Paragraph proof: in the figure AC and BD LIE IN PLANE j and BY AND CX lie on k C and D are collinear. 2.) points a c and x are coplanar. 3.) X and Y are collinear

Why do you assume we can see the figure?

You don't have to be rude

To prove the given statements, we can start by visualizing the figure described. We have a plane represented as "j" and two lines represented as "AC" and "BD" laying on this plane. Then, there are two other lines, "BY" and "CX," lying on another plane represented as "k." Additionally, we are told that point C and point D are collinear, meaning they lie on the same line.

Now, let's go through the paragraph proof step by step:

1. Given: AC and BD lie on plane j, and BY and CX lie on plane k.
2. Given: C and D are collinear.
3. Given: A, C, and X are coplanar.
4. To prove: X and Y are collinear.

To prove that X and Y are collinear, we need to show that they lie on the same line.

Proof:
Since A, C, and X are coplanar (given in statement 3), they lie on the same plane. Therefore, we can form a plane that contains points A, C, and X. Let's call this plane "m."

Now, looking at plane m:

a) Since C and D are collinear (given in statement 2), they lie on the same line. Let's call this line "p."
b) Since BY and CX lie on plane k (given in statement 1), they are coplanar and lie on the same plane.
c) Points C and X lie on plane m.
d) So, the line that contains C and X must lie on plane m. Let's call this line "n."

Since line "p" lies on plane m, and line "n" lies on plane m as well, they must intersect at a point. Let's call this point "Z."

Now, consider the line CZ:
- CZ lies on plane m since C and Z are collinear points on plane m.
- CZ lies on plane k since Z lies on line CX, which lies on plane k.

Therefore, CZ lies on both plane m and plane k, meaning CZ is the intersection of plane m and plane k.

Now, since CZ lies on plane m and plane k, CZ must lie on their intersection line, which we'll call "q."

Since line p and line q both contain point C, they must be the same line. Therefore, point D lies on line q.

To summarize, we have shown that point D lies on line q, which is the intersection line of plane m (which contains A, C, and X) and plane k (which contains BY and CX). Therefore, D lies on line BY as well since BY lies on plane k.

Finally, since X and Y are both points on line BY, we can conclude that X and Y are collinear.

Hence, we have proved that X and Y are collinear.