Angle Postulates and the Supplement and Complement Theorems | Geometry | LevelKite

You already know how to add angle measures and how to find complements and supplements. In this lesson, you turn those ideas into reasons in proofs, just as you did for segments with the Ruler and Segment Addition Postulates.

Every statement in a proof needs a reason. A reason is the exact name of a definition, a postulate, a property of equality or congruence, or a theorem already proved, or the word Given. A step is never justified by its own conclusion, by an unstated fact, or by how a figure looks.

Two postulates, which are statements accepted without proof, connect angles to numbers.

Protractor Postulate: every angle has exactly one measure, a real number greater than $0$ and at most $180$ . It does for angles what the Ruler Postulate does for segments. The symbol $m\angle ABC$ means the measure of $\angle ABC$ in degrees. Because a measure is a number, you can use it in an equation even before you know its value.

Angle Addition Postulate: if $D$ is in the interior of $\angle ABC$ , then $m\angle ABD + m\angle DBC = m\angle ABC$ . In words, the measures of the two parts add to the measure of the whole, in either order. You may cite this postulate only after the interior fact is stated.

Angle

ABD

measures

25^\circ

and angle

CBE

measures

25^\circ

, as the proof below states. The interior facts also come from the Given, not from the drawing.

Worked example

Walk the plan of a proof

Given: $D$ is in the interior of $\angle ABC$ , $C$ is in the interior of $\angle DBE$ , $m\angle ABD = 25$ , and $m\angle CBE = 25$ . Prove: $m\angle ABC = m\angle DBE$ .

Reason: Given. Start with the interior facts, because the Angle Addition Postulate depends on them. They are stated, not read from the figure.
$D \text{ is in the interior of } \angle ABC,\quad C \text{ is in the interior of } \angle DBE$
Reason: Given. Here is the plan: angle DBC is part of both large angles, so show that each large angle equals 25 plus the same number.
$m\angle ABD = 25,\quad m\angle CBE = 25$
Reason: Protractor Postulate. Angle DBC has exactly one measure, a real number. You do not know its value, but you can still use it in equations.
$m\angle DBC \text{ is a real number}$
Reason: Angle Addition Postulate. You may cite it because D is stated to be in the interior of angle ABC.
$m\angle ABD + m\angle DBC = m\angle ABC$
Reason: Angle Addition Postulate. You may cite it because C is stated to be in the interior of angle DBE. The two parts can be added in either order.
$m\angle CBE + m\angle DBC = m\angle DBE$
Reason: Substitution Property of Equality. Replace the measure of angle ABD with 25.
$25 + m\angle DBC = m\angle ABC$
Reason: Substitution Property of Equality. Replace the measure of angle CBE with 25.
$25 + m\angle DBC = m\angle DBE$
Reason: Substitution Property of Equality. The last two equations start with the same sum. Use the first of them to replace that sum in the second with the measure of angle ABC.
$m\angle ABC = m\angle DBE$

Answer

Both large angles equal $25$ plus the same measure, so $m\angle ABC = m\angle DBE$ , even though you never needed the value of $m\angle DBC$ .

The finished two-column proof

Statement	Reason
$D$ is in the interior of $\angle ABC$	Given
$C$ is in the interior of $\angle DBE$	Given
$m\angle ABD = 25$	Given
$m\angle CBE = 25$	Given
$m\angle DBC$ is a real number	Protractor Postulate
$m\angle ABD + m\angle DBC = m\angle ABC$	Angle Addition Postulate
$m\angle CBE + m\angle DBC = m\angle DBE$	Angle Addition Postulate
$25 + m\angle DBC = m\angle ABC$	Substitution Property of Equality
$25 + m\angle DBC = m\angle DBE$	Substitution Property of Equality
$m\angle ABC = m\angle DBE$	Substitution Property of Equality

Adjacent angles share a vertex and one side and have no interior points in common. Their other two sides are the noncommon sides. A linear pair is two adjacent angles whose noncommon sides are opposite rays, which means the two sides have the same endpoint and together form a line. Those sides form a straight angle of $180^\circ$ , and the shared side splits it into two parts, so the two measures add to $180$ .

That reasoning gives a theorem, a statement that has been proved. The Supplement Theorem says: if two angles form a linear pair, then they are supplementary, meaning their measures add to $180$ . In a proof, use two reasons in order: the theorem gives you the word supplementary, and the Definition of supplementary angles gives you the equation. Try it in this short proof.

Given: $\angle ABD$ and $\angle DBC$ form a linear pair.

Prove: $m\angle DBC = 180 - m\angle ABD$ .

Points

A

,

B

and

C

lie on line

AC

with

B

between

A

and

C

, so rays

BA

and

BC

are opposite rays. Ray

BD

is the shared side of the linear pair,

\angle ABD

and

\angle DBC

.

A proof that cites the Supplement Theorem

Statement	Reason
$\angle ABD$ and $\angle DBC$ form a linear pair	Given
$\angle ABD$ and $\angle DBC$ are supplementary	Supplement Theorem
$m\angle ABD + m\angle DBC = 180$	Definition of supplementary angles
$m\angle DBC = 180 - m\angle ABD$	Subtraction Property of Equality

Worked example

Use the Supplement Theorem to find measures

Angles $ABD$ and $DBC$ form a linear pair, as in the figure above, with $m\angle ABD = 2x + 30$ and $m\angle DBC = 3x + 20$ . Find $x$ and both measures.

