Alternate Interior Angles
In short: Alternate interior angles lie inside the strip between two lines but on opposite sides of the crossing line; they match in size precisely when the two lines are parallel.
8th GradeGeometry8.G.A.5
What is Alternate Interior Angles?
Two lines cut by a third trap a strip in the middle, and a pair of alternate interior angles sits inside that strip, one at each crossing, on opposite sides of the cutting line, which traces a letter Z.
When the outer two lines are parallel, this pair is congruent. The statement also works backward: one equal pair is enough to prove the lines never meet.
How to spot it
- The picture shows a Z or reversed Z shape hidden among two lines and a crossing line.
- A problem gives expressions like \(4x – 9\) and \(2x + 31\) for two angles inside a strip and asks for x.
- You must prove two lines are parallel using only angle measures.
- The wording says between the lines and on opposite sides of the transversal.
The method
- Shade the strip between the two lines; both angles must lie inside it.
- Pick the two inside angles that are on opposite sides of the crossing line, never on the same side.
- If the lines are parallel, set the two angle expressions equal to each other.
- Solve for the unknown and substitute back to get the angle measure.
Alternate Interior Angles: solved example
Problem
Two parallel bike lanes are crossed by a diagonal path. Inside the strip between the lanes, one angle measures \((4x – 9)^{\circ}\) and its alternate partner measures \((2x + 31)^{\circ}\). Find x and the angle.
Solution
The lanes are parallel, so the pair is congruent: \(4x – 9 = 2x + 31\).
Subtract \(2x\) and add 9: \(2x = 40\), so \(x = 20\). Each angle is \(4(20) – 9 = 71\) and also \(2(20) + 31 = 71\). The angles measure \(71^{\circ}\).
Try it
Question 1
Two parallel lines are cut by a transversal. One interior angle is \(117^{\circ}\). What is its alternate interior partner?
Show the answer
It is also \(117^{\circ}\), since parallel lines make these angles congruent.
Question 2
A pair of alternate interior angles measures \(82^{\circ}\) and \(98^{\circ}\). Are the two lines parallel?
Show the answer
No. The angles are not equal, so the lines cannot be parallel.
Alternate Interior Angles vs. Same-Side Interior Angles
Same-side interior angles are inside the strip too, but on the same side of the transversal, and for parallel lines they add up to \(180^{\circ}\). Alternate ones sit on opposite sides and are equal. Look for a Z for equal angles and a C shape for a sum of \(180^{\circ}\).
Keep going
Lessons
Worksheets and quizzes
- Angles in Triangles and Parallel Lines – Worksheet
- Angles in Triangles and Parallel Lines – Quiz
- Parallel Lines and Transversals – Worksheet
- Parallel Lines and Transversals – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry