| 41 |
When two line segments meet, they form an
angle.
We ordinarily think of an angle as formed by two rays,
$OA$ and $OB,$ that extend from a common point $O$
called the
vertex.
The rays are called
the sides of the angle
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| 42 |
If we think of the ray $OA$ as being fixed
and $OB$ as rotating about the vertex, $OA$ is called the
initial side
and $OB$ the
terminal side.
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| 42 |
The amount of rotation of its terminal side is the
measure of an angle.
Two angles have
equal size (measure)
if they are formed by the same degree of rotation.
Note the wording, "equal size" not "equal angles".
Angle size is usually measured in degress or radians.
The measure of an angle is not unique because there
are many possible rotations of its terminal side
to achieve the same angle. However, if a measure of
an angle $A$ is given, we may denote it $m(A).$
Just remember, another measure $m'(A)\ne m(A)$ is
also possible for the same angle $A.$ The notation
$m(A)$ refers to just one of these measures.
The distinction between an angle and one of its
measures is often blurred so that, given a measure
$m(A)$ we write $A=m(A)$, such as
$A=180^\circ.$
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| 42 |
degree:
A unit of angular measure defined as $\frac{1}{360}$
of the measure of an angle formed by one complete
revolution of the terminal side about its vertex
(360 degrees). A positive degree indicates a
counterclockwise rotation of the terminal side, a
negative degree a clockwise one.
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| 42 |
radian:
A unit of angular measure defined as
the measure of an angle whose vertex is at the
center of a circle of radius $r$ (i.e. a central
angle) and whose rays subtend an arc on the
circle whose length is $r.$
|
| 42 |
straight angle:
An angle of $180^\circ,$ or of
half of one complete revolution of the terminal
side about the angle’s vertex, i.e. half
of $360^\circ.$
|
| 42 |
right angle:
An angle of $90^\circ,$ or of one fourth of one complete
revolution of the terminal side about the angle’s vertex,
i.e. one fourth of $360^\circ.$
|
| 42 |
acute angle:
An angle smaller in size than a right angle.
|
| 42 |
obtuse angle:
An angle larger than a right angle but smaller than a straight angle.
|
| 43 |
congruent, coterminal angles.
Angles $A$ and $B$ are said to be coterminal
(congruent) if
\[
\begin{align*}
m(A) &\equiv m(B)\pmod{360^\circ}\\
m(A) &\equiv m(B)\pmod{2\pi}
\end{align*}
\]
or equivalently, if
\[
\begin{align*}
m\left(A\right) &=m\left(B\right)+360^\circ n\\
m\left(A\right) &=m\left(B\right)+2\pi n
\end{align*}
\]
for some integer $n,$ and we write
\[
A\equiv B
\]
Note the distinction between equal measures and coterminal
(congruent) angles. For example, if $\alpha=m(A)=0$ and
$\beta=m(B)=2\pi$
then
\[
A\equiv B
\]
since
\[
\alpha\equiv\beta\pmod{2\pi}
\]
but
\[
\alpha\ne\beta
\]
since
\[
0\ne2\pi
\]
In this discussion, as in the book, "equal angles"
have not been defined, but in practice, we forgive
writing
\(
A=B
\)
instead of
\(
A\equiv B
\)
so that equality of angles is synonymous with
congruence. That is, when it is understood that
$A$ and $B$ refer to the angles themselves and not
their measures.
Another way to say this, given in the book,
is that two angles with the same initial and
terminal sides are coterminal (congruent),
leaving meaning of "sameness" to the reader.
In general, coterminal angles differ in measure
by an integral multiple of
$360^\circ$ or $2\pi\text{ radians}.$
|
| 43 |
positive angle.
If the angle is measured through
counterclockwise rotation of the terminal side
then it's sign is positive.
|
| 43 |
negative angle.
If the angle is measured through
clockwise rotation of the terminal side
then it's sign is negative.
|
| 44 |
Principal angle.
The measure of an angle $A$ is not unique.
