Angle Bisectors and Angle Algebra | Geometry | LevelKite

Picture an angle cut from paper. Fold it through its vertex, the point where its two rays start, so that one ray lands exactly on the other. The crease splits the angle into two matching parts.

To bisect an angle means to divide it into two angles of equal measure. The ray along the crease is called an angle bisector. It lies in the angle's interior, the region between the angle's two rays. In this lesson, $m\angle ABC$ means the measure of $\angle ABC$ in degrees.

Rays

BA

and

BC

form

\angle ABC

. Ray

BD

starts at vertex

B

and makes

\angle ABD

and

\angle DBC

. This drawing alone does not tell you that those two angles are equal.

Key concept

What an angle bisector is

An angle bisector is a ray that starts at an angle's vertex, lies in its interior, and divides it into two angles of equal measure. The converse of a statement swaps its if and then parts. A converse is not always true, but a definition holds both ways.

Forward: if ray $BD$ bisects $\angle ABC$ , then $D$ lies in the interior of $\angle ABC$ and $m\angle ABD = m\angle DBC$ .
Reverse (converse): if $D$ lies in the interior of $\angle ABC$ and $m\angle ABD = m\angle DBC$ , then ray $BD$ bisects $\angle ABC$ .

These facts must be stated in the text, because a figure is not drawn to scale and equal angles are never read from it. Every angle measure must be greater than $0^\circ$ and at most $180^\circ$ , and every segment length must be positive.

\text{ray } BD \text{ bisects } \angle ABC \;\Rightarrow\; m\angle ABD = m\angle DBC

To identify a bisector, test the facts the text states, not the look of the drawing, because a figure is not drawn to scale. You need two facts together: $D$ lies in the interior of the angle, and the two parts have equal measure. Equal measures alone are not enough, as the second row shows. All measures in the table are greater than $0^\circ$ and at most $180^\circ$ , as every angle measure must be.

Facts stated in the text	Does ray BD bisect angle ABC?	Reason
$D$ is in the interior of $\angle ABC$ , and $m\angle ABD = m\angle DBC = 25^\circ$ .	Yes	Both facts hold, so the reverse direction of the definition applies.
$D$ is not in the interior of $\angle ABC$ , $m\angle ABC = 60^\circ$ , and $m\angle ABD = m\angle DBC = 150^\circ$ .	No	Ray BD points away from the angle, so the interior fact fails and the definition is not met.
Only a drawing that looks evenly split, with no measures or position of $D$ stated.	Cannot tell	A figure is not drawn to scale, so you cannot read equal angles from it.

Worked example

Use a bisector with numbers

Given: ray $BD$ bisects $\angle ABC$ , and $m\angle ABD = 34^\circ$ . Find: $m\angle DBC$ and $m\angle ABC$ .

The text states that ray BD bisects angle ABC, so use the definition of angle bisector in the forward direction: angle DBC and angle ABD have equal measure. Then substitute the given 34° (Substitution Property of Equality).
$m\angle DBC = m\angle ABD = 34^\circ$
The forward direction also says D lies in the interior of angle ABC, so the Angle Addition Postulate applies. Add the measures of the two parts to get the whole.
$m\angle ABC = m\angle ABD + m\angle DBC = 34^\circ + 34^\circ = 68^\circ$
Check the conditions: every angle measure must be greater than 0° and at most 180°. Both 34° and 68° meet them, so you keep both.
$0^\circ < 34^\circ < 68^\circ \le 180^\circ$

Answer

The measure of $\angle DBC$ is $34^\circ$ , and the measure of $\angle ABC$ is $68^\circ$ .

Sometimes the text gives one part and the whole, so you must find the other part before you can test it. In the next example, the drawing looks evenly split, but it is not drawn to scale, so that look proves nothing. The same drawing serves two different cases, and only the stated measures decide each one.

In both cases, angle A B D measures 38°. In case (a), angle A B C measures 76°; in case (b), it measures 80°. One drawing fits both because it is not to scale.

Worked example

Identify a bisector from stated measures

Given: $D$ lies in the interior of $\angle ABC$ , and angle A B D measures 38°. Decide whether ray $BD$ bisects $\angle ABC$ in each case: (a) $m\angle ABC = 76^\circ$ ; (b) $m\angle ABC = 80^\circ$ .

Case (a). D is given in the interior of angle ABC, so the Angle Addition Postulate applies. Substitute the given measures.
$38^\circ + m\angle DBC = 76^\circ$
Subtract 38° from both sides (Subtraction Property of Equality).
$m\angle DBC = 38^\circ$
Compare the parts. Both measure 38°, which is greater than 0° and at most 180°, and D is in the interior. Both facts hold, so the reverse direction of the definition says ray BD bisects angle ABC.
$m\angle ABD = m\angle DBC = 38^\circ$
Case (b). Use the Angle Addition Postulate the same way, then subtract 38° from both sides (Subtraction Property of Equality).
$38^\circ + m\angle DBC = 80^\circ \;\Rightarrow\; m\angle DBC = 42^\circ$
Compare the parts. Both 38° and 42° are greater than 0° and at most 180°, but they are not equal. The forward direction of the definition says a bisector always makes equal parts, so ray BD is not a bisector here.
$m\angle ABD = 38^\circ \ne 42^\circ = m\angle DBC$

Answer

In case (a), ray $BD$ bisects $\angle ABC$ . In case (b), ray $BD$ does not bisect $\angle ABC$ , even though the drawing is the same.

Often the measures are written as expressions with a variable. When the text states that ray $BD$ bisects $\angle ABC$ , you can write either of two equations. Never write them just because a drawing looks even, since a figure is not drawn to scale.

