Parallel-Line Equations and Finding Mistakes | Geometry | LevelKite

You already know how to find an angle measure when two lines are parallel. Now some measures are written with a variable, such as $(3x + 10)^\circ$ . An expression like this stands for a measure you do not know yet.

A parallel-line fact tells you how two measures are related. When you write that relationship using the expressions, you get an equation you can solve. In this lesson you build and solve those equations, and then you learn to find mistakes in this kind of work.

Two terms will help. A transversal is a line that crosses two other lines at different points. The symbol $m\angle 1$ means the measure of angle 1.

Key concept

Parallel lines turn an angle pair into an equation

When two lines are given parallel, a transversal forms pairs of angles whose measures are related. Call a pair $\angle 1$ and $\angle 2$ .

Corresponding, alternate interior and alternate exterior pairs are congruent, which means their measures are equal.
Consecutive interior pairs are supplementary, which means their measures add to $180^\circ$ .

Write the matching equation and solve it. Then substitute into both expressions. Keep a value only if each measure is greater than $0^\circ$ and at most $180^\circ$ and the pair fits its reason.

m\angle 1 = m\angle 2 \quad \text{or} \quad m\angle 1 + m\angle 2 = 180^\circ

Here is the figure for Examples 1, 2 and 4. The arrows on line $AB$ and line $CD$ state that these lines are parallel. Line $EF$ is a transversal that crosses line $AB$ at $G$ and line $CD$ at $H$ . The figure is not drawn to scale, so only the arrows and the text are givens. Nothing is read from the drawing. The two marked angles are the ones Example 1 uses: angle $AGH$ measures $(3x + 10)^\circ$ and angle $GHD$ measures $(5x - 30)^\circ$ .

Arrows mark line

AB

parallel to line

CD

. The marked angles are angle

AGH

and angle

GHD

, whose values are in the text above.

What each angle pair lets you write, using the labels in the figure above

Angle pair	Equation in the figure	Reason to name	Must the lines be given parallel?
Corresponding: same position at each crossing	$m\angle EGB = m\angle GHD$	Corresponding Angles Postulate	Yes
Alternate interior: between the lines, opposite sides of the transversal	$m\angle AGH = m\angle GHD$	Alternate Interior Angles Theorem	Yes
Alternate exterior: outside the lines, opposite sides of the transversal	$m\angle EGA = m\angle DHF$	Alternate Exterior Angles Theorem	Yes
Consecutive interior: between the lines, same side of the transversal	$m\angle AGH + m\angle CHG = 180^\circ$	Consecutive Interior Angles Theorem	Yes
Linear pair: adjacent angles whose outer rays form a line	$m\angle AGH + m\angle BGH = 180^\circ$	Supplement Theorem	No
Vertical: opposite angles where two lines cross	$m\angle AGH = m\angle EGB$	Vertical Angles Theorem	No

Worked example

Example 1: A pair with equal measures

In the figure above, line $AB$ is given parallel to line $CD$ , and transversal $EF$ crosses line $AB$ at $G$ and line $CD$ at $H$ . Angle $AGH$ measures $(3x + 10)^\circ$ and angle $GHD$ measures $(5x - 30)^\circ$ . Find $x$ and both measures.

Name the pair and the reason. The angles lie between the two lines, on opposite sides of the transversal, so they are alternate interior angles. The lines are given parallel, so use the Alternate Interior Angles Theorem in the forward direction: the angles are congruent, so their measures are equal.
$3x + 10 = 5x - 30$
Subtract 3x from both sides (Subtraction Property of Equality), which gathers the variable on the right.
$10 = 2x - 30$
Add 30 to both sides (Addition Property of Equality), which leaves only the variable term on the right.
$40 = 2x$
Divide both sides by 2 (Division Property of Equality).
$20 = x$
Substitute 20 for x in each expression to get the measures (Substitution Property of Equality).
$m\angle AGH = 3(20) + 10 = 70^\circ, \quad m\angle GHD = 5(20) - 30 = 70^\circ$
Check the result. Each measure is greater than 0° and at most 180°, and the two measures are equal, as the theorem requires. Keep x = 20.
$70 = 70 \quad \text{and} \quad 0 < 70 \le 180$

Answer

The value of $x$ is $20$ , and angle $AGH$ and angle $GHD$ each measure $70^\circ$ .

Worked example

Example 2: A supplementary pair

Use the labels from the figure above, with line $AB$ given parallel to line $CD$ . This time angle $AGH$ measures $(x + 50)^\circ$ and angle $CHG$ measures $(3x + 10)^\circ$ . Find $x$ and both measures.

These angles lie between the two lines, on the same side of the transversal, so they are consecutive interior angles. The lines are given parallel, so use the Consecutive Interior Angles Theorem in the forward direction: the angles are supplementary, so their measures add to 180°.
$(x + 50) + (3x + 10) = 180$
Combine like terms on the left: x + 3x = 4x and 50 + 10 = 60.
$4x + 60 = 180$
Subtract 60 from both sides (Subtraction Property of Equality).
$4x = 120$
Divide both sides by 4 (Division Property of Equality).
$x = 30$
Substitute 30 for x in each expression and check. Each measure is greater than 0° and at most 180°, and the two measures add to 180°.
$m\angle AGH = 30 + 50 = 80^\circ, \quad m\angle CHG = 3(30) + 10 = 100^\circ, \quad 80 + 100 = 180$

Answer

The value of $x$ is $30$ , angle $AGH$ measures $80^\circ$ , and angle $CHG$ measures $100^\circ$ .

