# The first criterion of parallel lines (via equality of alternate angles)

When two lines cut by a transversal alternate angles are equal, then these lines are parallel:

The first criterion of parallel lines (via equality of alternate angles)

## Step 1

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

As a result of the intersection of line c and lines a and b, alternate angles are formed: ∠1 and ∠2.

Let:

Let us prove that a ׀׀ b.

Proof of the first criterion of parallel lines. Step 1

## Step 2

Consider a case when alternate angles are equal – right angles:

If angles 1 and 2 are right angles, then a and b are perpendicular to line c and, consequently, are parallel.

Proof of the first criterion of parallel lines. Step 2

## Step 3

Consider a case when angles 1 and 2 are not right angles.

Denote the points of intersection of transversal c and lines a and b by letters A and B.

Divide segment AB into two equal parts. And denote the point, which divides AB by half, by letter O.

Proof of the first criterion of parallel lines. Step 3

## Step 4

Draw a perpendicular line OH to line а to form point O on segment AB.

Proof of the first criterion of parallel lines. Step 4

## Step 5

On line b from point B, extend segment ВН1 equal to segment АН.

Proof of the first criterion of parallel lines. Step 5

## Step 6

Draw a segment ОН1.

Proof of the first criterion of parallel lines. Step 6

## Step 7

Consider triangles ОНА and ОН1А:

АН=Н1В – by construction;

ОА=ОВ – by construction;

∠1=∠2 – by condition.

Proof of the first criterion of parallel lines. Step 7

## Step 8

According to the property of congruent triangles:

From equality ∠AOH and ∠BOH1 it follows that point H1 lies on the continuation of ray ОН, i.e. points Н, О and Н1 lie on the same line.

And from equality ∠Н and ∠Н1 it follows that angle Н1 is the right angle (because angle Н is the right angle by construction).

Hence lines a and b are perpendicular to line НН1; therefore, they are parallel.

The first criterion of parallel lines (via equality of alternate angles) is proved.

Proof of the first criterion of parallel lines. Step 8