Angle Pair Equations and Word Problems | Geometry | LevelKite

In this lesson, you turn facts about pairs of angles into equations, then solve those equations to find angle measures. Start with two terms you already know:

Complementary angles are two angles whose measures add to $90^\circ$ .
Supplementary angles are two angles whose measures add to $180^\circ$ .

Each definition is a statement about numbers, which is why it gives you an equation. The two angles do not have to touch. Only their measures matter.

Key concept

Every angle relationship gives an equation

To find an unknown angle measure, make four moves:

Name the relationship, and give the exact reason it holds.
Write the equation that relationship gives.
Solve the equation, showing every step.
Check every measure. Each one must be greater than $0^\circ$ and at most $180^\circ$ . Reject a value that breaks this condition, including any value that makes a measure zero or negative.

A figure is not drawn to scale. Only the facts stated in the text and any tick, arrow or right-angle marks are givens. You never read a measure from the picture, and you never decide from the picture that two angles are equal.

Word problems often describe an angle through its complement or its supplement. An angle's complement is the angle that adds with it to $90^\circ$ , and its supplement is the angle that adds with it to $180^\circ$ . So if an angle measures $a^\circ$ :

its complement measures $(90 - a)^\circ$ , which exists only when $0 < a < 90$ ;
its supplement measures $(180 - a)^\circ$ , which exists only when $0 < a < 180$ .

These limits keep every measure greater than $0^\circ$ . To translate the words, let a variable stand for the angle that the other angle is described from. For example, $7$ more than twice $x$ is $2x + 7$ , and $7$ less than three times $x$ is $3x - 7$ .

Worked example

Word problem: complementary angles

A frame maker joins two strips of wood to form a corner. The joint splits the $90^\circ$ corner into two angles, so the two angles are complementary. One angle must measure $15^\circ$ more than twice the other. Find both measures.

Separate what you know from what you need. Given: the angles are complementary, and one is 15 degrees more than twice the other. To find: both measures. Since one angle is described from the other, let x stand for the measure of the other angle, in degrees.
$\text{one angle} = (2x + 15)^\circ, \quad \text{the other} = x^\circ$
Write the equation. By the definition of complementary angles, the two measures add to 90 degrees.
$x + (2x + 15) = 90$
Combine like terms on the left side, because x + 2x = 3x.
$3x + 15 = 90$
Subtract 15 from both sides (Subtraction Property of Equality), then divide both sides by 3 (Division Property of Equality).
$3x = 75 \quad\Rightarrow\quad x = 25$
Substitute x = 25 into 2x + 15 (Substitution Property of Equality).
$2(25) + 15 = 50 + 15 = 65$
Check. The measures add to 90 degrees, and 65 is 15 more than twice 25. Both 25 and 65 are greater than 0 and at most 180 degrees, so keep them.
$25 + 65 = 90$

Answer

The two angles at the corner measure $25^\circ$ and $65^\circ$ .

Worked example

Word problem: a supplement

The supplement of an angle measures $20^\circ$ less than three times the angle. Find the measure of the angle and the measure of its supplement.

Given: the description of the supplement. To find: the angle and its supplement. Let a be the measure of the angle, in degrees. By the definition of supplementary angles, its supplement measures 180 − a degrees.
$\text{supplement} = (180 - a)^\circ$
Translate the sentence. The word "measures" becomes the equals sign. The phrase "20 less than three times the angle" means subtract 20 from 3a, so write 3a − 20 and not 20 − 3a.
$180 - a = 3a - 20$
The variable is on both sides. Add a to both sides (Addition Property of Equality).
$180 = 4a - 20$
Add 20 to both sides (Addition Property of Equality), then divide both sides by 4 (Division Property of Equality).
$200 = 4a \quad\Rightarrow\quad a = 50$
Substitute a = 50 to find the supplement (Substitution Property of Equality), then test the value in the original sentence.
$180 - 50 = 130, \quad 3(50) - 20 = 130$
Check. The measures add to 180 degrees. Both 50 and 130 are greater than 0 and at most 180 degrees, so keep them.
$50 + 130 = 180$

Answer

The angle measures $50^\circ$ and its supplement measures $130^\circ$ .

