Geometry

Reflection of a Figure (The Flip)

- A Reflection changes the orientation of the figure - A Reflection is an isometry

Polygon

A 2 dimensional figure which is formed by a number of line segments, each connected end to end to form a closed shape.

Right Circular Cone

A cone is a solid that has a circular base and a a single vertex. If the vertex is over the center of the base, it is called a right cone. If it is not, it is called an oblique cone.

Right Circular Cylinder

A cylinder is a closed solid that has two parallel circular bases connected by a curved surface.

Dilation of a figure

A dilation is a transformation that produces an image that is the similar to the pre-image (original figure). A dilation stretches or shrinks the pre-image. The description of a dilation includes the scale factor (or ratio) and the center of the dilation.

Special Quadrilateral

A four-sided polygon.

Net

A net is a two dimensional figure that can be folded to make a three dimensional object.

Supplementary Angle Pairs

A pair of angles whose sum is 180 degrees

Complementary Angle Pairs

A pair of angles whose sum is 90 degrees

Rhombus

A parallelogram with all congruent sides The diagonals bisect opposite angles The diagonals are perpendicular bisectors

Square

A parallelogram with four congruent sides and four congruent angles The diagonals are congruent The diagonals are perpendicular bisectors

Coordinate Plane

A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.

Center of Dilation

A point about which the figure is dilated.

Convex Polygon

A polygon that has all interior angles measure less than 180°. (All the vertices point 'outwards')

Regular Polygon

A polygon with all angles Congruent and all sides congruent.

Concave Polygon

A polygon with one or more interior angles greater than 180°. It looks like a vertex has been 'pushed in' towards the inside of the polygon.

Irregular Polygon

A polygon with sides that are not all congruent or angles that are not all congruent

Polyhedron

A polyhedron is a three-dimensional solid figure in which each side is a flat surface. These flat surfaces are polygons and are joined at their edges. The word polyhedron is derived from the Greek poly (meaning many) and the Indo-European hedron (meaning seat or face).

Proportion

A proportion is an equation with a ratio (fraction) on each side.

Trapezoid

A quadrilateral with exactly one pair of parallel sides.

Radical Expression

A radical expression is defined as any expression containing a radical (√) symbol.

Square Roots and Perfect Squares Chart

A square root is a number that, when multiplied by itself, will result in a given number.

Right Square Pyramid

A three dimensional shape that has a square base and four congruent faces that meet at a point above the base called the vertex. The vertical height (h) is the perpendicular distance from the top down to the base. The 4 lateral faces are congruent isosceles triangles. The slant height (l) is the height of the lateral faces. s is the side of the base (square).

Vertical Line

A vertical line is a line that goes straight up and down, parallel to the y-axis of the coordinate plane. All points on the line will have the same x-coordinate. The slope of a vertical line is undefined.

Classify Angles

Acute Obtuse Right Straight Reflex

Alternate Exterior Angle Pairs

Alternate Exterior Angle pairs are congruent.

Opposite Angles

Angels that do not share a side.

Angle Bisector

Angle bisector divides an angle into two congruent angles.

Circle - Chord

Chord: A line segment with the endpoint on the circumference of the circle. Diameter: A cord that passes through the center of the circle.

Concentric circles

Circles that have their centers at the same point.

Irregular Shape

Irregular shapes have sides and angles of any length and size.

Naming Angles

Name an angle in four different ways: <ABC, <CBA, <1, <B

Coordinates of a point

Point (-3, 2)

Circle (Radius, Diameter, Circumference)

Radius is the distance from the center to the edge. Diameter starts at one side of the circle, goes through the center and ends on the other side. Circumference is the distance around the edge of the circle.

Similarity Ratio

Ratio of corresponding sides

Clockwise Rotation

Right Turn

Alternate Interior Angle Pairs

Same Side Exterior Angle Pairs are supplementary (add up to 180 degrees).

Same Side Interior Angle Pairs

Same Side Interior angle pairs are supplementary (add up to 180 degrees).

Special Right Triangle 30-60-90

Short Leg is opposite the smallest angle (30°). Long Leg is opposite the 60° angle. Long Leg = (short leg) *√3. Hypotenuse = 2 * Short Leg

Slope is Negative

Slope drops from left to right

Slope is Zero

Slope of a Horizontal Line Equation of a horizontal line y=constant Examples: y=3, y=-4

Slope is Positive

Slope rises from left to right

Special Right Triangles

Special Right Triangle are right triangles 1. Special Right Triangle 45º- 45º- 90º 2. Special Right Triangle 30º- 60º- 90º

Sphere

Surface Area: The number of square units that will exactly cover the surface of a sphere. Volume: The number of cubic units that will exactly fill a sphere.

