Question

given: ad = cf
bc = de
prove: ab = ef
statement\treason

1. ad = cf\tgiven
2. ad = ab + bc + cd

cf = cd + de + ef\tsegment addition

1. ab + bc + cd = cd + de + ef\ttransitive property of equality
2. ab + bc = de + ef
3. bc = de\tgiven
4. ab = ef\tsubtraction property of equality

what is the reason for the fourth statement in this proof?
a. definition of collinear points
b. segment addition
c. subtraction property of equality
d. substitution property of equality

Explanation:

The third statement is \(AB + BC+CD = CD + DE + EF\). To get to the fourth statement \(AB + BC=DE + EF\), we subtract \(CD\) from both sides of the equation in the third - statement. According to the Subtraction Property of Equality, if \(a=b\), then \(a - c=b - c\). Here \(a = AB + BC+CD\), \(b = CD + DE + EF\), and \(c = CD\).

Answer:

C. Subtraction Property of Equality