What are alternate interior angles?

How to find the right pair

This is one of the most important patterns in parallel-line geometry because, when the crossed lines are parallel, alternate interior angles are congruent. That makes them a frequent reason in proofs and a common shortcut in missing-angle problems.

The phrase can still describe a positional pair when the lines are not parallel, but the equal-measure result no longer follows automatically. The parallel condition is what upgrades the pattern into a theorem tool.

Start at one intersection and shade the region between the two lines. Then move to the other intersection and look on the opposite side of the transversal. That two-step search is more reliable than trying to remember the name from a complicated picture.

When the lines are parallel, the two angles have equal measures. If the lines are not parallel, they may still be alternate interior by position, but you cannot claim equality without another reason.

• Alternate interior angles lie between the lines on opposite sides of the transversal.
• Interior means the angles lie between the two target lines.
• Alternate means the pair lies on opposite sides of the transversal.
• In the parallel case, alternate interior angles are equal in measure.

Where alternate interior angles help

• Use alternate interior angles in parallel-line proofs and algebraic angle equations.
• Use them when testing whether two lines are parallel from given angle information.
• Use them to interpret textbook diagrams that show inside-opposite angle positions.

Three alternate-interior checks

• Do not call a pair alternate interior if the angles are inside the lines but on the same side of the transversal.
• Do not assume equality unless the lines are known or proven to be parallel.
• Do not confuse alternate interior angles with corresponding angles, which occupy matching rather than opposite positions.