If are cut by a , then the pairs of are congruent.
Consider the following diagram.
To prove that corresponding angles are congruent, it will be shown that
∠1 and
∠5 are congruent.
Notice that, by definition,
∠1 and
∠4 are , as are
∠4 and
∠5. Thus, the following equations can be written.
m∠1+m∠4=180∘m∠4+m∠5=180∘
By , the equations can be set equal to each other.
m∠1+m∠4=m∠4+m∠5
It follows then that
m∠1=m∠5. Thus,
∠1≅∠5. The same reasoning applies to all pairs of corresponding angles. Thererefore, when a pair of parallel lines is cut by a transversal, the pairs of corresponding angles are congruent.
This can be summarized in the following flowchart proof.