Determine whether each statement is always sometimes or never true. If the answer is sometimes or never draw a counterexample.

Two congruent triangels are similar :

Two squares are similar :

Two isosceles triangles are similar :

Two obtuse triangles are similar :

Two equilateral triangles are similar

Okay I got that. How would I draw a counterexample?

LOL, just draw tow isoceles triangle, one equilateral 60 60 60 , one 90 45 45

Two congruent triangels are similar :

ALWAYS

Two squares are similar :
ALWAYS

Two isosceles triangles are similar :
SOMETIMES

Two obtuse triangles are similar :
SOMETIMES

Two equilateral triangles are similar
ALWAYS

To determine whether each statement is always, sometimes, or never true, we need to understand what it means for two figures to be similar.

Two figures are similar if their corresponding angles are equal, and the ratios of corresponding sides are equal.

Let's analyze each statement one by one:

1. Two congruent triangles are similar:

Explanation: Congruent triangles have all corresponding angles equal and all corresponding sides equal in length. If two triangles are congruent, then they are also similar, because all corresponding angles are equal, and the ratios of corresponding sides (their lengths divided by each other) will also be equal. Therefore, this statement is always true.

Conclusion: Always true.

2. Two squares are similar:

Explanation: In squares, all angles are right angles and all sides are equal in length. If two squares have equal side lengths, then their corresponding angles are equal as well. However, similarity also requires the ratios of corresponding sides to be equal. Since two squares will rarely have sides that are in a ratio of 1:1, the statement is sometimes false.

Counterexample: Consider a square with side length 2 and another square with side length 4. Although they have all right angles and equal side lengths, the ratio of their sides is 2:4, which is not equal to 1:1. Therefore, this is a counterexample, showing that two squares can be congruent but not similar.

Conclusion: Sometimes false.

3. Two isosceles triangles are similar:

Explanation: Isosceles triangles have two equal sides and two equal angles. If two isosceles triangles have the same angle measures and the proportional side lengths, they will be similar. However, similarity also depends on the ratios of the corresponding sides. Thus, two isosceles triangles are not automatically similar. This makes the statement sometimes false.

Counterexample: Consider an isosceles triangle with side lengths 3, 3, and 2 and another isosceles triangle with side lengths 6, 6, and 4. These triangles have equal angles, but their side lengths do not have the same ratio, as 3/6 ≠ 2/4. Therefore, this is a counterexample, showing that two isosceles triangles can be congruent but not similar.

Conclusion: Sometimes false.

4. Two obtuse triangles are similar:

Explanation: Obtuse triangles have one angle greater than 90 degrees. Two obtuse triangles, even if they have equal angle measures, will not automatically have proportional side lengths, as their side lengths can vary greatly. Thus, the statement is never true.

Conclusion: Never true.

5. Two equilateral triangles are similar:

Explanation: Equilateral triangles have all sides equal in length and all angles equal to 60 degrees. If two equilateral triangles have equal side lengths, then all their angles are equal and their corresponding sides are in a ratio of 1:1. Therefore, two equilateral triangles are always similar.

Conclusion: Always true.

Remember, counterexamples are useful to show that a statement is sometimes or never true. They provide concrete examples that contradict the statement and help verify its validity.