Given one of these measures,
$m\left(A\right),$
we may express its set of all possible measures
by
\[
m\left(A\right)+2\pi n
\]
for every integer $n.$ Let
\begin{equation}
\label{eq_PrincipalAngle}
m'(A)=m\left(A\right)+2\pi n
\end{equation}
such that $n$ satisfies
\begin{equation}
\label{eq_PrincipalAnglePredicate}
0\le m\left(A\right)+2\pi n\lt2\pi
\end{equation}
That the linear inequality
\eqref{eq_PrincipalAnglePredicate}
has a unique solution in $n$ is guaranteed by
algebra. Thus, angle $A$ is guaranteed to have
a unique measure between $0$ and $2\pi,$
denoted by $m'(A)$
in equation \eqref{eq_PrincipalAngle}.
Allowing
$m(A')=m'(A)$ then $A'$ is known as
the principal angle
of $A$ and we may restate equation \eqref{eq_PrincipalAngle}
as
\begin{equation}
\label{eq_PrincipalAngleRevised}
m(A')=m\left(A\right)+2\pi n
\end{equation}
subject to \eqref{eq_PrincipalAnglePredicate}.
To find the measure of the principal angle $A'$, first find
the integer $n$ that satisfies the inequality
\[
-\frac{m\left(A\right)}{2\pi}\le n\lt\frac{2\pi-m\left(A\right)}{2\pi}
\]
-
If $n\gt0$ then $n$ is the integer part of
$-\frac{m\left(A\right)}{2\pi},$
the left-hand term.
-
If $n\lt0$ then $n$ is the integer part of
$\frac{2\pi-m\left(A\right)}{2\pi},$
the right-hand term.
Next, substitute that value of $n$ in equation \eqref{eq_PrincipalAngleRevised}
and compute $m(A')$.
To solve for degrees, substitute
"$360^\circ$"
for
"$2\pi$"
in the steps given.
|
| 45 |
complementary angle:
Two angles are complementary if and only if the
sum of their measures is $90^\circ.$
|
| 45 |
supplementary angle:
Two angles are supplementary if and only if the
sum of their measures is $180^\circ.$
|
| 45 |
minute:
A unit of angular measure equal to $\frac{1}{60}^\circ,$
normally written as $1'.$
|
| 45 |
second:
A unit of angular measure equal to
$\frac{1}{60}^\prime$,
normally written as $1''.$
|
| 46 |
standard position:
An angle $A$ is in standard position if
its vertex is at the origin and its initial
side is on the positive half of the
$x$-axis.
The measure of such an angle is taken to be the
positive measure from the initial side (the
positive
$x$-axis)
to the terminal side.
|
| 47 |
first quadrant angle:
An angle whose terminal side is in the first quadrant.
|
| 47 |
second quadrant angle:
An angle whose terminal side is in the second quadrant.
|
| 47 |
third quadrant angle:
An angle whose terminal side is in the third quadrant.
|
| 47 |
fourth quadrant angle:
An angle whose terminal side is in the fourth quadrant.
|
| 47 |
quadrantal angle:
An angle whose terminal side does not lie in any quadrant but lies on one of the axes.
|
| 49 |
equiangular triangle:
A triangle whose angles are exactly the same.
|
| 49 |
equilateral triangle:
A triangle whose sides all have the same length.
|
| 49 |
isosceles triangle:
A triangle two of whose sides are equal in length.
|
| 49 |
right triangle:
A triangle one of whose angles is a right angle.
|
| 49 |
oblique triangle:
A triangle none of whose angles are right angles.
|
|
opposite side:
The side of a triangle connecting the two sides adjacent to the vertex to which the side is opposite.
|
| 50 |
adjacent side:
A side of a triangle that ends at the vertex to which it is adjacent.
|
| 50 |
hypotenuse:
The opposite to the vertex of the right angle in a right triangle.
|
| 51 |
Special Right Triangles
The $30^\circ\text{-}60^\circ$ Right Triangle
The ratios of the sides are
\[
\quad 1:\sqrt{3}:2
\]
where the first side is opposite the $30^\circ$
angle, the second side is adjacent the $30^\circ$
angle, and the third side is the hypotenuse.
The $45^\circ\text{-}45^\circ$ Right Triangle
The ratios of the sides are
\[
\quad 1:1:\sqrt{2}
\]
where the first and second sides are opposite
the $45^\circ$ angles and the third side is
the hypotenuse.
|
| 52 |
Pythagorean triple:
Three integers that have the property that the sum of the
square of two of the integers equals the square
of the remaining integer.
|
| 53 |
similar triangles:
Two triangles that have the same shape (not the same
size), i.e. whose angles are equal.
|
Alex Notes