Name, Measure, and Add Angles | Geometry | LevelKite

An angle describes an opening between two directions that leave the same point. To say this exactly, start with a ray.

A ray is the part of a line that starts at one point, called its endpoint, and goes on forever in one direction. You name a ray with its endpoint first: $\overrightarrow{BA}$ starts at $B$ and passes through $A$ .

An angle is a figure made of two different rays with the same endpoint. That shared endpoint is the vertex of the angle, and the two rays are its sides. The figure below shows one angle, with its vertex at $B$ .

Figures in this lesson are not drawn to scale. Only the marks on a figure and the facts stated in the text are given, so you never read a measure, a right angle or equal angles from how a drawing looks.

Rays

\overrightarrow{BA}

and

\overrightarrow{BC}

are the sides of

\angle ABC

, and

B

is its vertex. No measure is given for this angle.

You already use letters to name points and lines. You can name an angle in three ways, and the first two use those letters.

Three letters: write a point on one side, then the vertex, then a point on the other side. The vertex always goes in the middle. In the figure above, $\angle ABC$ and $\angle CBA$ name the same angle. But $\angle BAC$ does not name it, because its middle letter says the vertex is $A$ .
Vertex only: write $\angle B$ , but only when exactly one angle in the figure has vertex $B$ . That is true in the figure above.
Number: a figure may place a number inside the opening, and then $\angle 1$ names that angle.

Three rays start at

B

:

\overrightarrow{BA}

,

\overrightarrow{BD}

and

\overrightarrow{BC}

. No measures are given.

Worked example

Name the angles at one vertex

In the figure above, $\overrightarrow{BA}$ , $\overrightarrow{BD}$ and $\overrightarrow{BC}$ all start at $B$ . Name every angle formed by two of these rays. Then decide whether $\angle B$ is a clear name.

Find the vertex. The vertex of an angle is the shared endpoint of its sides, and all three rays start at B, so B is the vertex of every angle here.
Pair up the rays, because by definition an angle is made of two different rays with the same endpoint. Three pairs are possible: rays BA and BD, rays BD and BC, and rays BA and BC.
Write a name for each pair, with the vertex B in the middle and a point from each side on the outside.
$\angle ABD,\quad \angle DBC,\quad \angle ABC$
Test the name angle B. A vertex-only name is allowed only when exactly one angle has that vertex. Three angles have vertex B here, so that name does not say which one you mean.

Answer

The three angles are $\angle ABD$ , $\angle DBC$ and $\angle ABC$ , and $\angle B$ is not a clear name because all three have vertex $B$ .

The measure of an angle is the size of the turn from one side to the other, going the shorter way around the vertex. You write $m\angle ABC$ for the measure of $\angle ABC$ .

Measures use degrees. One degree, $1^\circ$ , is $\frac{1}{360}$ of a full turn, so a quarter turn is $90^\circ$ and a half turn is $180^\circ$ . In this course, an angle measure is always greater than $0^\circ$ and at most $180^\circ$ .

A postulate is a statement accepted without proof. The Protractor Postulate turns the idea of a turn into numbers.

Choose a line through the vertex, and choose one side of that line. A protractor, the tool for measuring angles, has a straight edge that plays the part of this line. Each ray from the vertex that lies on the line or on the chosen side matches exactly one number from $0$ to $180$ , and each number matches exactly one such ray. The measure of an angle formed by two of these rays is the absolute value of the difference of their numbers. Absolute value means distance from zero, so the order of subtraction does not matter. If the rays match $r$ and $s$ , the measure in degrees is $|r - s|$ .

In this lesson the protractor readings are stated in the text, because the drawings are not to scale.

You classify an angle by its measure, using the definitions in the table below. A definition works in both directions: if $m\angle ABC = 90^\circ$ , then $\angle ABC$ is a right angle, and if $\angle ABC$ is a right angle, then $m\angle ABC = 90^\circ$ . The same holds for every row.

A small square drawn at a vertex is a right-angle mark. It states that the angle is a right angle. Classify only from a measure you are given or have found from stated facts, or from a mark, never from how a drawing looks.

Angle types by measure

Angle type	Measure	Meaning
Acute	Greater than $0^\circ$ and less than $90^\circ$	Less than a quarter turn
Right	Exactly $90^\circ$	Exactly a quarter turn
Obtuse	Greater than $90^\circ$ and less than $180^\circ$	More than a quarter turn and less than a half turn
Straight	Exactly $180^\circ$	Exactly a half turn, so the two sides form a line

Worked example

Measure and classify from protractor readings

A protractor is centered at $B$ . On one scale, $\overrightarrow{BA}$ reads $30^\circ$ , $\overrightarrow{BD}$ reads $75^\circ$ and $\overrightarrow{BC}$ reads $150^\circ$ . Find the measures of $\angle ABD$ , $\angle DBC$ and $\angle ABC$ , and classify each angle.

