### Key Concepts

### 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 $S=\left(n-2\right){180}^{\circ}.$ The measure of a single interior angle of a regular polygon with $n$ sides is determined using the formula $a=\frac{\left(n-2\right){180}^{\circ}}{n}.$
- 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 $A=\frac{1}{2}bh,$ where $b$ is the base and $h$ is the height.
- The area of a parallelogram is found using the formula $A=bh,$ where $b$ is the base and $h$ is the height.
- The area of a rectangle is found using the formula $A=lw,$ 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 $A=\frac{1}{2}ap,$ 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 $SA$ of a right prism is found using the formula $SA=2B+ph,$ 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 $V=Bh,$ 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 $SA=2\pi {r}^{2}+2\pi rh,$ 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 ${30}^{\circ}\text{-}{60}^{\circ}\text{-}{90}^{\circ}$ triangle, use the ratio $x:x\sqrt{3}:2x,$ 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 ${45}^{\circ}\text{-}{45}^{\circ}\text{-}{90}^{\circ}$ 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 $\mathrm{sin}\theta =\frac{opp}{hyp},$ $\mathrm{cos}\theta =\frac{adj}{hyp},$ and $\mathrm{tan}\theta =\frac{opp}{adj}.$
- 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.