The converse has to earn its own argument
Alternate angles are equal. Are the two lines parallel?
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- A transversal cuts two lines. Two alternate angles are both 72°.
- At the bottom crossing, the angle facing one of them is also 72°.
- Now two corresponding angles are equal. The lines are parallel.
The rule
Yes. Equal alternate angles give equal corresponding angles. Those make the lines parallel.
Axiom. A transversal cuts two lines. If corresponding angles are equal, the lines are parallel. This is the converse of the corresponding angles axiom. We accept it without proof.
Theorem. A transversal cuts two lines. If alternate angles are equal, the lines are parallel.
Proof. The alternate angles are equal. In the picture, both are 72°. Take a 72° angle at one crossing. Its vertically opposite angle is also 72°. That opposite angle and the other alternate angle are corresponding angles. So corresponding angles are equal. By the axiom, the lines are parallel.
The same works for co-interior angles. Call them x and y, and x + y = 180°. At x's crossing, let z be in the same corner as y. Then x and z make a linear pair. So z = 180° − x = y. The angles z and y are corresponding angles. So the lines are parallel.
Example. A transversal makes alternate angles of 65° and 65°. By the theorem, the lines are parallel.
Watch out!
Two lines may not meet on your page. They may still meet far away. Equal angles are the test, not the page.