Multi-Step Angle Problems and Error Analysis | Grade 7 Math | LevelKite

In a busy figure, the angle you want is often not next to anything you know. You reach it with a chain. First you use one angle fact to find a new angle. Then you use that new angle in the next fact. Each link needs a reason you can point to: a mark in the figure, or a position that the problem states.

You can't measure your way there, because these figures are not drawn to scale. In this lesson, $m\angle ABC$ means the measure of angle $ABC$ in degrees. The middle letter always names the vertex.

Angle facts and the positions that make them true

Relationship	Position that makes it apply	Fact
Complementary angles	Two angles side by side that together fill a right angle shown by a square mark	Measures add to $90^\circ$
Supplementary angles, such as a linear pair	A linear pair is two angles that share a side, with their other sides forming one straight line	Measures add to $180^\circ$
Vertical angles	Two angles directly opposite each other where two lines cross	Measures are equal
Angles of a triangle	The three inside angles of one triangle	Measures add to $180^\circ$
Angles of a quadrilateral	The four inside angles of one four-sided figure	Measures add to $360^\circ$

Key concept

Every link needs a stated reason

Solve a multi-step angle problem one link at a time. For each new angle, name the position in the figure that makes your fact true. That position might be a square mark, two lines crossing, two angles side by side on a line, or the corners of one triangle. Then write the equation that the fact gives you.

When you check a solution, including your own, re-read the figure beside each equation. Ask whether that position is really there.

Angle

FED

is the given angle and measures

35^\circ

. The square mark at

E

makes angle

FEB

a right angle.

Worked example

A chain of three facts

In the figure, lines $AB$ and $CD$ cross at point $E$ , and ray $EF$ starts at $E$ . The square mark shows that angle $FEB$ is a right angle. Ray $ED$ lies inside angle $FEB$ , and angle $FED$ measures $35^\circ$ . Find the measures of angle $AEC$ and angle $CEB$ .

Ray $ED$ splits the right angle $FEB$ into angles $FED$ and $DEB$ . Two angles that together fill a marked right angle are complementary, so their measures add to $90^\circ$ . Subtract the known part from $90^\circ$ .
$35^\circ + m\angle DEB = 90^\circ \;\Rightarrow\; m\angle DEB = 90^\circ - 35^\circ = 55^\circ$
Lines $AB$ and $CD$ cross at $E$ , and angle $AEC$ sits directly opposite angle $DEB$ . That makes them vertical angles, and vertical angles are equal.
$m\angle AEC = m\angle DEB = 55^\circ$
Angles $DEB$ and $CEB$ share ray $EB$ . Their other sides, $ED$ and $EC$ , form line $CD$ . So they are a linear pair, which means they are supplementary.
$m\angle CEB = 180^\circ - 55^\circ = 125^\circ$
Check your answers with a different pair. Angles $AEC$ and $CEB$ share ray $EC$ , and their other sides form line $AB$ . They are also a linear pair, so their measures should add to $180^\circ$ .
$55^\circ + 125^\circ = 180^\circ$

Answer

Angle $AEC$ measures $55^\circ$ , and angle $CEB$ measures $125^\circ$ .

Sometimes a given angle is an expression, such as $(2x + 10)^\circ$ . Here $x$ is an unknown number, and the whole expression is the angle's measure in degrees. You still chain the facts in the same way. At the last link, the fact becomes an equation in $x$ , and you solve it as you did in the previous lesson.

In the next figure, line $AB$ is crossed by two lines, and those two lines meet each other at $G$ .

The three marked angles are the givens: angle

AEG

is

120^\circ

, angle

CGD

is

50^\circ

, and angle

EFG

is

(2x+10)^\circ

.

Worked example

A chain that ends in an equation

Line $AB$ is crossed by line $EC$ at $E$ and by line $FD$ at $F$ . Point $G$ lies on line $EC$ between $E$ and $C$ , and on line $FD$ between $F$ and $D$ . So $E$ , $F$ and $G$ are the corners of triangle $EFG$ . Angle $AEG$ measures $120^\circ$ , angle $CGD$ measures $50^\circ$ , and angle $EFG$ measures $(2x+10)^\circ$ . Find $x$ and the measure of angle $EFG$ .

