Key takeaway: Alternate interior angles are two angles between a pair of lines on opposite sides of a transversal, and they are equal exactly when the two lines they lie between are parallel.
Grade 88.G.A.5Geometry and Measurement
Definition of Alternate Interior Angles
Alternate interior angles form when a transversal cuts two lines. Both angles lie in the strip between the two lines (interior) and sit on opposite sides of the transversal (alternate). They make a Z shape, or a backwards Z.
When the two lines are parallel, these angles are equal in measure. The reverse also holds: if a transversal makes a pair of equal angles of this kind, the two lines must be parallel. This single fact lets you find missing angles in a map of crossing streets or in a roof truss.
Formula and notation
With parallel lines \(a\) and \(b\) cut by transversal \(t\), and the angles numbered so that \(\angle 3\) and \(\angle 6\) are alternate interior angles:
\[a \parallel b \ \Rightarrow\ m\angle 3 = m\angle 6\]
Here \(m\angle 3\) means the measure of angle 3 in degrees.
Key facts
- There are two pairs of alternate interior angles for every transversal and two lines.
- Look for a Z or backwards Z: its two corner angles are the pair.
- Equal angles in a Z pattern prove the two lines are parallel.
- These angles are not supplementary unless both are right angles.
Alternate Interior Angles example
Two parallel bike lanes are crossed by a service road. One of the Z-pair angles measures \((4x + 9)^{\circ}\) and the other measures \((6x – 25)^{\circ}\). Find \(x\) and the angle measure.
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Parallel lines make the pair equal: \(4x + 9 = 6x – 25\).
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Subtract \(4x\) and add 25: \(34 = 2x\), so \(x = 17\).
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Angle: \(4(17) + 9 = 77\). Check: \(6(17) – 25 = 77\).
\(x = 17\) and each angle measures \(77^{\circ}\).
Common mistake
Watch out
Dana sets \((4x + 9) + (6x – 25) = 180\) because she thinks angles on a transversal always add to 180, and gets \(x = 19.6\).
Why it is wrong
Alternate interior angles on parallel lines are equal, not supplementary. The sum rule of 180 belongs to angles that share a side on a straight line, or to same-side interior angles. Equal angles give \(x = 17\) and a check that both expressions give \(77^{\circ}\).
Related terms and lessons
- Grade 8 · Angles, Parallel Lines and TrianglesLessonWorksheetQuiz
- Geometry · Parallel Lines, Transversals and ProofsLessonWorksheetQuiz
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement