Question

how can the segment addition postulate be used to show that ( ae = ab + bc + cd + de )? image of a line with points a, b, c, d, e choose the correct answer below

a. by the segment addition postulate, ( ab = bc ) and ( cd = de ). so, using simple addition, ( ae = ab + bc + cd + de ).

b. by the segment addition postulate, ( ad = ab + cd ) and ( ce = cd + de ). so, using substitution, ( ae = ab + bc + cd + de ).

c. by the segment addition postulate, ( cd = ab + bc ) and ( de = bc + cd ). so, using substitution, ( ae = ab + bc + cd + de ).

d. by the segment addition postulate, ( ae = ac + ce ), ( ac = ab + bc ), and ( ce = cd + de ). so, using substitution, ( ae = ab + bc + cd + de ).

Explanation:

The Segment Addition Postulate states that if points are collinear (on a straight line), then the length of a segment is the sum of the lengths of its sub - segments. For the line segment \(AE\), we can break it into \(AC\) and \(CE\) (so \(AE = AC+CE\) by the postulate). Then, \(AC\) can be broken into \(AB\) and \(BC\) (so \(AC = AB + BC\) by the postulate), and \(CE\) can be broken into \(CD\) and \(DE\) (so \(CE=CD + DE\) by the postulate). By substituting \(AC\) and \(CE\) in the equation \(AE = AC + CE\) with \(AB + BC\) and \(CD+DE\) respectively, we get \(AE=AB + BC+CD + DE\).

• Option A is incorrect because the Segment Addition Postulate does not state that \(AB = BC\) and \(CD = DE\) (there is no indication that these segments are equal in length).
• Option B is incorrect. The correct breakdown for \(AD\) should be \(AD=AB + BC+CD\) (not \(AB + CD\)) and \(CE = CD+DE\) is correct, but the first part of the breakdown of \(AD\) is wrong.
• Option C is incorrect. The Segment Addition Postulate does not give \(CD=AB + BC\) or \(DE = BC + CD\).
• Option D follows the correct application of the Segment Addition Postulate by breaking down \(AE\) into \(AC\) and \(CE\), then further breaking down \(AC\) and \(CE\) into their respective sub - segments and substituting.

Answer:

D. By the Segment Addition Postulate, \(AE = AC + CE\), \(AC=AB + BC\), and \(CE = CD + DE\). So, using substitution, \(AE=AB + BC + CD+DE\)