Classrooms Curricula References Textbooks Site Map

Transversals

A transversal is a line that intersects two or more coplanar lines in two or more different points.

 Not a transversalThe two lines that m intersects aren't coplanar Not a transversalNone of the lines intersects the other two at two different points. This is a transversal

Note that the definition doesn't say anything about the two intersected lines except that they have to be in the same plane. In particular, they don't, necessarily, have to be parallel.

When a transversal crosses two lines the transversal makes eight angles. These angles are grouped into pairs and classified. Concise definitions of these angles are usually pretty wordy so most books usually just give examples.

 Angles 1 and 2 are corresponding angles. Angles 3 and 4 are alternate interior angles. Angles 5 and 6 are alternate exterior angles.

The "alternate" part of the names refers to the angles being on alternate sides of the transversal.

Finally there are special relationships between these angles if the lines are parallel.

 The Parallel Postulate Through a point on a line there is exactly one line parallel to the given line. Postulate 1 If two parallel lines are crossed by a transversal then alternate interior angles are congruent. Theorem 1 If two parallel lines are crossed by a transversal then corresponding angles are congruent. Theorem 2 If two parallel lines are crossed by a transversal then interior angles on the same side of the transversal are supplementary.

Example 1 - Proving Lines Are Parallel

If l || m then what are m∠GEF and m∠BEF?

GEF and ∠GBC are corresponding angles. The corresponding angles formed by two parallel lines are congruent so m∠GEF = m∠GBC = 81°.

BEF and ∠GBC are interior angles on the same side of transveral. When those lines are formed by two parallel lines, they have to be supplementary.

m∠BEF + m∠GBC = 180°
m∠BEF + 81° = 180°
m∠BEF = 99°

Example 2 - Proving Lines Are Parallel

If l || m then what is m∠ABC?

The two marked angles in the diagram are alternate interior angles and, according to Postulate 1, they have to be congruent.

5x = 4x + 42
x = 42

Which means that ABC = 5 · x = 5 · 42 = 210°.