Angle Relationships: Measures, Equations, and Errors | Grade 8 Math | LevelKite

In this lesson you use angle relationships to find unknown angle measures, write and solve equations, and find and fix mistakes. Every relationship you need is stated in words. You work only from those stated facts, never from how an angle might look. An angle has two sides that meet at a point called its vertex. All measures are in degrees, and every angle here has a positive measure, greater than $0^\circ$ . You write $m\angle 1$ for the measure of $\angle 1$ . For example, $m\angle 1 = 40^\circ$ means that $\angle 1$ measures $40$ degrees.

What each stated relationship lets you write

Stated relationship	What it means	What you can write
$\angle 1$ and $\angle 2$ are complementary	Their measures add to $90^\circ$ .	$m\angle 1 + m\angle 2 = 90^\circ$
$\angle 1$ and $\angle 2$ are supplementary	Their measures add to $180^\circ$ .	$m\angle 1 + m\angle 2 = 180^\circ$
$\angle 1$ and $\angle 2$ are vertical angles	They are opposite each other where two lines cross, sharing the vertex but no side. Their measures are equal.	$m\angle 1 = m\angle 2$ , an equation with no number of degrees in it
$\angle 1$ and $\angle 2$ are adjacent	They share a vertex and a side and do not overlap.	No equation by itself. If the whole angle they fill has a stated measure, their measures add to that measure.

Being adjacent tells you where two angles sit, not how big they are. Two adjacent angles might add to $50^\circ$ , to $150^\circ$ , or to many other totals. So being adjacent gives you no equation by itself. You get an equation when the whole angle they fill has a stated measure.

Suppose $\angle ABD$ and $\angle DBC$ are adjacent, sharing vertex $B$ and side $BD$ . Together they form $\angle ABC$ , and $m\angle ABC = 130^\circ$ . The two parts add to that stated whole. If $m\angle ABD = 85^\circ$ , you write $85^\circ + m\angle DBC = 130^\circ$ . Subtracting $85^\circ$ from both sides keeps the equation true and gives $m\angle DBC = 130^\circ - 85^\circ = 45^\circ$ . Check: both measures are positive, and $85^\circ + 45^\circ = 130^\circ$ .

Complementary and supplementary are the opposite kind of fact. They describe size only, so two such angles may be adjacent or apart.

Key concept

Write only what a stated fact gives, then check

Write only the equation that a stated fact gives you.

Complementary: the two measures add to $90^\circ$ .
Supplementary: the two measures add to $180^\circ$ .
Vertical: the two measures are equal. This is an equation with no number of degrees in it.
Adjacent: no equation by itself. When the whole angle they fill has a stated measure, the two measures add to that measure.

Then solve. Check that every angle measure is positive and fits every stated fact.

Worked example

Find measures from stated facts

You are told that $m\angle 1 = 34^\circ$ , that $\angle 1$ and $\angle 2$ are complementary, that $\angle 1$ and $\angle 3$ are supplementary, and that $\angle 3$ and $\angle 4$ are vertical angles. Find $m\angle 2$ , $m\angle 3$ , and $m\angle 4$ .

Start with the complementary pair. Complementary angles have measures that add to $90^\circ$ . Write that sum and put in the measure you know.
$34^\circ + m\angle 2 = 90^\circ$
Subtract $34^\circ$ from both sides. Doing the same thing to both sides keeps the equation true.
$m\angle 2 = 90^\circ - 34^\circ = 56^\circ$
Supplementary angles have measures that add to $180^\circ$ . Write that sum, then subtract $34^\circ$ from both sides, as before.
$34^\circ + m\angle 3 = 180^\circ \quad\Rightarrow\quad m\angle 3 = 180^\circ - 34^\circ = 146^\circ$
Vertical angles have equal measures. That equation has no number of degrees in it, so you use the measure you just found for $\angle 3$ .
$m\angle 4 = m\angle 3 = 146^\circ$
Check every stated fact. All the measures are positive. The complementary pair adds to $90^\circ$ , the supplementary pair adds to $180^\circ$ , and the vertical pair is equal.
$34^\circ + 56^\circ = 90^\circ, \quad 34^\circ + 146^\circ = 180^\circ, \quad 146^\circ = 146^\circ$

Answer

The missing measures are $m\angle 2 = 56^\circ$ , $m\angle 3 = 146^\circ$ , and $m\angle 4 = 146^\circ$ .

