Transversal Angle Pairs and Missing Measures | Grade 8 Math | LevelKite

Two lines are parallel when they never meet, however far they extend. In every figure in this lesson, line $AB$ and line $CD$ carry matching arrow marks, which state that they are parallel. A transversal is a line that crosses two or more lines at different points. Here the transversal is line $EF$ . It crosses line $AB$ at point $G$ and line $CD$ at point $H$ , and each crossing makes four angles.

You can place any of these angles with two checks: where it sits relative to the parallel lines, and which side of the transversal it is on. The region between the parallel lines is the interior, and the two regions outside them are the exterior. The transversal splits the figure into two sides. The four pairs you will name each take one angle at $G$ and one at $H$ .

The figures are not drawn to scale. Use a figure to see where each angle sits, but take every angle measure and every parallel fact from a mark or a value stated in the text, never from how the drawing looks.

Arrow marks state that line

AB

is parallel to line

CD

, and line

EF

is the transversal. The marked angles are

\angle EGB

at

G

and

\angle GHD

at

H

.

Look at the two marked angles, $\angle EGB$ at $G$ and $\angle GHD$ at $H$ , and apply both checks. $\angle EGB$ is outside the parallel lines and $\angle GHD$ is between them, but each lies above its own parallel line. Both lie on the same side of the transversal, the side with points $B$ and $D$ . So the two angles hold the same position at their two crossings. Two angles in matching positions like this are corresponding angles. The table below applies the same two checks to all four pair names.

Four pair names, each with one angle at G and one at H

Pair name	Relative to the parallel lines	Relative to the transversal
Corresponding	One between and one outside: both above, or both below, their own line	Same side
Alternate interior	Both between the parallel lines	Opposite sides
Alternate exterior	Both outside the parallel lines	Opposite sides
Same-side interior	Both between the parallel lines	Same side

The marked angles are

\angle BGH

at

G

and

\angle CHG

at

H

, one at each crossing of the transversal.

Worked example

Name a pair by its position

In the figure above, the marked angles are $\angle BGH$ and $\angle CHG$ . What kind of angle pair are they?

First check, the parallel lines: ∠BGH lies below line AB, toward line CD, and ∠CHG lies above line CD, toward line AB. So both angles are between the parallel lines, which makes both of them interior angles.
Second check, the transversal: ∠BGH is on the side with point B, and ∠CHG is on the side with point C. Those are opposite sides of the transversal.
Match both results to the table. Between the parallel lines and on opposite sides of the transversal is the description of alternate interior angles.

Answer

$\angle BGH$ and $\angle CHG$ are alternate interior angles.

Now for measures. Write $m\angle EGB$ for the measure of $\angle EGB$ .

Imagine sliding the crossing at $G$ along the transversal until $G$ lands on $H$ . This slide is a translation, so it keeps every angle measure the same. It moves line $AB$ to a line through $H$ that is parallel to line $AB$ , and line $CD$ , marked parallel and passing through $H$ , is the only such line. So the crossing at $G$ lands exactly on the crossing at $H$ , and each angle lands on its corresponding angle. For example, $\angle EGB$ lands on $\angle GHD$ , so $m\angle EGB = m\angle GHD$ .

Two facts you already know extend this: vertical angles (the opposite angles at one crossing) have equal measures, and a linear pair (two angles that share a side and whose other sides form a straight line) has measures that add to $180^\circ$ . For example, $\angle BGH$ and $\angle AGE$ are vertical angles, and $\angle AGE$ and $\angle CHG$ are corresponding angles, so the alternate interior angles $\angle BGH$ and $\angle CHG$ have equal measures too.

Key concept

Angle pair facts for parallel lines

When a transversal crosses two lines marked parallel:

corresponding angles are congruent, meaning equal in measure;
alternate interior angles are congruent;
alternate exterior angles are congruent;
same-side interior angles are supplementary, meaning their measures add to $180^\circ$ .

Each fact depends on the parallel marks. Without them, you cannot use these facts to find measures.

The marked given angle is

\angle AGE

; the text states that it measures

64^\circ

. The other marked angle,

\angle DHF

, is the unknown.

Worked example

Use one pair fact

Line $AB$ is parallel to line $CD$ , as the arrow marks state. The marked given angle in the figure, $\angle AGE$ , measures $64^\circ$ . Find $m\angle DHF$ .

Name the pair first. ∠AGE lies above line AB and ∠DHF lies below line CD, so both are outside the parallel lines. They are on opposite sides of the transversal, so they are alternate exterior angles.
The arrow marks state that line AB is parallel to line CD, so alternate exterior angles are congruent. Use the stated value, 64°, not the look of the drawing.
$m\angle DHF = m\angle AGE = 64^\circ$

Answer

$\angle DHF$ measures $64^\circ$ .

The marked given angle is

\angle BGH

; the text states that it measures

72^\circ

. The other marked angle,

\angle GHD

, is the unknown.

Worked example

Chain a pair fact and a linear pair

Line $AB$ is parallel to line $CD$ , as the arrow marks state. The marked given angle in the figure, $\angle BGH$ , measures $72^\circ$ . Find $m\angle GHD$ .

Link the given angle to crossing H. ∠CHG, at H between rays HC and HG, is the alternate interior partner of ∠BGH that you named earlier. The lines are marked parallel, so these two angles are congruent.
$m\angle CHG = m\angle BGH = 72^\circ$
∠CHG and ∠GHD share ray HG, and their other sides, rays HC and HD, form line CD. So they are a linear pair, and their measures add to 180°.
$m\angle CHG + m\angle GHD = 180^\circ$
Replace m∠CHG with 72°, its value from the first step. Equal values can replace each other in an equation.
$72^\circ + m\angle GHD = 180^\circ$
Subtract 72° from both sides. Subtracting the same amount from both sides keeps the two sides equal.
$m\angle GHD = 180^\circ - 72^\circ = 108^\circ$

Answer

$\angle GHD$ measures $108^\circ$ . The same-side interior angles $\angle BGH$ and $\angle GHD$ together measure $72^\circ + 108^\circ = 180^\circ$ .

Watch out: $\angle BGH$ and $\angle GHD$ both lie between the parallel lines, on the same side of the transversal, so it is tempting to treat them like a congruent pair and write $m\angle GHD = 72^\circ$ . That fails. Because the lines are marked parallel, $\angle CHG$ , the alternate interior partner of $\angle BGH$ , also measures $72^\circ$ . $\angle CHG$ and $\angle GHD$ form a linear pair on line $CD$ , so their measures must total $180^\circ$ . A value of $72^\circ$ would give $72^\circ + 72^\circ = 144^\circ$ , not $180^\circ$ . When the lines are parallel, same-side interior angles are supplementary, so they are congruent only in the special case where each measures $90^\circ$ .

Takeaway: Two checks name each of the four angle pairs: where each angle sits relative to the parallel lines, and whether the two angles are on the same side or on opposite sides of the transversal. When the lines are marked parallel, the name tells you whether the two angles are equal or add to $180^\circ$ , and a chain of stated facts carries a given angle to an unknown angle at either crossing. Next, you will use these relationships to find missing angles in triangles, meet the exterior angle theorem and solve equations built from angle relationships.