# 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:

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# Criteria for Parallel lines. Criterion 1

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

**Proof of the first criterion of parallel lines (via equality of alternate angles)**

**Step 1**

**Step 2**

**Step 3**

**Step 4**

**Step 5**

**Step 6**

**Step 7**

**Step 8**

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- Criteria for Parallel lines. Criterion…

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)

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

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

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

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

On line *b* from point B, extend segment ВН_{1} equal to segment АН.

Proof of the first criterion of parallel lines. Step 5

Draw a segment ОН_{1}.

Proof of the first criterion of parallel lines. Step 6

Consider triangles ОНА and ОН_{1}А:

АН=Н_{1}В – by construction;

ОА=ОВ – by construction;

∠1=∠2 – by condition.

Proof of the first criterion of parallel lines. Step 7

According to the property of congruent triangles:

From equality ∠AOH and ∠BOH_{1} it follows that point H_{1 }lies on the continuation of ray ОН, i.e. points Н, О and Н_{1} lie on the same line.

And from equality ∠Н and ∠Н_{1} it follows that angle Н_{1} is the right angle (because angle Н is the right angle by construction).

Hence lines *a* and *b *are perpendicular to line НН_{1}; 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

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