Points $A$ , $E$ and $F$ lie on line $AB$ . So angles $AEG$ and $GEF$ form a linear pair, and their measures add to $180^\circ$ .
$m\angle GEF = 180^\circ - 120^\circ = 60^\circ$
Lines $EC$ and $FD$ cross at $G$ . Angle $CGD$ and angle $EGF$ sit opposite each other there, so they are vertical angles and have equal measures.
$m\angle EGF = m\angle CGD = 50^\circ$
Angles $GEF$ , $EGF$ and $EFG$ are the three angles of triangle $EFG$ , so their measures add to $180^\circ$ . Write that fact as an equation in $x$ .
$60 + 50 + (2x + 10) = 180$
Add the numbers on the left: $60 + 50 + 10 = 120$ . Then subtract $120$ from both sides. Doing the same thing to both sides keeps them equal.
$2x + 120 = 180 \;\Rightarrow\; 2x = 60$
Divide both sides by $2$ , which also keeps them equal. Then substitute $x = 30$ into the expression for angle $EFG$ .
$x = 30,\quad m\angle EFG = 2(30) + 10 = 70^\circ$
Check your answer. The measure is positive, as every angle of a triangle must be. The three angles of the triangle should also add to $180^\circ$ .
$60^\circ + 50^\circ + 70^\circ = 180^\circ$

Answer

$x = 30$ , so angle $EFG$ measures $70^\circ$ .

Now you will check someone else's work. In the figure below, lines $AB$ and $CD$ cross at $E$ . Angle $AEC$ measures $(3x+10)^\circ$ , and angle $CEB$ measures $(5x-30)^\circ$ . A student wrote this flawed solution to find $x$ and the measure of angle $BED$ :

Angles $AEC$ and $CEB$ are equal, so $3x + 10 = 5x - 30$ .
Subtract $3x$ from both sides and add $30$ to both sides: $40 = 2x$ , so $x = 20$ .
Angle $BED$ is vertical to angle $AEC$ , so it measures $3(20) + 10 = 70^\circ$ .

The algebra in step 2 is done correctly. Look for the step whose equation does not match the figure.

Angle

AEC

is

(3x+10)^\circ

and angle

CEB

is

(5x-30)^\circ

. No mark says that any angle here is a right angle.

Worked example

Find and fix the flawed step

Lines $AB$ and $CD$ cross at $E$ . Angle $AEC$ measures $(3x+10)^\circ$ , and angle $CEB$ measures $(5x-30)^\circ$ . Find the faulty step in the flawed solution above, then find the correct value of $x$ and the measure of angle $BED$ .

Re-read the figure. Angles $AEC$ and $CEB$ share ray $EC$ , and their other sides, $EA$ and $EB$ , form line $AB$ . So they are a linear pair, not vertical angles. Nothing in the figure says that they are equal.
So step 1 is the faulty step: it treats a linear pair as equal. Its answer fails a quick test. With $x = 20$ , both angles would measure $70^\circ$ , but a linear pair must add to $180^\circ$ .
$70^\circ + 70^\circ = 140^\circ \ne 180^\circ$
Write the corrected equation. A linear pair is supplementary, so the two measures add to $180$ .
$(3x + 10) + (5x - 30) = 180$
Combine like terms: $3x + 5x = 8x$ and $10 - 30 = -20$ . Add $20$ to both sides, then divide both sides by $8$ . Each move keeps the two sides equal.
$8x - 20 = 180 \;\Rightarrow\; 8x = 200 \;\Rightarrow\; x = 25$
Substitute $x = 25$ into both expressions. Angle $BED$ sits opposite angle $AEC$ where the two lines cross, so the two angles are vertical and equal. The reason in step 3 was right; only its value of $x$ was wrong.
$m\angle AEC = 3(25) + 10 = 85^\circ,\quad m\angle CEB = 5(25) - 30 = 95^\circ,\quad m\angle BED = 85^\circ$
Check your answer. Both measures are positive, and the linear pair adds to $180^\circ$ .
$85^\circ + 95^\circ = 180^\circ$

Answer

The error is in step 1, which set a linear pair equal. The correct equation gives $x = 25$ , so angle $BED$ measures $85^\circ$ .

Takeaway: A multi-step angle problem is a chain of small facts, and each fact rests on a position the figure or the text actually states. The same idea guides error analysis. A flawed solution usually has one equation that the figure does not support, for example:

a linear pair set equal,
a sum written for the wrong pair of angles, or
an angle treated as right only because it looks right.

Find that step, write the equation the figure really supports, solve it, and check your answer.