Question

Докажите что если прямая пересекает одну из двух параллельных прямых то она пересекает и другую

Answer 1

Из учебника набираю

Действительно, пусть прямые а и b паралл. и прямая с пересекает прямую а в точке М. Докажем, что прямая с пересеккает и b. если бы прямая не пересекала прямую b, то через точку М проходили бы две прямые  параллельные прямой b. но это противоречит аксиоме паралл прямых, и, значит, прямая с пересекает прямую b.

Answer 2

Proof that if a line intersects one of two parallel lines, then it intersects the other

To prove that if a line intersects one of two parallel lines, then it intersects the other, we can use the following reasoning:

Let's consider two parallel lines, line A and line B. We will assume that line A intersects line C, and we want to prove that line A also intersects line B.

To begin, we can draw a transversal line, line D, that intersects both line A and line B. This is possible because line A and line B are parallel, and according to the parallel postulate, any transversal line will intersect them.

Now, let's consider the angles formed by the intersection of line D with line A and line B. We will label these angles as angle 1 and angle 2, respectively.

According to the alternate interior angles theorem, if two parallel lines are intersected by a transversal line, then the alternate interior angles are congruent. In this case, angle 1 and angle 2 are congruent.

Since line A intersects line C, we can draw another transversal line, line E, that intersects both line A and line C. Again, this is possible because line A and line C are not parallel.

Now, let's consider the angles formed by the intersection of line E with line A and line C. We will label these angles as angle 3 and angle 4, respectively.

According to the corresponding angles theorem, if two parallel lines are intersected by a transversal line, then the corresponding angles are congruent. In this case, angle 3 and angle 4 are congruent.

Now, let's compare angle 1 and angle 3. Since angle 1 and angle 3 are congruent (as they are corresponding angles), and angle 1 and angle 2 are congruent (as they are alternate interior angles), we can conclude that angle 2 and angle 3 are congruent.

Since angle 2 and angle 3 are congruent, this means that line D intersects line C. Therefore, line A intersects both line B and line C.

In conclusion, if a line intersects one of two parallel lines, then it intersects the other as well.