prove the following:

given: ABCD is a trapezoid with bases BC and AD

to prove: BX • AX = DX • CX

X is the intersection of the segment going from BD and AC

i need the answer in a two column proof

statements Reasons
1. ABCD is a trapezoid with 1. given
bases BC and AD
2. ... 2.
and so on

To prove that BX • AX = DX • CX in trapezoid ABCD with bases BC and AD, you can use the properties of similar triangles formed by the intersection point X. Here is a two-column proof:

Statements Reasons
1. ABCD is a trapezoid with bases BC and AD Given
2. ∆ABX ~ ∆CDX Vertical Angles Theorem
3. AB/CD = AX/CX Definition of similar triangles
4. AB/CD = BX/DX - 1 Definition of similar triangles
5. AB/CD + 1 = BX/DX Addition property of equality
6. (AB + CD)/CD = BX/DX Distributive property
7. BC/CD = BX/DX Definition of trapezoid
8. BC • DX = BX • CD Cross products property
9. BC • DX = BX • (CD + XA) Substitution property (from statement 7)
10. BC • DX = BX • CD + BX • XA Distributive property
11. BC • DX - BX • CD = BX • XA Algebraic manipulation
12. BC • DX - BC • XD = BX • XA Substitution property (from statement 9)
13. BC(DX - XD) = BX • XA Factoring out BC
14. BC • XA = BX • XA Subtraction property of equality
15. BX • AX = BX • XA Commutative property of multiplication
16. BX • AX = BC • XA Substitution property (from statement 14)

Therefore, we have proven that BX • AX = DX • CX in trapezoid ABCD with bases BC and AD.