10.9.1: Key Concepts

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10.1 Points, Lines, and Planes

Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles associated with the line, the point, and the plane.
Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
The union of two sets, $A$ and $B$ , contains all points that are in both sets and is symbolized as $A \cup B .$
The intersection of two sets $A$ and $B$ includes only the points common to both sets and is symbolized as $A \cap B .$

10.2 Angles

Angles are classified as acute if they measure less than $90^{\circ},$ obtuse if they measure greater than $90^{\circ}$ and less than $180^{\circ},$ right if they measure exactly $90^{\circ},$ and straight if they measure exactly $180^{\circ} .$
If the sum of angles equals $90^{\circ}$ , they are complimentary angles. If the sum of angles equals $180^{\circ}$ , they are supplementary.
A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles

10.3 Triangles

The sum of the interior angles of a triangle equals $180^{\circ} .$
Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle.
Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.

10.4 Polygons, Perimeter, and Circumference

Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides.
The perimeter of a polygon is the measure of the outline of the shape. We determine a shape's perimeter by calculating the sum of the lengths of its sides.
The sum of the interior angles of a regular polygon with $n$ sides is found using the formula $S=(n-2) 180^{\circ}$. The measure of a single interior angle of a regular polygon with $n$ sides is determined using the formula $a=\frac{(n-2) 180^{\circ}}{n}$.
The sum of the exterior angles of a regular polygon is $360^{\circ}$. The measure of a single exterior angle of a regular polygon with $n$ sides is found using the formula $b=\frac{360^{\circ}}{n}$.
The circumference of a circle is $C=2 \pi r$, where $r$ is the radius, or $C=\pi d$, and $d$ is the diameter.

10.5 Tessellations

A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps.
Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern.

10.6 Area

The area $A$ of a triangle is found with the formula $A=\frac{1}{2} b h$, where $b$ is the base and $h$ is the height.
The area of a parallelogram is found using the formula $A=b h$, where $b$ is the base and $h$ is the height.
The area of a rectangle is found using the formula $A=l w$, where $l$ is the length and $w$ is the width.
The area of a trapezoid is found using the formula $A=\frac{1}{2} h\left(b_1+b_2\right)$, where $h$ is the height, $b_1$ is the length of one base, and $b_2$ is the length of the other base.
The area of a rhombus is found using the formula $A=\frac{d_1 d_2}{2}$, where $d_1$ is the length of one diagonal and $d_2$ is the length of the other diagonal.The area of a regular polygon is found using the formula $A=\frac{1}{2} a p$, where $a$ is the apothem and $p$ is the perimeter.
The area of a circle is found using the formula $A=\pi r^2$, where $r$ is the radius.

10.7 Volume and Surface Area

A right prism is a three-dimensional object that has a regular polygonal face and congruent base such that that lateral sides form a $90^{\circ}$ angle with the base and top. The surface area $S A$ of a right prism is found using the formula $S A=2 B+p h$, where $B$ is the area of the base, $p$ is the perimeter of the base, and $h$ is the height. The volume $V$ of a right prism is found using the formula $V=B h$, where $B$ is the area of the base and $h$ is the height.
A right cylinder is a three-dimensional object with a circle as the top and a congruent circle is the base, and the side forms a $90^{\circ}$ angle to the base and top. The surface area of a right cylinder is found using the formula $S A=2 \pi r^2+2 \pi r h$, where $r$ is the radius and $h$ is the height. The volume is found using the formula $V=\pi r^2 h$, where $r$ is the radius and $h$ is the height.

10.8 Right Triangle Trigonometry

The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formula $a^2+b^2=c^2$, where c is the hypotenuse.
To find the measure of the sides of a special angle, such as a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, use the ratio $x: x \sqrt{3}: 2 x$, where each of the three sides is associated with the opposite angle and $2 x$ is associated with the hypotenuse, opposite the $90^{\circ}$ angle.
To find the measure of the sides of the second special triangle, the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, use the ratio $x: x: x \sqrt{2}$, where each of the three sides is associated with the opposite angle and $x \sqrt{2}$ is associated with the hypotenuse, opposite the $90^{\circ}$ angle.
The primary trigonometric functions are $\sin \theta=\frac{o p p}{h y p}, \cos \theta=\frac{a d j}{h y p}$, and $\tan \theta=\frac{o p p}{a d j}$.
Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles.

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