Transversals and the Angle Pairs They Form | Geometry | LevelKite

When one line crosses two other lines at two different points, it forms eight angles, four at each crossing. You already know names for pairs of angles at one crossing: adjacent angles, vertical angles and linear pairs. In this lesson, you learn to identify the crossing line and to name pairs of angles that sit at different crossings.

Lines are coplanar when they lie in the same plane. A transversal is a line that intersects two or more coplanar lines at different points. To test whether a line is a transversal of two lines, make two checks:

The two lines are coplanar.
The line meets them at two different points.

The definition never mentions parallel lines, so a transversal can cross lines that are parallel or lines that are not.

Here is a figure to work with, along with the facts you can use:

Lines $AB$ , $CD$ and $EF$ lie in one plane.
Line $EF$ meets line $AB$ at $G$ and meets line $CD$ at a different point, $H$ . Call $G$ and $H$ the crossing points.
Along line $EF$ , the points come in the order $E, G, H, F$ . Point $G$ lies between $A$ and $B$ , and point $H$ lies between $C$ and $D$ .
Points $A$ and $C$ lie on one side of line $EF$ , and points $B$ and $D$ lie on the other side.

No arrows appear on lines $AB$ and $CD$ , so you are not told that they are parallel.

This figure shows the facts above: line

EF

crosses line

AB

at

G

and line

CD

at

H

. It has no arrows, so it gives no parallel fact.

Worked example

Is the line a transversal?

(a) Using the facts above, is line $EF$ a transversal of lines $AB$ and $CD$ ? (b) In a different case, three lines in one plane all pass through a single point $P$ . Is any one of them a transversal of the other two?

Part (a), check 1. The definition of a transversal needs coplanar lines. You are told that all three lines lie in one plane, so lines AB and CD are coplanar.
Part (a), check 2. The definition also needs different intersection points. Line EF meets line AB at G and line CD at H, and you are told that G and H are different points. Both checks hold, so the definition is met.
$G \neq H$
Part (b). The lines are coplanar, so check 1 holds. But any one of the lines meets the other two at the same point, P, not at different points. Check 2 fails, so the definition is not met.

Answer

Line $EF$ is a transversal of lines $AB$ and $CD$ . When three lines all pass through one point, none of them is a transversal of the other two.

An angle's arms are the two rays that form it. Line $EF$ forms eight angles with lines $AB$ and $CD$ , and each one has one arm on the transversal and one arm on the other line:

At $G$ : angles $AGE$ , $BGE$ , $AGH$ and $BGH$ .
At $H$ : angles $CHG$ , $DHG$ , $CHF$ and $DHF$ .

At one crossing point, the names you already know still apply. Angles $AGE$ and $BGH$ are vertical angles, and angles $AGE$ and $BGE$ form a linear pair. The new names compare one angle at $G$ with one angle at $H$ , so they never describe two angles at the same crossing point.

To name a pair, ask two questions about position.

Question 1: interior or exterior? Interior suggests "between the two lines," and exterior suggests "outside them." To decide exactly, look at the angle's arm on the transversal. If that arm points toward the other crossing point, the angle is interior. If it points away, the angle is exterior. Arm $GH$ of angle $AGH$ points toward $H$ , so angle $AGH$ is interior. Arm $GE$ of angle $AGE$ points away from $H$ , so angle $AGE$ is exterior. A pair is an interior pair when both angles are interior, and an exterior pair when both are exterior.

Question 2: same side or opposite sides? Look at each angle's other arm. Angle $AGH$ has arm $GA$ , and angle $DHG$ has arm $HD$ . You are told that $A$ and $D$ lie on opposite sides of line $EF$ , so these two angles lie on opposite sides of the transversal.

Both questions use only the stated order of the points and the stated sides of the line. Neither one uses how large an angle looks.

Key concept

Position alone gives the name

Take one angle at each of the two points where a transversal crosses two coplanar lines. Decide whether each angle is interior or exterior, and whether the two angles lie on the same side or on opposite sides of the transversal. Those answers give the pair its name, if it has one. The name describes position only. It holds whether or not the two lines are parallel, and by itself it never tells you that two angles are equal. A figure is not drawn to scale, so take measures and parallel or perpendicular lines only from marks and stated facts.

Those two answers decide the name, and the words hint at the meaning. Alternate angles take opposite sides of the transversal, one on each side. Consecutive angles follow one another along the same side; some books say same-side instead. Corresponding angles sit in matching positions at the two crossing points. The table gives the exact definitions, with one example pair from lines $AB$ , $CD$ and $EF$ in each row. One combination has no special name.

Definitions for one angle at G and one angle at H, with line EF as the transversal

Name	Interior or exterior?	Side of the transversal	Example pair
Corresponding	One interior, one exterior	Same side	$\angle AGH$ and $\angle CHF$
Alternate interior	Both interior	Opposite sides	$\angle BGH$ and $\angle CHG$
Alternate exterior	Both exterior	Opposite sides	$\angle BGE$ and $\angle CHF$
Consecutive interior	Both interior	Same side	$\angle AGH$ and $\angle CHG$
Consecutive exterior	Both exterior	Same side	$\angle AGE$ and $\angle CHF$
No special name	One interior, one exterior	Opposite sides	$\angle BGH$ and $\angle CHF$

These are the same lines and points. The text gives the two marked angles: angle

BGE

measures

115^\circ

and angle

DHG

measures

105^\circ

. No arrows appear.

