Question

Докажите что биссектрисы е и d внутренних накрест лежащих углов,образованных параллельными прямыми а и b и секущей с, параллельны,то есть лежат на параллельных прямых.

Answer 1

биссектрисы e и d делят внутренние накрест лежащие углы (которые равны) на 4 равных угла, 2 из которых являются также  внутренними накрест лежащими для прямых e и d и секущей с. из равенства этих углов следует, что прямые e и d параллельны.

Answer 2

Proof that the bisectors e and d of the interior alternate angles formed by parallel lines a and b and a transversal c are parallel

To prove that the bisectors e and d of the interior alternate angles formed by parallel lines a and b and a transversal c are parallel, we can use the properties of parallel lines and the angle bisector theorem.

Let's consider the following diagram:

          a
          |
          |
    c-----+-----d
          |
          |
          b

We have parallel lines a and b, and a transversal c that intersects them. Let's denote the points of intersection between the transversal c and the lines a and b as A and B, respectively.

We want to prove that the bisectors e and d of the interior alternate angles formed by the lines a and b and the transversal c are parallel.

To prove this, we will use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the adjacent sides of the triangle.

Let's consider the angle formed by the lines a and c. The bisector e divides this angle into two equal angles. Similarly, the angle formed by the lines b and c is bisected by the line d.

Now, let's consider the triangles formed by the lines a, c, and e, and the lines b, c, and d. According to the angle bisector theorem, the segments formed by the bisectors e and d on the opposite sides of the triangles are proportional to the adjacent sides.

Since the lines a and b are parallel, the corresponding angles formed by the transversal c are congruent. Therefore, the triangles formed by the lines a, c, and e, and the lines b, c, and d are similar triangles.

By the properties of similar triangles, we know that corresponding sides of similar triangles are proportional. Therefore, the segments formed by the bisectors e and d on the opposite sides of the triangles are proportional.

Since the segments formed by the bisectors e and d are proportional, and they divide the opposite sides of the triangles, it follows that the bisectors e and d are parallel.

Therefore, we have proved that the bisectors e and d of the interior alternate angles formed by parallel lines a and b and a transversal c are parallel.