Question

the pythagorean theorem states that in a right triangle, the square of the ______ is equal to the sum of the squares of the legs.
the triangle proportionality theorem states that a line parallel to one side of a triangle divides the other two sides ______.
the converse of the triangle proportionality theorem proves that two lines are ______ when the sides are divided proportionally.
the ______ postulate states that if all three sides of two triangles are equal, the triangles are congruent.

Explanation:

First Question:

The Pythagorean Theorem is a fundamental principle in geometry (a subfield of Mathematics) for right - angled triangles. It is given by the formula \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the side opposite the right angle) and \(a\) and \(b\) are the legs (the other two sides) of the right triangle. So, the square of the hypotenuse is equal to the sum of the squares of the legs.

The Triangle Proportionality Theorem (also known as Thales' theorem) states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. For example, if in \(\triangle ABC\), a line \(DE\) is parallel to \(BC\) and intersects \(AB\) at \(D\) and \(AC\) at \(E\), then \(\frac{AD}{DB}=\frac{AE}{EC}\).

The converse of the Triangle Proportionality Theorem states that if a line divides two sides of a triangle proportionally, then that line is parallel to the third side. So, when the sides are divided proportionally, the two lines (the line dividing the sides and the third side of the triangle) are parallel.

Answer:

hypotenuse

Second Question: