# Parallel lines and Alternate angles

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

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# Theorems on angled forms by two parallel lines and a transversal. Theorem 1

# Parallel lines and Alternate angles

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

** **

**Step 1**

**Step 2**

**Step 3**

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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.

Therefore,

The theorem is proved.

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

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