Question

Отрезки EFиPQ пересекаются в их середине M.докажите,что PE||QF

Answer 1

Давай 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.

Рисунок корявый, но думаю, поймёшь)

Answer 2

Given Information

We are given that the line segments EF and PQ intersect at their midpoint M. We need to prove that PE is parallel to QF.

Proof

To prove that PE is parallel to QF, we can use the property that if two lines are parallel, then the alternate interior angles formed by a transversal are congruent.

Let's consider the line segments EF and PQ intersecting at their midpoint M. Since M is the midpoint of both line segments, we can say that EM = MF and PM = MQ.

Now, let's draw a transversal line through M that intersects the line segments EF and PQ. Let's call the point of intersection of the transversal with EF as X and the point of intersection of the transversal with PQ as Y.

Since M is the midpoint of EF, we can say that EX = XF. Similarly, since M is the midpoint of PQ, we can say that PY = YQ.

Now, let's consider the alternate interior angles formed by the transversal line. The alternate interior angles are angle PEX and angle QFY.

Since EM = MF and PM = MQ, we can say that triangle EMP is congruent to triangle FMQ by the Side-Angle-Side (SAS) congruence criterion.

By the congruence of triangles EMP and FMQ, we can say that angle PEX is congruent to angle QFY.

Therefore, we have proved that PE is parallel to QF, as the alternate interior angles formed by the transversal are congruent.

Note: The proof assumes that the given line segments EF and PQ are straight lines and that the given information is accurate.