Part equals part: $m\angle ABD = m\angle DBC$ , by the definition of angle bisector (forward direction).
Twice a part equals the whole: $D$ lies in the interior, so $m\angle ABD + m\angle DBC = m\angle ABC$ by the Angle Addition Postulate. Replacing $m\angle DBC$ with $m\angle ABD$ (Substitution Property of Equality) gives $m\angle ABD + m\angle ABD = m\angle ABC$ , so $2\,m\angle ABD = m\angle ABC$ .

Pick the equation that matches the expressions you are given. Solving for $x$ does not finish the problem, because $x$ is not a measure. Substitute its value back, then check that every measure is greater than $0^\circ$ and at most $180^\circ$ .

Worked example

Set the two parts equal

Given: ray $BD$ bisects $\angle ABC$ , $m\angle ABD = (6x - 13)^\circ$ , and $m\angle DBC = (4x + 5)^\circ$ . Find: $x$ , $m\angle ABD$ , $m\angle DBC$ , and $m\angle ABC$ .

The text states that ray BD bisects angle ABC, and the two expressions name its parts. The definition of angle bisector (forward direction) says the parts have equal measure, so set the expressions equal. Both are in degrees, so you can work with the numbers alone.
$m\angle ABD = m\angle DBC \;\Rightarrow\; 6x - 13 = 4x + 5$
Subtract 4x from both sides (Subtraction Property of Equality).
$2x - 13 = 5$
Add 13 to both sides (Addition Property of Equality).
$2x = 18$
Divide both sides by 2 (Division Property of Equality).
$x = 9$
Substitute x = 9 into each expression (Substitution Property of Equality). Both parts come out equal, which agrees with the definition.
$m\angle ABD = (6(9) - 13)^\circ = 41^\circ,\quad m\angle DBC = (4(9) + 5)^\circ = 41^\circ$
D lies in the interior because ray BD is a bisector (forward direction), so the Angle Addition Postulate gives the whole. Check: 41° and 82° are each greater than 0° and at most 180°, so you keep x = 9.
$m\angle ABC = 41^\circ + 41^\circ = 82^\circ$

Answer

The value of $x$ is $9$ , $m\angle ABD = m\angle DBC = 41^\circ$ , and $m\angle ABC = 82^\circ$ .

Worked example

Use a part and the whole

Given: ray $BD$ bisects $\angle ABC$ , $m\angle ABD = (3x + 6)^\circ$ , and $m\angle ABC = (2x + 60)^\circ$ . Find: $x$ , $m\angle ABD$ , and $m\angle ABC$ .

One expression is a part and the other is the whole, so use twice a part equals the whole. It comes from the definition of angle bisector (forward direction) and the Angle Addition Postulate. Substitute the given expressions (Substitution Property of Equality).
$2\,m\angle ABD = m\angle ABC \;\Rightarrow\; 2(3x + 6) = 2x + 60$
Apply the Distributive Property.
$6x + 12 = 2x + 60$
Subtract 2x from both sides (Subtraction Property of Equality).
$4x + 12 = 60$
Subtract 12 from both sides (Subtraction Property of Equality).
$4x = 48$
Divide both sides by 4 (Division Property of Equality).
$x = 12$
Substitute x = 12 (Substitution Property of Equality), then check. Twice the part matches the whole, since 42° + 42° = 84°. Both measures are greater than 0° and at most 180°, so you keep x = 12.
$m\angle ABD = (3(12) + 6)^\circ = 42^\circ,\quad m\angle ABC = (2(12) + 60)^\circ = 84^\circ$

Answer

The value of $x$ is $12$ , $m\angle ABD = 42^\circ$ , and $m\angle ABC = 84^\circ$ .

Judge a solution by the measures it gives, not by the sign of $x$ . A negative $x$ is fine when every measure is greater than $0^\circ$ and at most $180^\circ$ . If any measure is $0^\circ$ or less, or more than $180^\circ$ , reject the value, because no such angle exists. A segment length is checked the same way: it must be positive.

In each row below, the text states that ray $BD$ bisects $\angle ABC$ . The definition (forward direction) sets the parts equal, the Subtraction and Addition Properties of Equality solve the equation, and the Angle Addition Postulate gives the whole.

Stated measures of the two parts	Solve the equation	Measures found	Decision
$m\angle ABD = (x + 20)^\circ$ , $m\angle DBC = (2x + 30)^\circ$	$x + 20 = 2x + 30$ , so $20 = x + 30$ , so $x = -10$	Each part $10^\circ$ ; whole $20^\circ$	Keep: all are greater than $0^\circ$ and at most $180^\circ$ .
$m\angle ABD = (x + 5)^\circ$ , $m\angle DBC = (2x + 15)^\circ$	$x + 5 = 2x + 15$ , so $5 = x + 15$ , so $x = -10$	Each part $-5^\circ$	Reject: a measure must be greater than $0^\circ$ .
$m\angle ABD = (2x + 10)^\circ$ , $m\angle DBC = (3x - 50)^\circ$	$2x + 10 = 3x - 50$ , so $10 = x - 50$ , so $x = 60$	Each part $130^\circ$ ; whole $260^\circ$	Reject: the whole must be at most $180^\circ$ .

Takeaway: you know a ray is an angle bisector from stated facts, never from the look of a drawing, because a figure is not drawn to scale. It starts at the vertex, lies in the interior, and makes two parts of equal measure. When the text gives you a bisector, write part equals part or twice a part equals the whole, solve for $x$ , substitute back, and keep only measures greater than $0^\circ$ and at most $180^\circ$ .