Scale Factor

The amount by which the image grows or shrinks is called the "Scale Factor". SF > 1 --> Image is enlarged (larger) 0 < SF < 1 --> Image is reduced (smaller) SF = 1 --> Image is the exact same size as the original (Isometry).

Midpoint Formula

The coordinates of the midpoint of a line segment are the average of the coordinates of its endpoints.

Circle - Measure of Major Arc

The measure of a major arc equals 360 minus the measure of the minor arc.

Circle - Measure of Minor Arc

The measure of a minor arc equals the measure of its central angle.

Circle - Measure of Minor and Major Arcs

The measure of an arc is equal to the measure of the central angle. Measure of Minor Arc < 180 degrees Measure of Major Arc > 180 degrees

Triangle Exterior Angle Theorem

The measure of an exterior angle is equal to the sum of the two remote interior angles.

Central Angle Theorem

The measure of the inscribed angle (∠ABC) is always half the measure of the central angle (∠AOC) when both angles intercept the same arc AC.

Midpoint of a Line Segment

The midpoint is equidistant from the endpoints.

Circle - Area of Sector

The number of square units it takes to exactly fill a sector of a circle.

Midpoint Definition

The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides a line segment into two equal segments. By definition, a midpoint of a line segment is the point on that line segment that divides the segment two congruent segments.

Slant Height of a Cone

The slant height is the distance from the vertex of the cone, down the side, to a point on the circumference of the base.

Slant height of a pyramid

The slant height is used to find the total surface area of a pyramid. The slant height is not the height of the pyramid.

Slope

The steepness of a line on a graph Rise over Run

Triangle Angle Sum Theorem

The sum of the three angle measures of any triangle is always180 degrees..

Transformation Image

The transformed figure

Three Dimensional Shapes or Solids

Three-Dimensional shapes can be made up of flat of curved or flat surfaces

Simplify Square Roots

To simplify a square root, you "take out" anything that is a "perfect square".

Solving a Proportion

To solve a proportion, cross multiple and solve the equation for the unknown.

Surface Area Cylinder

Total SA = LA + Area of 2 Bases

Transformation of a figure

Transformation is a change in position, shape, or size of a figure.

Perpendicular Lines

Two line that intersect at 90 degree angles.

Consecutive Sides

Two sides of a polygon with a common vertex.

Volume cylinder

Volume is the number of cubic units to exactly fill a Cylinder.

Volume of Rectangular Prism

Volume is the number of cubic units to exactly fill a prism.

Radical Expressions and how to Rationalize the Denominator

When you have a fraction with a radical in the denominator, you can use a technique called rationalizing a denominator to eliminate the radical. Multiply numerator and denominator by the radical that converts the denominator into a square root of a perfect square.

Pythagorean Triples

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

Composite figure

A figure (or shape) that can be divided into more than one of the basic figures is said to be a composite figure (or shape).

Horizontal Line

A horizontal line is one the goes left-to-right, parallel to the x-axis of the coordinate plane. All points on the line will have the same y-coordinate. A horizontal line has a slope of zero.

Tangent

A line that "just touches" the circle at one point.

Transversal line

A line that cuts across parallel lines at two distinct points.

Rectangle

A parallelogram with four right angles The diagonals are congruent and bisect each other

Ratio

A ratio is a comparison of two numbers. It is expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second.

Rotation of a figure (Turn)

A rotation is a transformation where a figure is turned about a given point (center of rotation). A Rotation is an isometry.

Circle - Sector

A sector is a region bounded by two radii of a circle and the intercepted arc of the circle.

Right Rectangular Prism

A solid (3-dimensional) object which has six faces that are rectangles. The two bases are parallel and congruent. It has the same cross-section along a length, which makes it a prism.

Isosceles Trapezoid

A trapezoid with congruent legs

Sum of Exterior Angles of a Polygon

Always equals 360 degrees

Angle

An Angle is a figure formed by two rays (called SIDES or LEGS) with a common endpoint called the VERTEX (plural VERTICES).

