Parallel lines and Alternate angles

 

If two parallel lines are cut by a transversal, then the alternate angles are equal.

Parallel lines and Alternate angles

Parallel lines and Alternate angles

Theorem on angled forms by two parallel lines and a transversal

Proof of the theorem on angled forms by two parallel lines and a transversal

 

Step 1

 

Consider two lines a and b. Let line MN cuts these lines, i.e. line MN is a transversal.

Let us prove that, alternate angles 1 and 2 are equal.

Proof of the theorem on angled forms by two parallel lines and a transversal. Step

Proof of the theorem on angled forms by two parallel lines and a transversal. Step 1

Step 2

 

We will prove the theorem using a method of proof by contradiction.

Assume that angles 1 and 2 are not equal.

From ray MN extend angle PMN equal to angle 2.

At that construct angle PMN so that angles PMN and 2 are alternate angles for lines MP and b and transversal с.

And as by construction the alternate angles are equal, then the lines will be parallel:

Proof of the theorem on angled forms by two parallel lines and a transversal

Proof of the theorem on angled forms by two parallel lines and a transversal. Step

Proof of the theorem on angled forms by two parallel lines and a transversal. Step 2

Step 3

 

As a result we have obtained that two lines pass through point M (lines а and МР) parallel to line b.

But it contradicts the parallel axiom.

Hence our assumption that alternate angles with parallel lines and a transversal are not equal is false.

Therefore,

Theorems on angled forms by two parallel lines and a transversal

The theorem is proved.

Proof of the theorem on angled forms by two parallel lines and a transversal. Step

Proof of the theorem on angled forms by two parallel lines and a transversal. Step 3