Parallel lines and Alternate angles
If two parallel lines are cut by a transversal, then the alternate angles are equal.
Theorem on angled forms by two parallel lines and a transversal
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 1
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. Step 2
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.
The theorem is proved.
Proof of the theorem on angled forms by two parallel lines and a transversal. Step 3
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