delta.

math sandwich

[delta-math]

Theorem: All points of each of two parallel lines are equidistant from each other.

Let's say we have two parallel lines a and b.

On the line a, mark two points - A and X. Draw from both perpendiculars to the line b (B and Y).

Since XY⊥b, then XY⊥a (because a and b are parallel).

Right-angled triangles △ABY and △YXA are equal (by hypotenuse and acute angle) due to the fact that angles ∠XAY and ∠AYB lie crosswise and AY is a common side, => XY=AB.

[delta-math]

Ex: Median AM of triangle ABC is equal to segment BM. Prove that one of the angles of triangle ABC is equal to the sum of the other two angles.

Since AM is the median, then BM=MC. But since AM=BM => AM=BM=MC.

If you look closely, you can see that ΔABM is isosceles => the angles at the base are equal. Same thing with ΔAMC.

We see that ∠BAC=∠BAM+∠MAC. But ∠BAM=∠ABM and ∠MAC=∠MCA.

=> ∠BAC=∠ABM+∠MCA or ∠A=∠B+∠C.

[delta-math]

If the sides of one angle are respectively parallel to the sides of another angle, then such angles are either equal or add up to 180°.

For some reason, this particular theorem causes some schoolchildren to have problems with its proof.

So we can consider that we are proving two theorems. Let's start with the first case.

If two angles are parallel, then they are equal.

Let's ignore what you already know it’s true. Let's just prove it. I will try to explain without water and unnecessary introductory notation.

Let's say we have two corners. Arrange them as you like, but so that their corners look in the same direction.

If we imagine each corner lying on invisible lines (light blue lines), we can notice that two parallel lines intersect other two parallel lines and in total they give 4 points of their intersection.

Then we recall the theorem on parallel lines and a secant: two crosswise lying angles are equal.

Therefore, we can represent angle ∠A as a cross to ∠B. However, ∠B can also be represented as a cross to ∠C.

Then ∠A=∠B=∠C

Well, from the fact that ∠A=∠C we can conclude that the angles are equal.

If two angles are parallel, then they add up to 180°.

Let's do the same thing as last time, only arrange the corners so that they look in opposite directions.

If you look closely, you can see that ∠C=∠B. Well, the angle ∠B is a one-sided angle to ∠A, and, according to the parallel lines and secant theorem, ∠A and ∠B should add up to 180°.

I hope there won't be any more problems. :)