Now you switch roles and check someone else's work. Follow the work in the order it was built, and ask four questions.

Given: Are the lines given parallel, by arrows or in words? If not, the Corresponding Angles Postulate and the parallel-line theorems cannot be used.
Reason: Is the pair named correctly, and does the equation match its reason?
Algebra: Is each step valid, with the same operation on both sides? Going from $10 = 2x - 30$ to $-20 = 2x$ is not valid, because $30$ must be added to both sides.
Result: Did the work substitute to find the measures? In Example 1, $20$ is the value of $x$ , and $70^\circ$ is the measure. Is each measure greater than $0^\circ$ and at most $180^\circ$ , and does the pair fit its reason?

A measure outside that range always means something is wrong. A measure inside that range does not prove the work is right, so ask all four questions.

Example 3 uses the figure below. It has the same labels as the first figure, but it has no arrows, and the text does not say the lines are parallel. Line $EF$ crosses line $AB$ at $G$ and line $CD$ at $H$ . The two marked angles carry these values: angle $AGH$ measures $(2x + 15)^\circ$ and angle $GHD$ measures $(4x - 45)^\circ$ .

There are no arrows, so the figure does not state that line

AB

is parallel to line

CD

. The marked angles are angle

AGH

and angle

GHD

.

Worked example

Example 3: Find the mistake in the given

A student looks at the figure above and writes: "The lines look parallel, so by the Alternate Interior Angles Theorem, $2x + 15 = 4x - 45$ . Then $x = 30$ , and both angles measure $75^\circ$ ." Find the mistake.

Ask the first question: are the lines given parallel? The figure has no arrows and the text does not say so. The forward direction of the theorem needs that given.
Now read the student's reason. How lines look is not a given, because the figure is not drawn to scale. So the student's first equation has no valid reason. That is the mistake.
Check whether anything rescues the equation. The algebra is valid and 75° is a possible measure, but valid algebra on an unjustified equation proves nothing. The Alternate Interior Angles Converse does not help, because it needs equal measures as a given, and they are not given. With arrows or a statement that the lines are parallel, the same equation would be correct.
$2x + 15 = 4x - 45 \;\Rightarrow\; 15 = 2x - 45 \;\Rightarrow\; 60 = 2x \;\Rightarrow\; x = 30$

Answer

The mistake is the student's first line: the lines are not given parallel, so the Alternate Interior Angles Theorem does not apply, and this information cannot determine $x$ .

Worked example

Example 4: Find the mistake in the reason

Use the labels from the first figure. Line $AB$ is given parallel to line $CD$ . Angle $EGA$ measures $(4x + 30)^\circ$ and angle $GHD$ measures $(2x + 6)^\circ$ . A student writes: "These are corresponding angles, so $4x + 30 = 2x + 6$ . Then $2x = -24$ and $x = -12$ ." Find the mistake, then solve correctly.

Test the student's value. Both expressions give −18°. A measure must be greater than 0°, so reject x = −12. The negative measure is the problem, not the negative x; a negative x is fine when the measures are valid. This tells you a mistake exists, but not where it is.
$4(-12) + 30 = -18, \quad 2(-12) + 6 = -18$
Ask the four questions. The lines are given parallel, and the algebra is valid, so look at the reason. The angle at G that corresponds to angle GHD is angle EGB, not angle EGA. Calling angles EGA and GHD corresponding is the mistake.
Build the correct reason. The lines are given parallel, so the Corresponding Angles Postulate in the forward direction gives m∠EGB = m∠GHD. Rays GA and GB are opposite, so angles EGA and EGB form a linear pair, and the Supplement Theorem says their measures add to 180°. Replace m∠EGB with m∠GHD (Substitution Property of Equality).
$m\angle EGA + m\angle GHD = 180^\circ$
Replace each measure with its expression (Substitution Property of Equality), then combine like terms: 4x + 2x = 6x and 30 + 6 = 36.
$(4x + 30) + (2x + 6) = 180 \;\Rightarrow\; 6x + 36 = 180$
Subtract 36 from both sides (Subtraction Property of Equality), then divide both sides by 6 (Division Property of Equality).
$6x = 144 \;\Rightarrow\; x = 24$
Substitute 24 for x and check. Each measure is greater than 0° and at most 180°, and the two measures add to 180°.
$m\angle EGA = 4(24) + 30 = 126^\circ, \quad m\angle GHD = 2(24) + 6 = 54^\circ, \quad 126 + 54 = 180$

Answer

The mistake is calling angles $EGA$ and $GHD$ corresponding. With the lines given parallel, they are supplementary, so $x = 24$ , angle $EGA$ measures $126^\circ$ , and angle $GHD$ measures $54^\circ$ .

Takeaway: When two lines are given parallel, name the angle pair and let its reason decide the equation: equal measures, or a sum of $180^\circ$ . Solve, substitute, and check each measure. To find a mistake in someone's work, question the given, the reason, the algebra and the result, in that order.