An angle is two rays that share an endpoint, called the vertex. You know angle pairs from Complementary, Supplementary and Vertical Angles; now we name and measure angles with the same care we used for segments in Points, Lines and Planes.
- Name an angle by three points with the vertex in the middle: $\angle ABC$ has its vertex at $B$. Use one letter, $\angle B$, only when no other angle shares that vertex.
- $\angle ABC$ is the figure; $m\angle ABC$ is its measure in degrees. Two angles are congruent ($\angle A \cong \angle B$) exactly when their measures are equal.
- Acute: less than $90^\circ$. Right: $90^\circ$. Obtuse: between $90^\circ$ and $180^\circ$. Straight: $180^\circ$.
Why it works
Measuring an angle is like reading a ruler that has been bent into a half circle. A protractor places ray $BA$ at one reading and ray $BC$ at another; the measure is the difference. If ray $BD$ lies between them, its reading is in the middle, so the two smaller differences add to the big one. That is the Angle Addition Postulate, the angle version of segment addition.
Key idea
If $D$ is in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$. A ray $\overrightarrow{BD}$ bisects $\angle ABC$ when it splits it into two congruent angles, each half of the whole.
Worked example: Angle addition
$D$ is in the interior of $\angle ABC$, $m\angle ABC = 112^\circ$ and $m\angle ABD = 47^\circ$. Find $m\angle DBC$.
- By the Angle Addition Postulate, ${}47 + m\angle DBC = 112$.
- $m\angle DBC = 112 - 47 = 65^\circ$.
Answer: $65^\circ$
Worked example: A bisector with algebra
$\overrightarrow{BD}$ bisects $\angle ABC$, $m\angle ABD = (4x + 6)^\circ$ and $m\angle DBC = (6x - 20)^\circ$. Find $m\angle ABC$.
- A bisector makes two congruent angles: $4x + 6 = 6x - 20$.
- Subtract $4x$ and add 20: $26 = 2x$, so $x = 13$.
- Each half is $4(13) + 6 = 58^\circ$, so $m\angle ABC = 2 \cdot 58 = 116^\circ$.
Answer: $116^\circ$
Common mistake
Answering with $x$ when the question asks for an angle. In the example, $x = 13$ is only a stepping stone; the angle is $116^\circ$. Reread the question before you write the answer. Another slip: naming an angle $\angle BAC$ when the vertex is $B$. The middle letter must be the vertex.