The check can change your answer. Suppose a problem says that two complementary angles have measures that differ by $100^\circ$ . Let the smaller measure be $x^\circ$ , so the larger measure is $(x + 100)^\circ$ . The definition of complementary angles gives $x + (x + 100) = 90$ .

Combine like terms to get $2x + 100 = 90$ . Subtract $100$ from both sides to get $2x = -10$ . Divide both sides by $2$ to get $x = -5$ .

The algebra is correct, but a measure of $-5^\circ$ is not greater than $0^\circ$ , so you reject this value. No pair of complementary angles fits that description, and saying so is the correct answer.

Now look at relationships that come from how lines and rays meet. Adjacent angles share a vertex and a side and do not overlap. Opposite rays share an endpoint and together form a line. A linear pair is two adjacent angles whose unshared sides are opposite rays, so together they fill one side of a line.

A postulate is a statement accepted without proof. The Linear Pair Postulate says: if two angles form a linear pair, then they are supplementary. You use it in that direction only. Its converse, which swaps the "if" and "then" parts, is false, because two supplementary angles do not have to be adjacent.

The symbol $m\angle AEC$ means the measure of angle $AEC$ . In the figure below, the position of $E$ and both measures come from the text, because the figure is not drawn to scale.

Point

E

lies between

A

and

B

on line

AB

. Angle

AEC

measures

(3x + 10)^\circ

and angle

CEB

measures

(5x - 6)^\circ

. The figure is not drawn to scale.

Worked example

Using a linear pair

Point $E$ lies between $A$ and $B$ on line $AB$ . Ray $EC$ starts at $E$ , and $C$ is not on line $AB$ . Angle $AEC$ measures $(3x + 10)^\circ$ and angle $CEB$ measures $(5x - 6)^\circ$ . The figure is not drawn to scale. Find both measures.

Name the relationship from the given facts, not from the drawing. Angles AEC and CEB share vertex E and side EC. Their unshared sides, rays EA and EB, are opposite rays because E lies between A and B on line AB. By the definition of a linear pair, these angles form a linear pair.
Use the Linear Pair Postulate, from linear pair to supplementary. By the definition of supplementary angles, the measures add to 180 degrees.
$(3x + 10) + (5x - 6) = 180$
Combine like terms: 3x + 5x = 8x and 10 − 6 = 4.
$8x + 4 = 180$
Subtract 4 from both sides (Subtraction Property of Equality), then divide both sides by 8 (Division Property of Equality).
$8x = 176 \quad\Rightarrow\quad x = 22$
Substitute x = 22 into each expression (Substitution Property of Equality).
$m\angle AEC = 3(22) + 10 = 76, \quad m\angle CEB = 5(22) - 6 = 104$
Check. The measures add to 180 degrees, and both are greater than 0 and at most 180 degrees, so keep x = 22.
$76 + 104 = 180$

Answer

Angle $AEC$ measures $76^\circ$ and angle $CEB$ measures $104^\circ$ .

Vertical angles are two angles whose sides form two pairs of opposite rays. When two lines intersect at one point, they form two pairs of vertical angles, and the angles in each pair are not adjacent. Decide this from stated facts, because a figure is not drawn to scale.

Here is the idea: each vertical angle forms a linear pair with the same neighboring angle, so each one measures $180^\circ$ minus that neighbor's measure. A theorem is a statement proved from definitions, postulates and earlier theorems. The Vertical Angles Theorem says: if two angles are vertical angles, then they have equal measures. You use it in that direction only. Its converse is false, because two angles with equal measures do not have to be vertical angles.

The proof below uses the two lines shown in the figure after it. Given: lines $AB$ and $CD$ intersect at $E$ , with $E$ between $A$ and $B$ and between $C$ and $D$ . To prove: $m\angle AEC = m\angle BED$ . The same steps work for the other pair of vertical angles, angles $CEB$ and $DEA$ .

