Angle Measures with Parallel Lines | Geometry | LevelKite

A transversal is a line that crosses two other lines at two different points. When those two lines are parallel, meaning they lie in the same plane and never meet, the angles at the two crossings follow a pattern. You already know how to name these angle pairs. In this lesson, you will use those names to find angle measures.

Here is the idea behind the pattern. Parallel lines run in the same direction, so a transversal crosses each one at the same tilt. That is why a measure at one crossing can tell you a measure at the other.

All of this depends on one condition: the lines must be given parallel. A given is a fact you are told, by a mark in the figure (such as arrows on two lines) or by a sentence in the text. Figures in this lesson are not drawn to scale, so you never read an angle measure, a length, or a parallel or perpendicular fact from how a drawing looks.

Start with corresponding angles. The Corresponding Angles Postulate says: if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. Congruent angles have equal measures. A postulate is a statement accepted without proof.

Notice the direction. The postulate starts with parallel lines and ends with congruent angles. That forward direction is the one you use in this lesson. The Converse of the Corresponding Angles Postulate runs the other way: it starts with corresponding angles given congruent and concludes that the lines are parallel.

The symbol $m\angle GEB$ means the measure of angle $GEB$ .

The arrows give line

AB

parallel to line

CD

, and line

GH

is the transversal. Angle G E B measures 65°; that value is stated here, not read from the drawing. Angle E F D is the angle to find.

Worked example

One known corresponding angle

In the figure above, the arrows give line $AB$ parallel to line $CD$ , and you are told that $m\angle GEB = 65^\circ$ . Find $m\angle EFD$ .

Check the given first. The arrows state that the lines are parallel, so you may use the Corresponding Angles Postulate in its forward direction, from parallel lines to congruent angles.
Name the pair. Angles GEB and EFD are corresponding angles because each lies above its own line and to the right of the transversal. The postulate says they are congruent.
$\angle GEB \cong \angle EFD$
Congruent angles have equal measures, so replace the measure of angle GEB with its given value.
$m\angle EFD = m\angle GEB = 65^\circ$
Check the result. An angle measure must be greater than 0° and at most 180°, and 65° meets both conditions.
$0^\circ < 65^\circ \le 180^\circ$

Answer

By the Corresponding Angles Postulate, $m\angle EFD = 65^\circ$ .

Worked example

Corresponding angles with algebra

Use the same figure, with line $AB$ still given parallel to line $CD$ and $m\angle GEB$ still $65^\circ$ . You are also told that $m\angle GEA = (3x + 25)^\circ$ and $m\angle EFC = (5x - 35)^\circ$ . Find $x$ and both measures.

Angles GEA and EFC are corresponding angles because each lies above its own line and to the left of the transversal. The lines are given parallel, so the Corresponding Angles Postulate, used in its forward direction, makes their measures equal.
$3x + 25 = 5x - 35$
Subtract 3x from both sides to collect the x terms on one side.
$25 = 2x - 35$
Add 35 to both sides to isolate the x term.
$60 = 2x$
Divide both sides by 2.
$x = 30$
Substitute x = 30 into each expression to get each measure.
$3(30) + 25 = 115, \quad 5(30) - 35 = 115$
Check. The measures match, and 115° is greater than 0° and at most 180°. Also, angles GEA and GEB form a linear pair: they share a ray, and their other rays form a line. The Supplement Theorem says a linear pair is supplementary, meaning the measures add to 180°, and they do.
$115^\circ + 65^\circ = 180^\circ$

Answer

The value of $x$ is $30$ , and $m\angle GEA = m\angle EFC = 115^\circ$ .

Now see why the given matters. The figure below is the same drawing with one change: the arrows are gone. Angle G E B still measures $65^\circ$ , but no mark and no sentence says that line $AB$ is parallel to line $CD$ .

So you cannot use the Corresponding Angles Postulate, and you cannot find $m\angle EFD$ . The lines may look parallel, but looks are not a given. Line $CD$ could be tilted a little, which would change $m\angle EFD$ while $m\angle GEB$ stays $65^\circ$ .

Same drawing, no arrows: lines

AB

and

CD

are not given parallel. Angle G E B measures 65°, but you cannot find angle E F D from it.

Go back to the first figure, where the lines are given parallel, $m\angle GEB = 65^\circ$ and $m\angle EFD = 65^\circ$ . More pairs follow a pattern. Vertical angles are the opposite angles at a crossing, and the Vertical Angles Theorem says they are congruent. The Supplement Theorem says a linear pair is supplementary.

Alternate interior: $\angle AEF$ is vertical to $\angle GEB$ , so it measures $65^\circ$ , the same as $\angle EFD$ .
Alternate exterior: $\angle CFH$ is vertical to $\angle EFD$ , so it measures $65^\circ$ , the same as $\angle GEB$ .
Consecutive interior: $\angle BEF$ forms a linear pair with $\angle GEB$ , so it measures $115^\circ$ . Then $\angle BEF$ and $\angle EFD$ add to $115^\circ + 65^\circ = 180^\circ$ .

