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.