Question

write a paragraph proof using the figure shown to prove that ab + cb > ac. to prove that ab + cb > ac, start by identifying relationships between line segments in the given figure. the segment bd is for △abc

Explanation:

Step1: Use the property of perpendicular lines

In a right - triangle, the hypotenuse is the longest side. In right - triangle \(ABD\), \(AB\) is the hypotenuse. So, \(AB>AD\) (by the hypotenuse - leg relationship in right - triangles: in a right - triangle, the hypotenuse is longer than either of the legs).

Step2: Use the property of perpendicular lines for another right - triangle

In right - triangle \(CBD\), \(CB\) is the hypotenuse. So, \(CB > CD\) (by the hypotenuse - leg relationship in right - triangles).

Step3: Add the two inequalities

If \(AB>AD\) and \(CB > CD\), then \(AB + CB>AD + CD\).

Step4: Simplify the right - hand side

Since \(AD+CD = AC\) (by the segment addition postulate: if \(D\) lies on \(AC\), then \(AC=AD + CD\)), we have \(AB + CB>AC\).

Answer:

\(AB + CB>AC\)