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Geometry – Quick Reference Sheet

Point, Lines and Planes

Definitions:

- Point (•): a position without length, width or depth.
- Line (↔): a series of points that create length, but have no thickness or depth. May be curved, but typically straight with infinite length.
- Parallel Lines (||): two or more lines that never intersect. Both lines will have the same slope.
- Transversal: the line that intersects to parallel lines.
- Perpendicular Lines (⊥): two lines that intersect at a 90° angle. The product of their slopes will equal -1.
- Line segment (-): the line between 2 points.
- Ray (→): a portion of a line that extends infinitely in one directions.
- Angle (∠): a form created when two rays share an endpoint.
- Complimentary Angles: two angles with a sum measurement of 90°.
- Supplementary Angles: two angles with a sum measurement of 180°.
- Vertical Angles: congruent, non-adjacent angles formed by intersecting lines.
- Plane: an infinite series of points or lines with no depth.

Formulas:

- Distance Formula ⇒ $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{1}-y_{1})^{2}}$
- Midpoint Formula ⇒ $(x_{m},\ y_{m})=(\frac{x_{2}-x_{1}}{2},\ \frac{y_{2}-y_{1}}{2})$
- Slope ⇒ $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
- Equation of a line (point-slope form) ⇒ $y-y_{1}=m(x-x_{1})$
- Equation of a line (slope-intercept form) ⇒ $y=mx+b$

Polygons

Definition:

- Polygon: a closed plane figure with a minimum of three sides. Sides are composed of line segments that intersect at endpoints that are non-colinear.

Types:

- Regular: a polygon in which sides are congruent (equilateral) and interior angles are congruent (equiangular)
- Convex Polygon: a polygon in which any line segment connecting two endpoints lies on the interior of the polygon.
- Concave Polygon: a polygon in which one or more line segments connecting two endpoints does not lie on the interior of the polygon.

Angle Measures for Regular Polygons

Shape	Number of Sides	Sum of Interior Angles	Interior Angle Measure	Exterior Angle Measure
Triangle	3	180°	60°	300°
Quadrilateral	4	360°	90°	270°
Pentagon	5	540°	108°	252°
Hexagon	6	720°	120°	240°
n-gon	n	$180(n-2)=m$	$\frac{m}{n}=m_{i}$	$360-m_{i}=m_{e}$

Circles

Definitions:

- Circle: the set of all points on a plane that are equidistant from the center.
- Radius: a line segment with endpoints on the center and any point on the circle.
- Diameter: a line segment that passes through the center of the circle and has endpoints on the circle
- Chord: a segment that joins any two points on the circle.
- Circumference: the distance around a circle.
- Arc: a part of the circle with endpoints on the circle.
- Secant: a line with two points on the circle.
- Tangent: a line with only one point on the circle. The radius through that point of the circle is perpendicular to the tangent line.
- Central Angle: the angle created by two radii.
- Inscribed Angle: an angle formed by two chords whose vertex lies on the circle,
- Intercepted Arc: an arc between two specific points on a circle.
- Concentric Circles: circles with the same center.

Angle and Arc Relationships

- A Central Angle is measured by the length of the intercepted arc
- An Inscribed Angle is measured by one half the length of the intercepted arc
- Chord and Tangent Angles: angles formed by a chord and a tangent lines are measured by half the length of the arcs they intercept.
- Interior Chord Angles: the angle formed by two intersecting chords within a circle; angle is measured by one half the sum of the intercepted arc lengths.
- Exterior Chord Angles: the angle formed by two chords that intersect outside the circle; angle is measured by one half the difference of the intercepted arc lengths

Formulas

- Equation of a circle: a circle with a center at point (h, k) and a radius of r will have the equation $r^{2}=(x-h)^{2}+(y-k)^{2}$ .
- Circumference: the distance around a circle is equal to pi times two times the radius or $C=\pi\cdot 2r$
- Area: the area of a circle is equal to pi times the radius squared or $A=\pi\cdot r^{2}$

Transformations

Definitions

- Image: the point, line or figure that results from a transformation
- Pre-Image: the original point, line or figure

Types

- Isometry: a transformation in which the image is congruent to the pre-image
- Reflection: points, lines or figures are mirrored or flipped across a point, line or plane
- Rotation: points, lines or figures are turned around a point at a specified angle
- Translation: points, lines or figures are moved a specific distance in a specific direction
- Composition: a series of transformations
- Symmetry: an isometry that maps a figure onto itself

Triangles

Types

- Right Triangle: a triangle with a 90° angle.
- Acute Triangle: a triangle in which all angles are less than 90°.
- Obtuse Triangle: a triangle with one angle greater than 90°.
- Equiangluar Triangle: a triangle in which all angles have the same measure.
- Equilateral Triangle: a triangle constructed of three equal sides.
- Isosceles Triangle: a triangle in which two sides are equal or congruent.
- Scalene Triangle: a triangle in which all sides are different lengths.

