Question

По разные стороны от отрезка BC взяты точки K и M так, что BK=CM,KC=BM.Докажите что треугольник BKC и BCM равны

Answer 1

Given Information

We are given a line segment BC and two points K and M on opposite sides of BC such that BK = CM and KC = BM. We need to prove that triangles BKC and BCM are congruent.

Proof

To prove that triangles BKC and BCM are congruent, we will use the Side-Angle-Side (SAS) congruence criterion.

1. Side BC is common to both triangles BKC and BCM. 2. Side BK is equal to side CM (given). 3. Side KC is equal to side BM (given).

Now, let's consider the angles:

4. Angle BKC is equal to angle BCM. This is because they are vertical angles formed by the intersection of lines BK and CM. 5. Angle KBC is equal to angle MCB. This is because they are alternate interior angles formed by the intersection of lines BK and CM.

Therefore, we have:

- Side BC is equal to side BC (common side). - Side BK is equal to side CM (given). - Side KC is equal to side BM (given). - Angle BKC is equal to angle BCM (vertical angles). - Angle KBC is equal to angle MCB (alternate interior angles).

By the SAS congruence criterion, triangles BKC and BCM are congruent.

Therefore, we have proven that triangles BKC and BCM are congruent.