Apply the Protractor Postulate to angle ABD. Its sides are rays BA and BD, so take the absolute value of the difference of their readings.
$m\angle ABD = |75^\circ - 30^\circ| = 45^\circ$
Apply the Protractor Postulate to angle DBC, whose sides are rays BD and BC.
$m\angle DBC = |150^\circ - 75^\circ| = 75^\circ$
Apply the Protractor Postulate to angle ABC, whose sides are rays BA and BC.
$m\angle ABC = |150^\circ - 30^\circ| = 120^\circ$
Check the values. Each measure is greater than 0° and at most 180°, so all three are valid.
$0^\circ < 45^\circ < 75^\circ < 120^\circ \le 180^\circ$
Classify with the definitions in the table, read from measure to type. Measures of 45° and 75° are greater than 0° and less than 90°, so those angles are acute. A measure of 120° is greater than 90° and less than 180°, so that angle is obtuse.

Answer

The measures are $m\angle ABD = 45^\circ$ and $m\angle DBC = 75^\circ$ , so both angles are acute, and $m\angle ABC = 120^\circ$ , so $\angle ABC$ is obtuse.

Look at the three measures you just found: $45^\circ + 75^\circ = 120^\circ$ . The reading for $\overrightarrow{BD}$ lies between the readings for the sides of $\angle ABC$ , so $\overrightarrow{BD}$ splits that opening into two smaller angles, and their measures add up to the whole. One example is not a proof, and geometry accepts the general statement as a postulate.

To state it exactly, you need one more term. The interior of $\angle ABC$ , when its measure is less than $180^\circ$ , is the region inside its opening: every point on the same side of line $BA$ as $C$ and on the same side of line $BC$ as $A$ . A straight angle has no such region. Whether a point is in the interior must come from the text, because you do not read it from a drawing.

Key concept

Angle Addition Postulate

If point $D$ is in the interior of $\angle ABC$ , then the measures of the two smaller angles add up to the measure of the whole angle. A postulate is accepted without proof, and this one holds only under its condition: $D$ must be stated to be in the interior. Check every measure you solve for: it must be greater than $0^\circ$ and at most $180^\circ$ , and you reject a value that fails.

m\angle ABD + m\angle DBC = m\angle ABC

The square at

B

states that angle A B D is a right angle, and angle D B C measures 40°. The text states that

D

is in the interior of angle A B C. The figure is not to scale.

Worked example

Find the whole angle

Given: in the figure above, the square mark states that $\angle ABD$ is a right angle, $m\angle DBC = 40^\circ$ , and $D$ is in the interior of $\angle ABC$ . Find: $m\angle ABC$ , and classify $\angle ABC$ .

The square mark states that angle ABD is a right angle. The definition of a right angle, read from type to measure, gives its measure as 90°.
$m\angle ABD = 90^\circ$
Use the Angle Addition Postulate. It applies because D is given to be in the interior of angle ABC.
$m\angle ABD + m\angle DBC = m\angle ABC$
Replace each known measure with its value, which the Substitution Property of Equality allows.
$90^\circ + 40^\circ = m\angle ABC$
Add the two measures on the left side.
$m\angle ABC = 130^\circ$
Check and classify. 130° is greater than 0° and at most 180°, so it is a valid measure. It is also greater than 90° and less than 180°, so the definition of an obtuse angle, read from measure to type, says angle ABC is obtuse.
$90^\circ < 130^\circ < 180^\circ$

Answer

The measure of $\angle ABC$ is $130^\circ$ , so $\angle ABC$ is obtuse.

Worked example

Find a missing part

Given: $D$ is in the interior of $\angle ABC$ , $m\angle ABC = 125^\circ$ , and $m\angle ABD = 73^\circ$ . Find: $m\angle DBC$ , and classify $\angle DBC$ .

Use the Angle Addition Postulate, since D is given to be in the interior of angle ABC.
$m\angle ABD + m\angle DBC = m\angle ABC$
Substitute the given measures, by the Substitution Property of Equality.
$73^\circ + m\angle DBC = 125^\circ$
Subtract 73° from both sides, which keeps the equation true by the Subtraction Property of Equality.
$73^\circ + m\angle DBC - 73^\circ = 125^\circ - 73^\circ$
Simplify each side.
$m\angle DBC = 52^\circ$
Check and classify. 52° is greater than 0° and at most 180°, so you keep it. It is also less than 90°, so by definition the angle is acute.
$0^\circ < 52^\circ < 90^\circ$

Answer

The measure of $\angle DBC$ is $52^\circ$ , so $\angle DBC$ is acute.

Two checks keep your work correct when you use the Angle Addition Postulate.

Check the condition. Suppose that on one protractor scale, $\overrightarrow{BA}$ reads $30^\circ$ , $\overrightarrow{BC}$ reads $70^\circ$ and $\overrightarrow{BD}$ reads $120^\circ$ . The reading for $\overrightarrow{BD}$ is not between the other two, so $D$ is not in the interior of $\angle ABC$ . The sum fails here: $m\angle ABD + m\angle DBC = 90^\circ + 50^\circ = 140^\circ$ , but $m\angle ABC = 40^\circ$ .

Check the value. Suppose you are told that $D$ is in the interior of $\angle ABC$ , $m\angle ABC = 60^\circ$ and $m\angle ABD = 75^\circ$ . Subtracting gives $m\angle DBC = -15^\circ$ . You reject this value because an angle measure must be greater than $0^\circ$ , so these stated facts cannot all be true.

An angle is two rays with a shared vertex. Name it with the vertex in the middle, find its measure from stated readings or marks, and classify it by that measure. When a point is stated to be in an angle's interior, the two smaller measures add up to the whole.

Next, you will identify and use angle bisectors and find angle measures using algebra.