Math Core

When a line cuts across two other lines, it creates eight angles. If the two lines are parallel, those eight angles are tied together so tightly that knowing a single one tells you all the rest. In this lesson you'll name the angle pairs precisely, see which relationship is accepted as a postulate, and prove the others from it.

Parallel, perpendicular and skew

Two lines are parallel if they lie in the same plane and never intersect. You write $m \parallel n$ . Two lines are perpendicular if they intersect to form right angles, written $m \perp n$ . Lines that are not in the same plane, such as the edge where the front wall of a classroom meets the ceiling and the edge where a side wall meets the floor, never meet either, but they are not parallel. They are called skew lines.

Euclid's famous Parallel Postulate says that through a point not on a line, there is exactly one line parallel to the given line. Everything in this unit rests on that fact.

Transversals and the eight angles

Definition

Transversal

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

In the diagram, transversal $t$ crosses lines $m$ and $n$ . At each intersection, four angles are formed, numbered $1$ through $4$ at line $m$ and $5$ through $8$ at line $n$ , in the same order: upper left, upper right, lower left, lower right.

Lines m and n cut by transversal t.

The strip between $m$ and $n$ is the interior; the two regions outside the strip make up the exterior. Each type of angle pair is named by two facts: interior or exterior, and same side or opposite sides of the transversal.

angle pair	location	pairs in the diagram
corresponding	same position at each intersection	$\angle 1, \angle 5$ ; $\angle 2, \angle 6$ ; $\angle 3, \angle 7$ ; $\angle 4, \angle 8$
alternate interior	interior, opposite sides of $t$	$\angle 3, \angle 6$ ; $\angle 4, \angle 5$
alternate exterior	exterior, opposite sides of $t$	$\angle 1, \angle 8$ ; $\angle 2, \angle 7$
consecutive (same-side) interior	interior, same side of $t$	$\angle 3, \angle 5$ ; $\angle 4, \angle 6$
same-side exterior	exterior, same side of $t$	$\angle 1, \angle 7$ ; $\angle 2, \angle 8$

These names describe position only. Any two lines cut by a transversal have corresponding angles, alternate interior angles and so on, whether or not the lines are parallel. The measures are only guaranteed to be related when the lines are parallel.

What happens when the lines are parallel

Geometry has to start somewhere, so one of the relationships is accepted without proof.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

From this postulate and the angle facts you already proved (vertical angles are congruent; a linear pair is supplementary), the other relationships follow. Here is the proof for alternate interior angles.

Given: $m \parallel n$ . Prove: $\angle 3 \cong \angle 6$ .

statement	reason
1. $m \parallel n$	Given
2. $\angle 3 \cong \angle 2$	Vertical Angles Congruence Theorem
3. $\angle 2 \cong \angle 6$	Corresponding Angles Postulate
4. $\angle 3 \cong \angle 6$	Transitive Property of Congruence

The proof for consecutive interior angles is just as short. Since $\angle 3 \cong \angle 7$ (corresponding) and $\angle 7$ and $\angle 5$ form a linear pair, $m\angle 5 + m\angle 7 = 180^\circ$ . Substituting $m\angle 3$ for $m\angle 7$ gives $m\angle 3 + m\angle 5 = 180^\circ$ .

Parallel lines cut by a transversal

If two parallel lines are cut by a transversal, then:

corresponding angles are congruent (postulate),
alternate interior angles are congruent,
alternate exterior angles are congruent,
consecutive interior angles are supplementary,
same-side exterior angles are supplementary.

Perpendicular Transversal Theorem: in a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

Because of these facts, a transversal crossing two parallel lines makes angles of only two sizes. In the diagram, $\angle 2, \angle 3, \angle 6, \angle 7$ are the acute angles, and $\angle 1, \angle 4, \angle 5, \angle 8$ are the obtuse ones. Any acute angle plus any obtuse angle is $180^\circ$ . (If the transversal is perpendicular, all eight angles are $90^\circ$ .)

