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:

lines are parallel

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)

Proof of 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:

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Let us prove that a ׀׀ b.

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Proof of the first criterion of parallel lines. Step 1

Step 2

 

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

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

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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.

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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.

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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.

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Proof of the first criterion of parallel lines. Step 5

Step 6

 

Draw a segment ОН1.

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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.

By SAS criterion:

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Proof of the first criterion of parallel lines. Step 7

Step 8

 

According to the property of congruent triangles:

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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.

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Proof of the first criterion of parallel lines. Step 8