Final answer:
To prove that AB + BD = AC + CD given B and C are between A and D, we use the commutative property of addition and the fact that both AB + BD and AC + CD represent the full length of AD, thus are equal to each other.
Step-by-step explanation:
The question is asking to prove that AB + BD = AC + CD given that B is between A and D, and C is between A and D. We can use the properties of addition to demonstrate this.We know from the commutative property of addition (A + B = B + A) that the order in which we add segments does not affect their sum. Additionally, we can apply the distributive property to deal with expressions involving addition and multiplication of lengths.
Let's start by writing down what we know:
- AD is the whole segment, and B and C are points within the segment.
- AB + BD = AD (since B is between A and D).
- AC + CD = AD (since C is between A and D).
Since both AB + BD and AC + CD equal the length of AD, by the transitive property of equality, we can conclude:
AB + BD = AC + CD
This proof is reliant on the basic properties of algebra and geometry, specifically concerning the addition of lengths and segments.