The converse has to earn its own argument
Show on a big screenAlternate 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.
Try the lab to finish this lesson.