If A, B, C, D, and E are collinear with B between A and C, C between B and D, and D between Cand E, and AC = BD – CE, then AB = BC = DE
To prove that AB = BC = DE, we can use the properties of collinear points and segment addition
To prove that AB = BC = DE, we can use the properties of collinear points and segment addition.
Given: A, B, C, D, and E are collinear.
B is between A and C.
C is between B and D.
D is between C and E.
AC = BD – CE.
To prove: AB = BC = DE.
Proof:
1. By the definition of collinear points, since A, B, and C are collinear and B is between A and C, we can write:
AC = AB + BC. (Segment Addition Postulate)
2. By the definition of collinear points, since B, C, and D are collinear and C is between B and D, we can write:
BD = BC + CD. (Segment Addition Postulate)
3. By the definition of collinear points, since C, D, and E are collinear and D is between C and E, we can write:
CD = CE + DE. (Segment Addition Postulate)
4. From the given information, AC = BD – CE. Substitute step 2 and step 3 into it:
AB + BC = BD – CE. (Substitution)
5. Rearrange the equation by adding CE to both sides:
AB + BC + CE = BD. (Addition Property of Equality)
6. Substitute step 1 into the above equation:
AC + CE = BD. (Substitution)
7. Since AC = BD – CE from the given information, we can substitute it into the equation:
BD – CE + CE = BD. (Substitution)
8. Simplify the equation:
BD = BD. (Simplification)
9. Now, we can conclude that AB = BC = DE by comparing the equations from steps 1, 5, and 8.
Therefore, we have proven that AB = BC = DE using the properties of collinear points and segment addition.
More Answers:
[next_post_link]