Worked example

Classify one pair

The figure above uses the same lines, points and given facts as before. You are also told that angle $BGE$ measures $115^\circ$ and angle $DHG$ measures $105^\circ$ . Classify this pair of angles.

Find the transversal. Each of these angles has one arm on the transversal, so look for the line that holds an arm of both: line EF holds arm GE and arm HG. The first example showed, by the definition of a transversal, that line EF is a transversal of lines AB and CD, and G and H are its crossing points.
Answer question 1. The given order is E, G, H, F, so arm GE points away from H, and angle BGE is exterior by the definition of an exterior angle. Arm HG points toward G, so angle DHG is interior by the definition of an interior angle.
$\angle BGE:\ \text{exterior} \qquad \angle DHG:\ \text{interior}$
Answer question 2. The other arms are GB and HD. You are told that B and D lie on the same side of line EF, so the two angles lie on the same side of the transversal.
Match the definition. One exterior angle and one interior angle on the same side of the transversal are corresponding angles by definition. You never needed the measures 115° and 105°, because the definition does not use them.
$\text{one exterior, one interior, same side} \Rightarrow \text{corresponding}$

Answer

Angles $BGE$ and $DHG$ are corresponding angles, with line $EF$ as the transversal.

Worked example

Five more pairs

Use the same lines, points and given facts. Classify each pair of angles: (a) $AGH$ and $DHG$ ; (b) $AGE$ and $DHF$ ; (c) $BGH$ and $DHG$ ; (d) $BGE$ and $DHF$ ; (e) $AGE$ and $DHG$ .

Pair (a). Arms GH and HG each point toward the other crossing point, so both angles are interior. A and D lie on opposite sides of line EF. By definition, these are alternate interior angles.
$\angle AGH,\ \angle DHG:\ \text{alternate interior}$
Pair (b). Arms GE and HF each point away from the other crossing point, so both angles are exterior. A and D lie on opposite sides of line EF. By definition, these are alternate exterior angles.
$\angle AGE,\ \angle DHF:\ \text{alternate exterior}$
Pair (c). Arms GH and HG point toward the other crossing point, so both angles are interior. B and D lie on the same side of line EF. By definition, these are consecutive interior angles.
$\angle BGH,\ \angle DHG:\ \text{consecutive interior}$
Pair (d). Arms GE and HF point away from the other crossing point, so both angles are exterior. B and D lie on the same side of line EF. By definition, these are consecutive exterior angles.
$\angle BGE,\ \angle DHF:\ \text{consecutive exterior}$
Pair (e). Arm GE points away from H, so angle AGE is exterior. Arm HG points toward G, so angle DHG is interior. A and D lie on opposite sides of line EF. The table lists this combination as having no special name.
$\angle AGE,\ \angle DHG:\ \text{no special name}$

Answer

Pair (a) is alternate interior, pair (b) is alternate exterior, pair (c) is consecutive interior, pair (d) is consecutive exterior, and pair (e) has no special name.

The arrows give that line

AB

is parallel to line

CD

. Lines

EG

and

FH

carry no arrows. The three marked angles are the ones named in the example below.

Worked example

Find the transversal first

Lines $AB$ , $CD$ , $EG$ and $FH$ lie in one plane. The arrows give that lines $AB$ and $CD$ are parallel. Along line $AB$ , the order is $A, E, F, B$ . Points $A$ and $C$ lie on the same side of line $EG$ , and points $G$ and $H$ lie on the same side of line $AB$ . For each pair, find the transversal and classify the pair: (a) angles $AEG$ and $CGE$ ; (b) angles $AEG$ and $EFH$ .

Pair (a): find the transversal. Angle AEG has arms on lines AB and EG. Angle CGE has arms on lines CD and EG. Only line EG holds an arm of each angle. It crosses lines AB and CD at the different points E and G, so it is their transversal by the definition of a transversal.
Pair (a): classify. Arm EG points toward G and arm GE points toward E, so both angles are interior by the definition of an interior angle. The other arms, EA and GC, lie on the same side of line EG, because A and C are given on the same side. By definition, these are consecutive interior angles.
$\angle AEG,\ \angle CGE:\ \text{consecutive interior}$
Pair (b): find the transversal. Angle AEG has arms on lines AB and EG. Angle EFH has arms on lines AB and FH. Only line AB holds an arm of each angle. It crosses lines EG and FH at the different points E and F, so this time line AB is the transversal.
Pair (b): classify. The given order A, E, F, B shows that arm EA points away from F, so angle AEG is exterior. Arm FE points toward E, so angle EFH is interior. The other arms, EG and FH, lie on the same side of line AB, because G and H are given on the same side. By definition, these are corresponding angles.
$\angle AEG,\ \angle EFH:\ \text{corresponding}$
Check the arrows. Neither classification used them. The arrows state a fact about lines AB and CD, but they do not decide which line is the transversal. In pair (b), the crossed lines EG and FH are not given as parallel, and the name still holds.

Answer

In pair (a), angles $AEG$ and $CGE$ are consecutive interior angles, with line $EG$ as the transversal. In pair (b), angles $AEG$ and $EFH$ are corresponding angles, with line $AB$ as the transversal.

Takeaway: A transversal is a line that crosses coplanar lines at different points, and the name of an angle pair it forms comes from position alone. Find the line that holds one arm of each angle, ask interior or exterior, and then ask same side or opposite sides. A figure is not drawn to scale, so any measure, equal angles or parallel lines must come from marks or stated facts, never from the look of the drawing.