Proof that vertical angles AEC and BED have equal measures

Statement	Reason
Rays $EA$ and $EB$ are opposite rays, and so are rays $EC$ and $ED$ .	Given: $E$ is between $A$ and $B$ and between $C$ and $D$ .
Angles $AEC$ and $CEB$ form a linear pair.	Definition of a linear pair: they share ray $EC$ , and rays $EA$ and $EB$ are opposite.
Angles $CEB$ and $BED$ form a linear pair.	Definition of a linear pair: they share ray $EB$ , and rays $EC$ and $ED$ are opposite.
$m\angle AEC + m\angle CEB = 180^\circ$	Linear Pair Postulate and definition of supplementary angles
$m\angle CEB + m\angle BED = 180^\circ$	Linear Pair Postulate and definition of supplementary angles
$m\angle AEC + m\angle CEB = m\angle CEB + m\angle BED$	Transitive Property of Equality: both sums equal $180^\circ$ .
$m\angle AEC = m\angle BED$	Subtraction Property of Equality: subtract $m\angle CEB$ from both sides.

Lines

AB

and

CD

intersect at

E

. Angle

AEC

measures

(3x + 20)^\circ

and angle

BED

measures

(5x - 16)^\circ

. The figure is not drawn to scale, so the equal measures come from the theorem.

Worked example

Using vertical angles and linear pairs

Lines $AB$ and $CD$ intersect at $E$ , with $E$ between $A$ and $B$ and between $C$ and $D$ . Angle $AEC$ measures $(3x + 20)^\circ$ and angle $BED$ measures $(5x - 16)^\circ$ . The figure is not drawn to scale. Find the measures of all four angles at $E$ .

Given: E lies between A and B and between C and D, so rays EA and EB are opposite, and rays EC and ED are opposite. By definition, angles AEC and BED are vertical angles. Use the Vertical Angles Theorem, from vertical angles to equal measures. The reason for this equation is the theorem, not how the angles look.
$3x + 20 = 5x - 16$
The variable is on both sides. Subtract 3x from both sides and add 16 to both sides (Subtraction and Addition Properties of Equality), then divide both sides by 2 (Division Property of Equality).
$20 = 2x - 16 \;\Rightarrow\; 36 = 2x \;\Rightarrow\; x = 18$
Substitute x = 18 into both expressions (Substitution Property of Equality). The equal results confirm the algebra.
$m\angle AEC = 3(18) + 20 = 74, \quad m\angle BED = 5(18) - 16 = 74$
Angles AEC and CEB form a linear pair, because they share ray EC and rays EA and EB are opposite. By the Linear Pair Postulate and the definition of supplementary angles, their measures add to 180 degrees. Subtract 74 from both sides (Subtraction Property of Equality).
$74 + m\angle CEB = 180 \;\Rightarrow\; m\angle CEB = 180 - 74 = 106$
Angles CEB and DEA are vertical angles, because rays EC and ED are opposite and rays EB and EA are opposite. Apply the Vertical Angles Theorem again.
$m\angle DEA = m\angle CEB = 106$
Check. Angles AEC and CEB add to 180 degrees, and so do the other linear pairs. All four measures are greater than 0 and at most 180 degrees, so keep x = 18.
$74 + 106 = 180$

Answer

Angles $AEC$ and $BED$ each measure $74^\circ$ , and angles $CEB$ and $DEA$ each measure $106^\circ$ .

Takeaway: You can turn each angle relationship into one equation. Complementary angles add to $90^\circ$ , supplementary angles and linear pairs add to $180^\circ$ , and vertical angles have equal measures. Name the relationship from stated facts, not from the look of a figure, which is not drawn to scale. Then solve, and check that every measure is greater than $0^\circ$ and at most $180^\circ$ .

Next, you will identify perpendicular lines and use their properties, and you will identify mistakes in angle relationship problems.