Corresponding angles
A line that passes through two distinct points on two lines in the same plane is called a transversal. A transversal forms four pairs of corresponding angles. Corresponding angles are pairs of angles that lie on the same side of the transversal in matching corners. One of the angles in the pair is an exterior angle and one is an interior angle.
In the diagram below transversal l intersects lines m and n. ∠1 and ∠5 are a pair of corresponding angles. Similarly, ∠2 and ∠6, ∠3 and 7, and ∠4 and ∠8 are also corresponding angles.
Corresponding angles postulate
The corresponding angles postulate states that if two parallel lines are cut by a transversal, the corresponding angles are congruent.
Parallel lines m and n are cut by transversal l above, forming four pairs of congruent, corresponding angles: ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ 7, and ∠4 ≅ ∠8.
The converse of the postulate is also true. If the corresponding angles of two lines cut by a transversal are congruent, then the lines are parallel.
Example:
Using corresponding angles and straight angles, find the measures of the angles formed by the intersection of parallel lines m and n cut by transversal l below.
- ∠2 ≅ ∠60° since they are corresponding angles, and m and n are parallel.
- ∠1 and ∠2 form a straight angle, so∠1=120°.
- ∠5 ≅ ∠120° since ∠1 and ∠5 are corresponding angles, and m and n are parallel.
- ∠7 and ∠5 form a straight angle, so∠7=60°.
- ∠3 ≅ ∠60° since ∠3 and ∠7 are corresponding angles, and m and n are parallel.
- ∠3 and ∠4 form a straight angle, so∠4=120°.
- ∠8 ≅ ∠120° since ∠4 and ∠8 are corresponding angles, and m and n are parallel.