Question

1. communicate and justify write a paragraph proof for theorem 5-11. use the figure shown and prove that ab + cb > ac.

Explanation:

Step1: Apply the triangle inequality theorem in \(\triangle ABD\)

In \(\triangle ABD\), by the triangle - inequality theorem (the sum of the lengths of any two sides of a triangle is greater than the length of the third side), we have \(AB>AD\).

Step2: Apply the triangle inequality theorem in \(\triangle CBD\)

In \(\triangle CBD\), by the triangle - inequality theorem, we have \(CB > CD\).

Step3: Add the two inequalities

Add the inequalities \(AB>AD\) and \(CB > CD\). We get \(AB + CB>AD + CD\).

Step4: Use the segment addition postulate

Since \(AD+CD = AC\) (by the segment addition postulate, which states that if \(D\) lies on \(AC\), then \(AC=AD + CD\)), substituting \(AD + CD\) with \(AC\) in the inequality \(AB + CB>AD + CD\) gives \(AB + CB>AC\).

Answer:

We have proved that \(AB + CB>AC\) by using the triangle - inequality theorem in \(\triangle ABD\) and \(\triangle CBD\) and the segment addition postulate.