Someone correct me if I am wrong, but I don't believe triangles can be "equal", only congruent. The measurements can be equal, but not the triangle itself.
The triangle congruency postulates and theorems are:
Side/Side/Side Postulate - If all three sides of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Angle/Side/Angle Postulate - If two angles and a side included within those angles of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Side/Angle/Side Postulate - If two sides and an angle included within those sides of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Angle/Angle/Side Theorem - If two angles and an unincluded side of a triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Hypotenuse/Leg Theorem - (right triangles only) If the hypotenuse and a leg of a right triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
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The Pythagorean Theorem is not a triangle. It's a statement that describes a relationship among the lengths of the sides in any right triangle.
The circumcenter of a triangle is equidistant from the vertices of a triangle.
Yes
A right angle triangle
You use Lami's theorem wen you triangle of forces to calculate the sides of the triangle or the angles opposite to a side.