Consecutive Interior Angles Theorem

Start by noticing that $∠ 1$ and $∠ 4$ form a linear pair as well as $∠ 2$ and $∠ 3 .$

Therefore, these pair of angles are supplementary.

{m ∠ 1 + m ∠ 4 = 18 0^{\circ} m ∠ 2 + m ∠ 3 = 18 0^{\circ} (I) (II)

On the other hand, since the lines

ℓ_{1}

and

ℓ_{2}

are parallel, by the Alternate Interior Angles Theorem, the alternate interior angles are congruent. This allows to write the following relations.

∠ 1 ≅ ∠ 2 ∠ 3 ≅ ∠ 4 \Rightarrow m ∠ 1 = m ∠ 2 \Rightarrow m ∠ 3 = m ∠ 4

Next, in Equation I, substitute

m ∠ 3

for

m ∠ 4 .

m ∠ 1 + m ∠ 4 = 18 0^{\circ}

$m ∠ 4 = m ∠ 3$

m ∠ 1 + m ∠ 3 = 18 0^{\circ}

The last equation implies that

∠ 1

and

∠ 3

are supplementary. Similarly, in Equation II, substitute

m ∠ 4

for

m ∠ 3 .

m ∠ 2 + m ∠ 3 = 18 0^{\circ}

$m ∠ 3 = m ∠ 4$

m ∠ 2 + m ∠ 4 = 18 0^{\circ}

As before, the last equation implies that

∠ 2

and

∠ 4

are supplementary, which completes the proof.

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