This one case shows the pattern but does not prove it. Three theorems state the pattern for any two parallel lines cut by a transversal, and the table below names them. A theorem is a statement that can be proved.

Use each row in its forward direction, and only when the two lines are given parallel.

Angle pair	Example in the first figure	Reason	What it lets you write
Corresponding	$\angle GEB$ and $\angle EFD$	Corresponding Angles Postulate	Congruent: $m\angle GEB = m\angle EFD$
Alternate interior	$\angle AEF$ and $\angle EFD$	Alternate Interior Angles Theorem	Congruent: $m\angle AEF = m\angle EFD$
Alternate exterior	$\angle GEB$ and $\angle CFH$	Alternate Exterior Angles Theorem	Congruent: $m\angle GEB = m\angle CFH$
Consecutive interior	$\angle BEF$ and $\angle EFD$	Consecutive Interior Angles Theorem	Supplementary: $m\angle BEF + m\angle EFD = 180^\circ$

Key concept

What given parallel lines tell you

When two lines cut by a transversal are given parallel, corresponding, alternate interior and alternate exterior angles are congruent, and consecutive interior angles are supplementary. So one known measure can lead you to the others. This runs in one direction: parallel lines come first, and the angle fact follows. Going from an angle fact to parallel lines takes a converse, such as the Alternate Interior Angles Converse. If the lines are not given parallel, you cannot use any of these facts.

The arrows give line

AB

parallel to line

CD

; parallel lines can slant. Line

GH

is the transversal. This figure supplies labels only, and each example below states its own measures.

Worked example

Alternate interior, alternate exterior and consecutive interior angles

In the figure above, the arrows give line $AB$ parallel to line $CD$ , and you are told that $m\angle BEF = 58^\circ$ . Find $m\angle EFC$ , $m\angle DFH$ and $m\angle EFD$ .

The lines are given parallel, so you can use each theorem in its forward direction. Angles BEF and EFC are alternate interior angles, so the Alternate Interior Angles Theorem makes them congruent.
$m\angle EFC = m\angle BEF = 58^\circ$
To reach an alternate exterior pair, first find an exterior angle at E. Angle GEA is vertical to angle BEF, so the Vertical Angles Theorem makes them congruent.
$m\angle GEA = m\angle BEF = 58^\circ$
Angles GEA and DFH are alternate exterior angles, so the Alternate Exterior Angles Theorem makes them congruent.
$m\angle DFH = m\angle GEA = 58^\circ$
Angles BEF and EFD are consecutive interior angles, so the Consecutive Interior Angles Theorem makes them supplementary.
$58^\circ + m\angle EFD = 180^\circ$
Subtract 58° from both sides.
$m\angle EFD = 122^\circ$
Check. Both 58° and 122° are greater than 0° and at most 180°. Also, angles EFC and EFD form a linear pair, so the Supplement Theorem says their measures must add to 180°, and they do.
$58^\circ + 122^\circ = 180^\circ$

Answer

The measures are $m\angle EFC = 58^\circ$ , $m\angle DFH = 58^\circ$ and $m\angle EFD = 122^\circ$ .

Worked example

Consecutive interior angles with a rejected value

Use the labels in the slanted figure above, with line $AB$ given parallel to line $CD$ . This is a new problem with new measures: $m\angle BEF = (3x + 30)^\circ$ and $m\angle EFD = (x^2 + 20)^\circ$ . Find both measures.

The lines are given parallel, and angles BEF and EFD are consecutive interior angles. So the Consecutive Interior Angles Theorem, in its forward direction, says their measures add to 180°.
$(3x + 30) + (x^2 + 20) = 180$
Combine like terms on the left side.
$x^2 + 3x + 50 = 180$
Subtract 180 from both sides so that one side is 0.
$x^2 + 3x - 130 = 0$
Factor: 13 and −10 multiply to −130 and add to 3. A product is 0 only when a factor is 0, so set each factor equal to 0.
$(x + 13)(x - 10) = 0 \implies x = -13 \text{ or } x = 10$
Test x = −13. It gives measures of −9° and 189°. A measure must be greater than 0° and at most 180°, so reject x = −13.
$3(-13) + 30 = -9, \quad (-13)^2 + 20 = 189$
Test x = 10. It gives measures of 60° and 120°. Both are greater than 0° and at most 180°, and they add to 180°, so keep x = 10.
$3(10) + 30 = 60, \quad 10^2 + 20 = 120$

Answer

The only valid value is $x = 10$ , so $m\angle BEF = 60^\circ$ and $m\angle EFD = 120^\circ$ .

The central idea: when two lines are given parallel, one angle measure can lead you to others, using each fact in its forward direction. Corresponding, alternate interior and alternate exterior angles are congruent, and consecutive interior angles are supplementary. Each time, confirm the parallel given, name your reason, and check that every solved measure is greater than $0^\circ$ and at most $180^\circ$ .

Next, you will prove angle relationships formed by parallel lines.