Question

to prove the triangle inequality theorem for (\triangle pqr) by contradiction, you could assume that (pq + pr) choose... (qr) and then prove that choose...

Explanation:

Step1: Understand the Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any triangle \( \triangle PQR\), \(PQ + PR>QR\), \(PQ + QR>PR\), and \(QR + PR>PQ\).

Step2: Apply the method of proof by contradiction

In a proof by contradiction, we assume the opposite of what we want to prove. Since we want to prove \(PQ + PR>QR\), we assume \(PQ + PR\leq QR\). Then, we try to show that this assumption leads to a contradiction. For example, if \(PQ + PR\leq QR\), then the three line - segments \(PQ\), \(PR\), and \(QR\) cannot form a triangle (because the sum of the lengths of two sides is not greater than the third side, which violates the basic property of triangle formation).

Answer:

First drop - down: \(\leq\)
Second drop - down: this assumption leads to a contradiction (such as non - formation of a triangle)