Sometimes an angle measure is written with a variable. Here $x$ stands for an unknown number, and a measure of $(2x + 8)^\circ$ means $2x + 8$ degrees. Only values of $x$ that make every angle measure positive can work. In these equations every number counts degrees, so you can leave the degree sign off until you state a measure.

To solve, do the same thing to both sides. Adding or subtracting the same number keeps an equation true, and so does dividing both sides by the same nonzero number. Finding $x$ is not the end. Put its value into each expression to get the angle measures. Then check them against every stated fact.

Worked example

Complementary angles with a variable

$\angle P$ and $\angle Q$ are complementary, with $m\angle P = (2x + 8)^\circ$ and $m\angle Q = (3x + 2)^\circ$ . Find both angle measures.

Complementary angles have measures that add to $90$ degrees. Add the two expressions and set the sum equal to $90$ .
$(2x + 8) + (3x + 2) = 90$
Combine like terms on the left. Add the $x$ -terms, $2x + 3x = 5x$ , and add the plain numbers, $8 + 2 = 10$ . This rewrites the left side without changing its value.
$5x + 10 = 90$
Subtract $10$ from both sides. Then divide both sides by $5$ , which is not zero. Each move does the same thing to both sides, so the equation stays true.
$5x = 80 \quad\Rightarrow\quad x = 16$
Put $x = 16$ into each expression, because the expressions give the measures. So $m\angle P = 40^\circ$ and $m\angle Q = 50^\circ$ .
$2(16) + 8 = 32 + 8 = 40, \quad 3(16) + 2 = 48 + 2 = 50$
Check: both measures are positive, and they add to $90^\circ$ , as complementary angles must.
$40^\circ + 50^\circ = 90^\circ$

Answer

The measures are $m\angle P = 40^\circ$ and $m\angle Q = 50^\circ$ .

Worked example

Vertical angles, then a supplementary pair

Two lines cross at point $O$ . You are told that $\angle 1$ and $\angle 3$ are vertical angles, with $m\angle 1 = (5x - 8)^\circ$ and $m\angle 3 = (3x + 22)^\circ$ , and that $\angle 1$ and $\angle 2$ are supplementary. Find $m\angle 1$ , $m\angle 3$ , and $m\angle 2$ .

Vertical angles have equal measures, so set the two expressions equal. The relationship itself gives no number of degrees. The numbers in this equation come from the expressions.
$5x - 8 = 3x + 22$
Subtract $3x$ from both sides to gather the $x$ -terms on the left. Then add $8$ to both sides. Both moves do the same thing to both sides, so the equation stays true.
$2x - 8 = 22 \quad\Rightarrow\quad 2x = 30$
Divide both sides by $2$ . Dividing both sides by the same nonzero number keeps the equation true.
$x = 15$
Put $x = 15$ into both expressions, because they give the two measures. Both give $67$ , so the vertical angles are equal, as they must be: $m\angle 1 = m\angle 3 = 67^\circ$ .
$5(15) - 8 = 75 - 8 = 67, \quad 3(15) + 22 = 45 + 22 = 67$
Now use the supplementary fact: the measures of $\angle 1$ and $\angle 2$ add to $180^\circ$ . Subtract $67^\circ$ from both sides.
$67^\circ + m\angle 2 = 180^\circ \quad\Rightarrow\quad m\angle 2 = 180^\circ - 67^\circ = 113^\circ$
Check every fact. All three measures are positive, the vertical pair is equal, and the supplementary pair adds to $180^\circ$ .
$67^\circ = 67^\circ, \quad 67^\circ + 113^\circ = 180^\circ$

Answer

The measures are $m\angle 1 = 67^\circ$ , $m\angle 3 = 67^\circ$ , and $m\angle 2 = 113^\circ$ .

