Question

c. the side lengths of a triangle
enter a conjecture and its converse in the space provided.
√x

Explanation:

A common conjecture about triangle side lengths is the Pythagorean theorem for right - angled triangles. The conjecture (for a right - angled triangle) is: If a triangle is a right - angled triangle with side lengths \(a\), \(b\) (the legs) and \(c\) (the hypotenuse), then \(a^{2}+b^{2}=c^{2}\). The converse of this conjecture is: If the side lengths of a triangle satisfy the equation \(a^{2}+b^{2}=c^{2}\) (where \(a\), \(b\), \(c\) are the side lengths of the triangle), then the triangle is a right - angled triangle.

Answer:

Conjecture: If a triangle is a right - angled triangle with side lengths \(a\), \(b\) (legs) and \(c\) (hypotenuse), then \(a^{2}+b^{2}=c^{2}\). Converse: If the side lengths of a triangle satisfy \(a^{2}+b^{2}=c^{2}\) (where \(a\), \(b\), \(c\) are the side lengths), then the triangle is a right - angled triangle.