Which tirangles must be similar

1. Two right triangles with a congruent acute angle
2. Two obtuse angles
3 Two isosceles triangles with congruent bases
4. Two scalene triangles with a congruent angle

#1. If the acute angle is equal, then all the angles are equal.

After drawing four squares, Chen completed this chart showing the length of the sides of each square and the area of each square. According to Chen’s chart, if A is the area and s is the length of a side, what can you conclude is the formula for the area of a square?

Side Length Area
1 1
2 4
3 9
4 16

A=s^2

To determine which triangles must be similar, we need to understand the conditions for triangle similarity.

Two triangles are considered similar if their corresponding angles are congruent, and their corresponding sides are in proportion. This is known as the Angle-Angle Similarity Postulate.

Let's analyze each scenario mentioned:

1. Two right triangles with a congruent acute angle:
In this case, the two right triangles could be similar if the remaining angles are congruent as well. Without this information, we cannot conclude that the triangles are similar.

2. Two obtuse triangles:
Two obtuse triangles cannot be similar since by definition, an obtuse triangle has one angle that is greater than 90 degrees. This means that the congruent angles in the triangles cannot exist.

3. Two isosceles triangles with congruent bases:
Isosceles triangles with congruent bases are always similar. This is known as the Angle-Angle-Side (AAS) Similarity Theorem. Since the bases are congruent, we already have one pair of congruent angles, and since isosceles triangles have two congruent sides, the corresponding sides are also proportional.

4. Two scalene triangles with a congruent angle:
Scalene triangles are those that have no congruent sides. However, if two scalene triangles have a congruent angle, we cannot automatically assume that they are similar. We would need additional information about the triangle's angles or sides to determine if they are similar.

In summary, the only scenario where we can conclude that the triangles must be similar is when we have two isosceles triangles with congruent bases.