Question

Две параллельные прямые пересечены секущей. докажите что биссектрисы накрест лежащих углов параллельны

Answer 1

Пусть a и b параллельные прямые, с - секущая. Тогда углы (обозначенные синим цветом) равны как накрест лежащие. m и n бисектриссы этих углов. Известно, что бисектрисса делит угол пополам. Если накрест лежащие углы равны, то также равны и их половинки, т. е. угол 1 равен углу 2.
Рассмотрим две прямые m и n и секущую с. Углы 1 и 2 (желтые) являются накрест лежащие для этих прямых и секущей и поскольку (как было сказано выше) угол 1 = 2, то прямые m и n параллельны.
Доказано.

Answer 2

Proof that the Bisectors of Opposite Angles of Intersecting Lines are Parallel

To prove that the bisectors of opposite angles formed by intersecting lines are parallel, we can use the following steps:

1. Let's consider two parallel lines, line AB and line CD, intersected by a transversal line EF, as shown below:

        A----------------B
        |                |
        |                |
        E----------------F
        |                |
        |                |
        C----------------D
   

2. We need to prove that the bisectors of angles AEF and BED are parallel.

3. Let's assume that the bisectors of angles AEF and BED intersect at point G.

4. Since the bisectors of angles AEF and BED are lines, they will intersect the transversal EF at points H and I, respectively.

5. Now, let's consider the triangles AEG and BEI. These triangles share the side EG = EI and the side AE = BE (since AB and CD are parallel lines).

6. By the Angle-Side-Angle (ASA) congruence criterion, we can conclude that triangles AEG and BEI are congruent.

7. Therefore, the corresponding angles of these congruent triangles are equal. In this case, angle AEG is congruent to angle BEI.

8. Since angle AEG is the bisector of angle AEF and angle BEI is the bisector of angle BED, we can conclude that angle AEF is congruent to angle BED.

9. Now, let's consider the triangles AEF and BED. These triangles share the side EF = ED and the side AE = BE (since AB and CD are parallel lines).

10. By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangles AEF and BED are congruent.

11. Therefore, the corresponding angles of these congruent triangles are equal. In this case, angle AFE is congruent to angle BDE.

12. Since angle AFE is the bisector of angle AEF and angle BDE is the bisector of angle BED, we can conclude that angle AFE is congruent to angle BDE.

13. From steps 8 and 12, we can conclude that angle AEF is congruent to angle AFE and angle BED is congruent to angle BDE.

14. By the Transitive Property of Congruence, we can conclude that angle AEF is congruent to angle AFE is congruent to angle BED is congruent to angle BDE.

15. Since angles AEF and BED are congruent, their bisectors, which are lines passing through the vertex of the angle, must be parallel.

Therefore, we have proved that the bisectors of opposite angles formed by intersecting lines are parallel.