Angles in the Plane

When two co-planar lines are intersected by a transversal (a third line that cuts through them), four pairs of angles are formed. In the image below, a transversal l intersects line PQ and XY, forming angles 1 through 8.

Corresponding angles are angles in similar locations with respect to the transversal and each of the two original lines. In the image above, angles 1 and 5, 2 and 6, 3 and 7, and 4 and 8 are corresponding angle pairs.

Two co-planar lines are parallel if and only if they have the same tilt, make the same angle with a transversal and do not intersect each other. In the image below, lines w and z are parallel while l is the transversal. 

There's more...

Alternate interior angles are found on opposite sides of a transversal and in between the two original lines.  When the two lines are parallel, these pairs of angles are congruent (and vice versa).

m∠3 = m∠6

m∠4 = m∠5

Consecutive interior angles are found on the same side of a transversal and in between the two original lines.  When the two lines are parallel, these pairs of angles are supplementary (and vice versa).

m∠3 + m∠5 = 180°

m∠4 + m∠6 = 180°

Alternate exterior angles are found on opposite sides of a transversal, but on the outside of the two original lines.  When the two lines are parallel, just like the with alternate interiors, these pairs of angles are congruent.

m∠1 = m∠8 

m∠2 = m∠7

Perpendicular Transversal Theorem

When a transversal line is perpendicular to one of two parallel lines, then the line is perpendicular to the other.

We're given that l ⊥ XY, so if lines PQ and XY are parallel,

then we also know that l ⊥ PQ.

Practice

In the picture below, PQ || XY.

1.)  Which of the following lists a pair of corresponding angles?

A.  ∠1 and ∠4
B.  ∠3 and ∠6
C.  ∠2 and ∠7
D.  ∠4 and ∠8

2.)  If m∠1 = 62°, find m∠8.

3.)  If m∠3 = 115°, find m∠5.

4.)  If m∠4 = 90°, find m∠7.

5.)  If m∠3 = 5x − 12 and m∠6 = 3(x + 12), find x.

Answers

1.)  D

2.)  62°

3.)  65°

4.)  90°

5.)  24