Question

Отрезки EF и PQ пересикаются в их середине M Докозать PE//QF

Answer 1

Рассмотрим треугольники EMP  и QMF
1)они равны по двум сторонам и углу между ними
EM=MF
PM=MQ(по условию)
угол  EMP  и угол QMF равны как вертикальные углы.
2)т.л треугольники равны,то угол PEF=УглуQFE следовательно PE//QF(тк накрестлежащие углы равны) 

Answer 2

The Problem:

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

Proof:

Let's consider the following facts: 1) Line segments EF and PQ intersect at point M, which means M is the midpoint of both EF and PQ. 2) For any two intersecting lines, the opposite angles formed are equal. 3) If two lines intersect, and a point on one line is the midpoint of the other line, then the lines are parallel.

Now, let's put these facts together: Since M is the midpoint of both EF and PQ, angles PEM and QFM must be equal (by fact 2 above). Also, M lies on both EF and PQ. Therefore, by fact 3, PE must be parallel to QF.

So we can conclude that PE // QF, as was required to be proven.