Geometry Final

Adjacent angles

Two angles in the same planes with a common vertex and a common side, but no interior points. Angle one and angle two are adjacent

Similarity Transformation

a dialation or composition of rigid movement and dilations

Pythagorean Theorem

a squared plus b squared equals c squared

Rotation Rules for 270°

(x,y)---> (y, -x)

Rotation Rules for 180°

(x,y)--->(-x,-y)

Rotation Rules for 360°

(x,y)--->(-x,-y)

Rotation Rules for 90°

(x,y)--->(-y,x)

To determine if a dilation is a reduction follow:

0<lKl<1

Glide Translation

1. Translation 2. Reflection

Sum of interior angles of a polygon

180(n-2)

Congruent Figures

2 figures are congruent IFFthere is a rigid movement on composition of rigid movemnt from one figure to the other and each figure has equal side lengths and interior angles

Similar Figures

2 figures are similar IFF there is a similarity transformation b/w them and they have the same shape with different side lengths

Angle bisector

A ray that divides an angle into two congruent angles

Translation

<a,b> ; (x,y)---> (x+a, y+b) It is also a rigid movement

Undefined term

A basic figure that is not defined in terms of other figures in undefined terms in geometry are point, line, and plane.

tranformation

A change is position, size, or shape of a figure or graph

Auxiliary line

A line drawn in a figure to aid a proof

transversal

A line that intersects two coplanar lines at two different points.

Segment bisector

A line, ray, or segment that divides a segment into two congruent segments

Line Segment

A part of a line consisting of two endpoints and all points between them.

Triangle

A polygon with three sides

Coordinate Proof

A proof that has variables and numbers

Vector

A quantity that has both magnitude and direction

Inequality

A statement that compares two expressions by using greater than, less than, greater than or equal to, less than or equal two, or not equal

Thoerem

A statement that has to be proven

dilation

A transformation that dilates the pre-image and changes the shape of the image; does not preserve rigid motion-Enlarging or reducing a figure by a scale factor (K)

Rigid movement

A transformation that does not change the size or shape of a figure

reflection

A transformation that reflects the pre-image over the x-axis or y-axis

rotation

A transformation that rotates the pre-image around a point (usually the origin point) at 90 degree intervals i.e. 90, 180, 270, 360

translation

A transformation that shifts or slides the position of every point of the pre-image

Super-Heronian Triangle

A triangle who's area, perimeter, and side lengths are integers and their side lengths are consecutive.

Heron's Triangle

A triangle who's area, perimeter, and side lengths are integers.

Isosceles

A triangle with 2 congruent legs, a base, and 2 equal base angles

Equilateral

A triangle with all equal side lengths and angles

Scalene

A triangle with different side lengths and angles

Coordinate proof

A type of proof that needs algebra to be proven

Right Angle Congruence Theorem

All right angles are congruent

Right Angle

An angle thats 90˚

Obtuse Angle

An angle thats <90˚

Acute Angle

An angle thats >90˚

Plane

An undefined term in geometry, it is a flat surface that has no thickness and extends forever.

Point

An undefined term in geometry, it names a location and has no size

Translation Vector

Go from <a,b> to √a²+b²

Side-Side-Side Angle Theorm

If three sides of one triangle are congruent to three sides of another triangle then the triangle are congruent.

Angle-Angle-Side Angle Theorem

If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included sides of another triangles, then the triangles are congruent.

Angle-Side-Angle Angle Theorem

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the the triangles are congruent.

Congruent supplements theorem

If two angles are supplementary to the same angles (or two congruent angles), then the two angles are congruent.

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are also congruent.

Corresponding angles postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Alternate Exterior Angles theorem

If two parallel lines are cut by a transversal, then the two pairs of Alternate Exterior angles are congruent.

Same-Side Interior Angles Theorem

If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary

Side-Angle-Side Angle Theorem

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Isosceles Triangle theorem

If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

Perpendicular Transversal Theorem

In a plane, if a transveral is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

perpendicular lines

Lines that intersect at 90 degrees

Base Angle Theorem

States that in an isosceles triangle, the base angles opposite the congruent sides are congruent.

Sum of exterior angles of a polygon

The angles will always add up to 360 degrees

Magnitude

The length of a vector

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

Vertical angles

The nonadjacent angles formed by two intersecting lines

Midpoint

The point that divides a segment into two congruent segments

Inductive reasoning

The process of reasoning that a rule or statement is true because specific cases are true

Deductive Reasoning

The process of using logic to draw conclusions from given facts definitions are properties

Angle

The space between two intersecting lines.

Triangle Sum Theorem

The sum of the angle measures of a triangle is 180 degrees

Linear pair

adjacent angle whose non-common sides opposite rays angle 1 and angle 2 are a linear pair.

properties of a square

all sides are congruent, there are four right angles

line

an undefined term in geometry, a straight path that has no thickness and extends forever.

Conditions of a sqaure

if it is a rectangle and a rhombus, then it is a square

conditions of a rectangle

if there is one right angle, then it is a rectangle. If the diagonals are congruent, then it is a rectangle

Alternate Interior Angles Theorem

if two parallel lines are cut by a transversal, then the pairs of Alternate Interior Angles are congruent.

To determine if a dilation is an enlargement follow:

lKl> 1

parallel lines

lines in a plane that do not intersect or touch each other at any point.

Norm

llVll = √a²+b² V= <a,b>

Ray

part of a line that starts at an endpoint and extends forever in one direction

Vertex

point where two or more curves, lines, or edges meet

Definition of a rectangle

quadrilateral with four right triangles

Rotations

rigid movement which a figure moves circular by an angle

Heron's Formula

s=1/2(a+b+c) A=√s(s-a)(s-b)(s-c)

Reflection

transformation which is reflected across a line, creating a mirror image.