Worked example: Naming angle pairs

Using the diagram, classify each pair: (a) $\angle 1$ and $\angle 8$ ; (b) $\angle 4$ and $\angle 6$ ; (c) $\angle 2$ and $\angle 6$ .

(a) Both are outside the strip, on opposite sides of $t$ : alternate exterior.

(b) Both are inside the strip, on the right side of $t$ : consecutive interior.

Worked example: Finding all eight angles

In the diagram, $m \parallel n$ and $m\angle 6 = 58^\circ$ . Find the other seven measures.

$\angle 6$ is acute, so every acute angle in the picture is $58^\circ$ : $m\angle 2 = m\angle 3 = m\angle 7 = 58^\circ$ . (For example, $\angle 2$ and $\angle 6$ are corresponding, and $\angle 3$ and $\angle 6$ are alternate interior.)

Every obtuse angle is $180^\circ - 58^\circ = 122^\circ$ : $m\angle 1 = m\angle 4 = m\angle 5 = m\angle 8 = 122^\circ$ . (For example, $\angle 5$ and $\angle 6$ form a linear pair.)

Worked example: Two variables

In the diagram, $m \parallel n$ , $m\angle 4 = (4x + 14)^\circ$ , $m\angle 5 = (6x - 34)^\circ$ and $m\angle 7 = (2y - 10)^\circ$ . Find $x$ and $y$ .

$\angle 4$ and $\angle 5$ are alternate interior angles, so they are congruent:

\begin{aligned} 4x + 14 &= 6x - 34 \\ 48 &= 2x \\ x &= 24 \end{aligned}

So $m\angle 5 = 6(24) - 34 = 110^\circ$ . Next, $\angle 5$ and $\angle 7$ form a linear pair, so $m\angle 7 = 180^\circ - 110^\circ = 70^\circ$ :

2y - 10 = 70 \quad\Longrightarrow\quad 2y = 80 \quad\Longrightarrow\quad y = 40.

Adding an auxiliary line

Some problems have a "bent" path between two parallel lines, with no transversal drawn. The trick is to add your own line: through the corner point, draw a line parallel to the other two. The Parallel Postulate guarantees that exactly one such line exists.

Worked example: An angle between two parallel lines

Lines $a$ and $b$ are parallel. Point $P$ lies between them. Segment $AP$ makes a $25^\circ$ angle with line $a$ , and segment $BP$ makes a $40^\circ$ angle with line $b$ , as shown. Find $m\angle APB$ .

Parallel lines a and b, with auxiliary line c drawn through P parallel to both.

Draw line $c$ through $P$ parallel to $a$ (and therefore parallel to $b$ ). This splits $\angle APB$ into two pieces.

$AP$ is a transversal of $a$ and $c$ . The $25^\circ$ angle at $A$ and the upper piece of $\angle APB$ are alternate interior angles, so the upper piece is $25^\circ$ .
$BP$ is a transversal of $b$ and $c$ . By the same reasoning, the lower piece is $40^\circ$ .

So $m\angle APB = 25^\circ + 40^\circ = 65^\circ$ .

Common mistake

Consecutive interior angles are supplementary, not congruent. Before writing an equation, check the diagram: if one angle is acute and the other obtuse, the measures add to $180^\circ$ ; if both are the same size, set them equal. And none of these relationships holds unless the lines are known to be parallel.

Tip

Check your answer by sorting angles into "small" and "large." Every small angle should have the same measure, every large angle should have the same measure, and small plus large should be $180^\circ$ .

Practice

Unless a problem says otherwise, it refers to the diagram in the section "Transversals and the eight angles," with $m \parallel n$ .

Practice 1

What kind of angle pair are $\angle 3$ and $\angle 6$ ?

Practice 2

Which pair of angles are same-side exterior angles?

Practice 3

If $m\angle 8 = 115^\circ$ , what is $m\angle 1$ , in degrees?