In an isosceles triangle, the perimeter is 75 cm and one of the sides is 25 cm. Find all its sides. Can you find all angles of the triangle? Explain your answer.

case1: One of the two equal sides is 25, then

2(25) + x =75
x = 25, and the triangle is equilateral, and all angles are 60°

case2: the base side is 25
then 2s + 25 = 75
2s = 50
s = 25
ahhh!, same thing.

To find the remaining sides of the isosceles triangle, we first need to determine the lengths of the two equal sides. Let's denote the length of the two equal sides as x.

Since it is an isosceles triangle, we know that one side is already given as 25 cm. Therefore, we have:
25 + x + x = 75 (using the perimeter formula)
25 + 2x = 75
Subtracting 25 from both sides gives us:
2x = 50

Dividing both sides by 2, we find that:
x = 25

Therefore, the length of the two equal sides is 25 cm each, and the third side is 25 cm as well.

Now let's find the angles of the triangle. Since it is an isosceles triangle, we can use the properties of isosceles triangles to find the angles. In an isosceles triangle, the two equal sides are opposite the two equal angles.

Denoting one of the equal angles as A, we know that it is opposite the side of length 25 cm. Therefore, it follows that angle A is the same as angle B, opposite the other side of length 25 cm.

Now, let's find the remaining angle, which we'll call angle C. The sum of the interior angles of a triangle is always 180 degrees. Therefore, we can write the equation:
A + B + C = 180

Since we know that A and B are equal, we can write the equation as:
A + A + C = 180
2A + C = 180

Substituting the value of A as angle B, we can rewrite the equation as:
2B + C = 180

Now, since B and C are equal angles, we can solve for both angles using one variable. Let's denote the value of B (and C) as y.

2y + y = 180
3y = 180

Dividing both sides by 3, we find that:
y = 60

Therefore, angle B (and angle C) is 60 degrees.

To summarize, in an isosceles triangle with sides measuring 25 cm, 25 cm, and 25 cm, the angles are 60 degrees, 60 degrees, and 60 degrees.