Choose the correct answer
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Choose the correct answer
We need to imagine that we have two isosceles triangles with equal vertex angles.
If the vertex angle is equal, we can know that the other two angles are also equal, since in a triangle the sum of angles equals 180, and in an isosceles triangle the base angles are equal to each other, so the ratios are maintained between the triangles.
However, the first statement is incorrect, because this is triangle similarity, not congruence. Triangle congruence requires at least one equal side.
The second statement is also incorrect, because we don't know which of their angles are equal, it's possible that a base angle of one triangle is identical to the vertex angle of the second triangle, therefore they are not equal.
The third statement is correct, according to the logic we defined at the beginning.
The fourth statement is incorrect, because not all isosceles triangles are similar.
Therefore, we understand that answer C is correct.
Equal isosceles triangles
The main angle is similar
Similar triangles have the same shape (equal angles) but can be different sizes. Congruent triangles have the same shape AND size (equal angles AND equal sides).
In an isosceles triangle, the two base angles are always equal. If the vertex angles are equal between two triangles, then all three corresponding angles are equal, making them similar by AA similarity.
Since triangle angles sum to , if vertex angle = , then each base angle = . Equal vertex angles mean equal base angles too!
No! Isosceles triangles are only similar if their vertex angles are equal. Different vertex angles create different triangle shapes, even if they're all isosceles.
In an isosceles triangle, the vertex angle is the angle between the two equal sides. The base angles are the two angles that touch the base (the side that's different length).
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