Central Angle

An angle formed at the center of a circle by two radii. The vertex of a central angle is always the center of the circle. The measure of a central angle is equal the measure of the intercepted arc.

Exterior Angle

An angle formed by one side of a polygon and the extension of an adjacent side

Interior Angle

An angle inside a polygon

Inscribed Angle

An angle is inscribed in a circle if the vertex of the angle is on the circle and the sides of the angle are chords of the circle

Consecutive or Adjacent Angles

Angles of a polygon that share a common side.

Area vs perimeter

Area is the number of square units enclosed by the 2D shape. Perimeter is the distance around the outside of a 2D shape.

Area

Area is the number of square units enclosed by the two dimensional shape.

Parallelogram

Both pairs of opposite sides are parallel and congruent Opposite angles are congruent Consecutive angles are supplementary The diagonals bisect each other

Classify Triangles

By Angles: Acute, Obtuse, Right, Equiangular By Sides: Equilateral, Isosceles, Scalene

Special quadrilaterals Chart

Categorize special quadrilaterals according to their properties.

Corresponding Angle Pairs

Corresponding angle pairs are congruent

Congruent Polygons

Corresponding angles (matching) are congruent . Corresponding sides are congruent.

Line Segment

Distance between two endpoints

Distance between two Points in the Coordinate Plane

Distance formula

Face, Vertex, Edge

Each flat surface is called a FACE. EDGE is a line segment that is the intersection of two faces. VERTEX is the point that is the intersection of three or more faces.

Volume of Composite 3D shapes

First calculate the volume of each of individual figures and then find the volume of the entire composite figure.

Equations of horizontal and vertical lines

Graph horizontal line: y=3, y=-4, etc Graph vertical libe: x=3, x=-7 etc

Hemisphere

Half a Sphere

Pythagorean Theorem

In mathematics, the Pythagorean theorem is a relation among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Triangle angles

Interior and exterior angles

Intersecting Lines

Intersecting lines intersect at one point.

Lateral Surface Area of a Rectangular Prism

Lateral surface area (LA) is the sum of the side faces.

Counterclockwise Rotation

Left Turn

Kite

No parallel sides Two pairs of adjacent sides are congruent Exactly one pair of opposite angles are congruent The diagonals meet at a right angle

Polygon Angle Sum Theorem

Number of triangles x 180 OR (n-2)180, n = number of sides (n-2)180

Parallel Lines

Parallel lines do not intersect. They have no common point.

Circle - Arc

Part of the circumference of the circle

Circle - Arc Length

Part of the circumference of the circle.

Perimeter

Perimeter is the distance around the outside of a 2D shape.

Similar Polygons

Polygons are similar when their corresponding angles are congruent and the ration of their corresponding sides are in proportion.

Special Right Triangle 45-45-90

Right Isosceles triangle. Both legs are equal. Hypotenuse= Leg * √ 2

Slope is Undefined

Slope of a Vertical Line. Equation of a vertical line x=constant Examples: x=3, x=-4

Circle - Measure of a Semicircle

The measure of a a semicircle equals 180 degrees.

Pre-image

The original figure

Comparing Ratios (Fractions)

The ratios are equal if their cross products are equal.

Volume Comparison: Cylinder and Cone

The volume of a cone is 1/3 the volume of a cylinder that has the same height and the same base.

Volume of a Cone

The volume of a cone is 1/3 the volume of a cylinder that has the same height and the same base.

Total Surface Area of Rectangular Prism

Total SA = LA + Areas of 2 Bases A right prism has a set of flat surfaces called the FACES. Total Surface Area (SA) is the sum of all faces. The two bases (top and bottom faces) are parallel and congruent.

Isometry

Transformation of a figure in which the pre-image of a figure and the image of the figure are congruent.

Translation of a Figure (sliding motion)

Translation is a transformation that moves points the same distance and in the same direction. - A Translation DOES NOT change the orientation of the figure. - A Translation is an isometry. Example: (x-7, y-3) algebraic notation or <-7,-3> vector notation

Trapezoid

Two dimensional shape with four sides and one pair of parallel and congruent sides.

Distance between two Points in the Coordinate Plane using the Pythagorean Theorem

Use the Pythagorean Theorem on the coordinate plane.

Vertical Angle Pair

Vertical angle pairs are congruent. They are formed by two intersecting lines.