To find an error in someone's work, start at the end. Put their angle measures into every stated fact. If any fact fails, something went wrong. Then test the work from the top:

Does the first equation say exactly what the stated relationship gives?
Does each later line follow correctly from the one before?

The first line that fails is the error. Fix it, redo the work from that point, and check the new measures against every fact.

Common mistake: treating adjacent angles as supplementary. You may be tempted to write $m\angle 1 + m\angle 2 = 180^\circ$ whenever two angles are adjacent. This fails because being adjacent tells you only where the angles sit, not their sum. Adjacency gives you a sum only when the whole angle they fill has a stated measure, and that measure can be many sizes. Say adjacent angles fill a stated $130^\circ$ angle and one measures $85^\circ$ . The mistake gives $180^\circ - 85^\circ = 95^\circ$ for the other. Then the check fails: $85^\circ + 95^\circ = 180^\circ$ , not $130^\circ$ . Use $180^\circ$ only when a stated fact gives it, such as the angles being stated as supplementary.

Worked example

Find and fix the error

$\angle ABD$ and $\angle DBC$ are adjacent, sharing vertex $B$ and side $BD$ . Together they form $\angle ABC$ , and $m\angle ABC = 120^\circ$ . Also, $m\angle ABD = (4x + 6)^\circ$ and $m\angle DBC = (2x + 12)^\circ$ . A student wrote $(4x + 6) + (2x + 12) = 180$ , then $6x + 18 = 180$ , $6x = 162$ , and $x = 27$ . The student answered $m\angle ABD = 114^\circ$ and $m\angle DBC = 66^\circ$ . Find the error and correct it.

Start at the end and test the student's measures against the stated facts. The parts should add to the stated whole, $120^\circ$ . They add to $180^\circ$ instead, so something went wrong.
$114^\circ + 66^\circ = 180^\circ \neq 120^\circ$
Now test the work from the top. Each algebra line follows correctly from the one before, and $x = 27$ does give the student's measures. So the error is the first equation. The student set the sum equal to $180$ as if the angles were supplementary, which was never stated. Being adjacent gives no equation by itself. The sum must come from the stated whole, $120^\circ$ .
$4(27) + 6 = 108 + 6 = 114, \quad 2(27) + 12 = 54 + 12 = 66$
Write the equation the facts do give: the two parts add to the stated whole.
$(4x + 6) + (2x + 12) = 120$
Combine like terms and subtract $18$ from both sides. Then divide both sides by $6$ , which is not zero. Each move keeps the equation true.
$6x + 18 = 120 \quad\Rightarrow\quad 6x = 102 \quad\Rightarrow\quad x = 17$
Put $x = 17$ into each expression, because the expressions give the two measures.
$4(17) + 6 = 68 + 6 = 74, \quad 2(17) + 12 = 34 + 12 = 46$
Check: both measures are positive, and the parts add to the stated whole.
$74^\circ + 46^\circ = 120^\circ$

Answer

The error is the first equation, which used $180^\circ$ instead of the stated whole, $120^\circ$ . The correct measures are $m\angle ABD = 74^\circ$ and $m\angle DBC = 46^\circ$ .

Takeaway: each stated fact tells you exactly what you may write. Complementary measures add to $90^\circ$ , and supplementary measures add to $180^\circ$ . Vertical angles have equal measures. Adjacent angles give an equation only when the whole angle they fill has a stated measure. Write the equation the facts give, solve it, find each angle measure, and check every measure against every fact. That same check is how you find and fix errors.

Next, you will find lengths and measures of bisected segments and angles, which are segments and angles split into two equal parts.