given isosceles triangle ABC, line AB = BC, and BD is the angle bisector of isosceles triangle ABC

prove AB x DC = BC x AD

Aren't the two halves congruent? (SAS)

If so, then DC is equal to AD.

Don't the ratios AB:AD=BC:DC by corresponding parts?

how do i write that out like on a table though like step by step

I won't do the table otherwise that defeats the purpose of the assignment. If you do the exercise and post your work for evaluation that's fine. We'll comment on it.
Start by drawing a diagram and labeling the parts. Then see what the given statement corresponds to in terms of the triangles parts. Be sure to check the book for theorems to use and examples to follow.

ok i wrote it all out but then when the last thing is AB X DC = BC X AD i don't remember what to write as the reason

If you proved that the triangles are congruent by some method such as SAS, then AB:BC=DC:AD i.e. corresponding parts are in the same ratio, and
AB*AD=DC*BC=BC*DC
because the product of the means = the product of the extremes, and the commutative property of multiplication.
I think that's sufficient, but check your text too. Hopefully I didn't use something not available to you here.

The reason for AB x DC = BC x AD can be explained using the concept of ratios and the properties of equality.

1. Start by proving that the triangles are congruent using the given information. If the triangle is isosceles with AB = BC, you can use the SAS (Side-Angle-Side) congruency theorem to establish congruence.

2. Once congruence is established, you can conclude that the corresponding parts of the triangles are in the same ratio. In this case, AB:BC=DC:AD.

3. Now, let's focus on the expression AB x DC = BC x AD. This is essentially stating that the product of the lengths of the corresponding sides are equal.

4. To justify this equality, you can use the property of equality that allows you to multiply both sides of an equation by the same value without changing its validity. In other words, you can multiply AB x DC and BC x AD by the same value (1 in this case).

5. By doing so, the equation becomes AB x DC = AB x DC, which is always true.

So, in conclusion, proving that AB:BC=DC:AD by congruence allows us to state that AB x DC = BC x AD. This can be understood using the concept of ratios and the properties of equality.