Terms

- Altitude: a perpendicular line from the vertex of one angle to the side opposite the angle.
- Median: a line segment connecting the vertext of one angle to the midpoint of the side opposite the angle.
- Incenter: the point at which the angle bisectors of a triangle intersect.
- Circumcenter: the point at which the perpendicular bisectors of each side of the triangle intersect.
- Centroid: the point at which all three medians of a triangle intersect.
- Orthocenter: the point at which all three altitudes of a triangle intersect.

Formulas

- Area: the area of a triangle is equal to one half the base times the height or $A=\frac{1}{2}(b\cdot h)$
- Perimeter: the distance around a triangle is equal to the sum of the three sides or $P=s_{1}+s_{2}+s_{3}$

Congruence

- Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
- Angle-Side-Angle (ASA): Triangles are congruent if two angles and the included side of one triangle are congruent to the corresponding angles and side of another triangle.
- Angle-Angle-Side (AAS): Triangles are congruent if two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle.
- Side-Angle-Side (SAS): Triangles are congruent if two sides and the included angle of one triangle are congruent to the corresponding sides and angle of another triangle.
- Side-Side-Side (SSS): Triangles are congruent if all three sides of one triangle and the corresponding sides of another triangle are congruent.

Similarity

- Angle-Angle-Angle (AAA): Triangles are congruent if all angles of one triangle are congruent to the corresponding angles of another triangle.
- Corresponding parts of similar triangles are proportionate to each other.

Inequality of Triangles

- Triangle Inequality Theorem: the sum of the measures of any two sides of a triangle is greater than the measure of the third side.
- Sides and Angles: if the measure of one side of a triangle is greater than the measure of a second side, then the angle opposite the first side is greater than the angle opposite the second side.
- Angles and Sides: if the measure of one angle in a triangle is greater than the measure of a second angle, then the side oppostie the first angle is greater than the side opposite the second angle.
- Exterior Angle Inequality Theorem: the measure of an exterior angle on a triangle is greater than the measure of either of the non-adjacent interior angles.
- Hinge Theorem: If two sides of a triangle A are congruent to two sides of triangle B and the angle between the two sides on triangle A is greater than the angle between the two sides on triangle B, then the third side of triangle A is greater than the third side of triangle B.

Polyhedron

Definitions and Formulas

- Polyhedron: a three dimensional figure constructed of polygons.
- Faces: the polygons that form the sides of the polyhedron
- Edges: the line segment formed where two polygons intersect
- Vertices: the point at which the edges intersect to form corners
- Euler’s Formula: the sum of the number of faces and vertices of a polyhedron is equal to two more than the number of edges or $F+V=E+2$
- Platonic Solids: the five polyhedron that are constructed of only regular polygons. See Table Below

	Polygon	Faces	Edges	Vertices
Tetrahedron	Triangle	4	6	4
Cube	Square	6	12	8
Octahedron	Triangle	8	12	6
Dodecahedron	Pentagon	12	30	20
Icosahedron	Triangle	20	30	12

Quadrilaterals

Types

- Parallelogram: a four sided figure made of two sets of parallel lines.
- Rhombus: a parallelogram with four equal sides.
- Rectangle: a parallelogram with four right angles.
- Square: a parallelogram with four right angles and four equal sides.
- Trapezoid: a quadrilateral with one set of parallel lines.
- Kite: a quadrilateral with two pairs of congruent sides that are adjacent.

Right Triangle

Pythagorean Theorem

- Definitions: the sum of each leg squared is equal to the hypotenuse squared.
- Formula: $c^{2}=a^{2}+b^{2}$
- Converse of the Pythagorean Theorem:
  - If $c^{2}=a^{2}+b^{2}$ , then triangle ABC is a right triangle
  - If $c^{2}<a^{2}+b^{2}$ , then triangle ABC is an acute triangle
  - If $c^{2}>a^{2}+b^{2}$ , then triangle ABC is an obtuse triangle

Special Right Triangle

Trigonometric Ratios

- $sin(\theta)=\frac{opposite}{hypotenuse}=\frac{y}{r}$
- $cos(\theta)=\frac{adjacent}{hypotenuse}=\frac{x}{r}$
- $tan(\theta)=\frac{sin(\theta)}{cos(\theta)}=\frac{opposite}{adjacent}=\frac{y}{x}$
- $csc(\theta)=\frac{1}{sin(\theta)}=\frac{hypotenuse}{opposite}=\frac{r}{y}$
- $sec(\theta)=\frac{1}{cos(\theta)}=\frac{hypotenuse}{adjacent}=\frac{r}{x}$
- $cot(\theta)=\frac{1}{tan(\theta)}=\frac{adjacent}{opposite}=\frac{x}{y}$