Question

Отрезки EF и PQ пересекаются в их середине M.Докажите что PE параллельна QF. Подробно и рисуночек пожалуйста

Answer 1

Авай рассмотрим треугольники РЕМ и QFM:
РМ=МQ, EM=FM, угол РМЕ = углу QМF (как вертикальные).
Отсюда делаем вывод, что треугольники РЕМ и QFM равны по двум сторонам и углу между ними.
Угол Р равен углу Q как НЛУ (накрест лежащие углы). Следовательно, РЕ||QF.

Answer 2

Proof that PE is parallel to QF

To prove that PE is parallel to QF, we can use the fact that the segments EF and PQ intersect at their midpoint M. Let's break down the proof step by step:

1. Given: Segments EF and PQ intersect at their midpoint M. 2. We need to prove: PE is parallel to QF.

Proof:

Step 1: Since M is the midpoint of both EF and PQ, we can say that EM = MF and PM = MQ.

Step 2: Let's consider the triangles PEM and QFM. We know that EM = MF (from Step 1) and PM = MQ (from Step 1).

Step 3: By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangles PEM and QFM are congruent.

Step 4: When two triangles are congruent, their corresponding angles are equal. Therefore, angle PEM = angle QFM.

Step 5: Since angle PEM and angle QFM are corresponding angles, and they are equal, we can conclude that lines PE and QF are parallel.

Therefore, we have proved that PE is parallel to QF.

Here is a diagram to illustrate the proof:

        E _______ M _______ F
         |\                 /|
         | \               / |
         |  \             /  |
         |   \           /   |
         |    \         /    |
         |     \       /     |
         |      \     /      |
         |       \   /       |
         |        \ /        |
         |         X         |
         |        / \        |
         |       /   \       |
         |      /     \      |
         |     /       \     |
         |    /         \    |
         |   /           \   |
         |  /             \  |
         | /               \ |
         |/________________\|
         P        Q

In the diagram, E and F represent the endpoints of segment EF, and P and Q represent the endpoints of segment PQ. The intersection point of EF and PQ is denoted by M. The line segments PE and QF are parallel, as proven above.

Please note that the diagram is for illustrative purposes only and may not be to scale.

I hope this explanation and diagram help you understand the proof. Let me know if you have any further questions!