and height allows the third vertex to be placed at different positions along a parallel line, which changes the side lengths and angles, making them non-congruent.