Question

1. choose the correct answer.

the triangle inequality theorems states that the sum of the lengths of any two sides of a triangle is ______ the length of the third side.

equal to
less than
greater than

1. choose the correct answer.

a(n) ______ is an argument that begins with the assumption that a conclusion is false, then uses logical reasoning to show that the assumption leads to a contradiction.

postulate
indirect proof
corollary
theorem

Explanation:

Question 2

The Triangle Inequality Theorem in Geometry (a subfield of Mathematics) states that for any triangle with side lengths \(a\), \(b\), and \(c\), the sum of the lengths of any two sides (\(a + b\), \(a + c\), or \(b + c\)) must be greater than the length of the remaining side. For example, if we have a triangle with sides 3, 4, and 5, \(3 + 4>5\), \(3 + 5>4\), and \(4 + 5>3\). If the sum were equal or less, a triangle couldn't be formed (e.g., 3, 4, 7: \(3 + 4 = 7\), which is a straight line, not a triangle; 3, 4, 8: \(3 + 4<8\), which can't form a triangle). So the correct relationship is "greater than".

• A postulate is a statement accepted without proof (e.g., through any two points, there is exactly one line).
• An indirect proof (also called proof by contradiction) starts by assuming the conclusion is false. Then, through logical reasoning, it shows this assumption leads to a contradiction, proving the original conclusion must be true.
• A corollary is a statement that follows easily from a theorem.
• A theorem is a statement proven to be true.

Since the description matches the definition of an indirect proof, that's the correct answer.

Answer:

greater than (the option with "greater than")

Question 3