Reason: Supplement Theorem. The linear pair is stated, so you use the theorem in its stated direction: from linear pair to supplementary.
$\angle ABD \text{ and } \angle DBC \text{ are supplementary}$
Reason: Definition of supplementary angles. Supplementary angles have measures that add to 180.
$m\angle ABD + m\angle DBC = 180$
Reason: Substitution Property of Equality. Replace each measure with its stated expression.
$(2x + 30) + (3x + 20) = 180$
Reason: Substitution Property of Equality. Replace 2x + 3x with 5x, and replace 30 + 20 with 50.
$5x + 50 = 180$
Reason: Subtraction Property of Equality. Subtract 50 from both sides.
$5x = 130$
Reason: Division Property of Equality. Divide both sides by 5.
$x = 26$
Reason: Substitution Property of Equality. Replace x with 26 in each stated expression to get the two measures.
$m\angle ABD = 2(26) + 30 = 82,\quad m\angle DBC = 3(26) + 20 = 98$
Check the solved measures. The Protractor Postulate requires each one to be greater than 0 and at most 180, and both are. Their sum is also 180, so keep x = 26.
$0 < 82 \le 180,\quad 0 < 98 \le 180,\quad 82 + 98 = 180$

Answer

The value of $x$ is $26$ , so angle $ABD$ measures $82^\circ$ and angle $DBC$ measures $98^\circ$ .

The Complement Theorem is the right-angle partner of the Supplement Theorem: if the noncommon sides of two adjacent angles form a right angle, with the shared side inside that right angle, then the angles are complementary, meaning their measures add to $90$ . The two angles fill the right angle exactly.

You can prove this theorem with the Angle Addition Postulate. After that, you may cite it by name.

Given: $\angle ABC$ is a right angle, and $D$ is in the interior of $\angle ABC$ .

Prove: $\angle ABD$ and $\angle DBC$ are complementary.

The square mark states that

\angle ABC

is a right angle, and the Given states that

D

is in its interior. Ray

BD

is the shared side of

\angle ABD

and

\angle DBC

; rays

BA

and

BC

are their noncommon sides.

A proof of the Complement Theorem

Statement	Reason
$\angle ABC$ is a right angle	Given
$D$ is in the interior of $\angle ABC$	Given
$m\angle ABC = 90$	Definition of right angle
$m\angle ABD + m\angle DBC = m\angle ABC$	Angle Addition Postulate
$m\angle ABD + m\angle DBC = 90$	Substitution Property of Equality
$\angle ABD$ and $\angle DBC$ are complementary	Definition of complementary angles

Worked example

Use the Complement Theorem and reject a value

In the figure above, $\angle ABC$ is a right angle and $D$ is in the interior of $\angle ABC$ . Also, $m\angle ABD = x^2$ and $m\angle DBC = 9x$ . Find the measures of $\angle ABD$ and $\angle DBC$ .

Reason: Complement Theorem. D is in the interior of right angle ABC, so angles ABD and DBC are adjacent, and their noncommon sides form that right angle. You use the theorem in its stated direction.
$\angle ABD \text{ and } \angle DBC \text{ are complementary}$
Reason: Definition of complementary angles. Complementary angles have measures that add to 90.
$m\angle ABD + m\angle DBC = 90$
Reason: Substitution Property of Equality. Replace each measure with its stated expression.
$x^2 + 9x = 90$
Reason: Subtraction Property of Equality. Subtract 90 from both sides so that one side is 0.
$x^2 + 9x - 90 = 0$
Factor the left side. The numbers 15 and -6 multiply to -90 and add to 9.
$(x + 15)(x - 6) = 0$
A product is 0 only when one of its factors is 0, so set each factor equal to 0 and solve. This gives two values to test.
$x + 15 = 0 \text{ or } x - 6 = 0,\ \text{ so } x = -15 \text{ or } x = 6$
Test x = -15 in both expressions. The Protractor Postulate requires a measure greater than 0 and at most 180, but one result is negative and the other is more than 180. Reject x = -15.
$m\angle ABD = (-15)^2 = 225,\quad m\angle DBC = 9(-15) = -135$
Test x = 6 in both expressions. Both results are greater than 0 and at most 180, and their sum is 90. Keep x = 6.
$m\angle ABD = 6^2 = 36,\quad m\angle DBC = 9(6) = 54$

Answer

Reject $x = -15$ and keep $x = 6$ , so angle $ABD$ measures $36^\circ$ and angle $DBC$ measures $54^\circ$ .

Key concept

A stated fact, a named reason, an equation

Each reason here turns a stated fact about angles into an equation. Cite it only when its condition is stated, never because of how a figure looks.

$D$ is in the interior of $\angle ABC$ : the Angle Addition Postulate gives $m\angle ABD + m\angle DBC = m\angle ABC$ .
Two angles form a linear pair: the Supplement Theorem and the Definition of supplementary angles give a sum of $180$ .
Two adjacent angles have noncommon sides that form a right angle, with the shared side inside it: the Complement Theorem and the Definition of complementary angles give a sum of $90$ .

Every solved measure must then be greater than $0$ and at most $180$ .

Your takeaway: an angle proof moves one named reason at a time, starting from stated facts and never from the look of a figure. When you solve for a measure, you also check it. Next, you will prove and use the Vertical Angles Theorem and use theorems about right angles and perpendicular lines.