Geometry Definitions and Theorems

Flashcards on Geometry Concepts

Midpoint

The point on a line segment that divides it into two equal and congruent segments

Midpoint Formula

Average value of endpoints yields the location of the segment midpoint: ((x1+x2)/2 , (y1+y2)/2)

Distance Formula

Derived from the Pythagorean Theorem. Gives distance of a point from the origin. d = √(x1+x2)2+(y1+y2)2+(z1+z2)2

Pythagorean Theorem

Leg2+Leg2=Hyp2

Hypotenuse

Side of a right triangle opposite the right angle

Perpendicular lines

Lines at a right angle relative to each other

Parallel lines

Lines at a straight angle relative to each other

Skew lines

Lines at a non-straight and non-right angle relative to each other

Perpendicular bisector

Line that bisects a line segment (splits into two congruent segments) and intersects the segment at a right angle

Perpendicular Bisector Theorem

Any point on the perpendicular bisector of a segment is equidistant from either endpoint of the segment

Converse of the Perpendicular Bisector Theorem

A point that is equidistant from the endpoints of a segment lies on the perpendicular bisector

Point

represents a location in space, has no dimensions

Line

a collection of points extending linearly to infinity with no endpoints

Plane

2D surface that extends to infinity for all 2D space

Line Segment

portion of a line with 2 endpoints

Ray

portion of a line with 1 endpoint

Transversal

a line intersecting two or more given lines in a plane at different points

Acute Angle

between 0 and 90 degrees

Obtuse Angle

between 90 and 180 degrees

Right Angle

90 degrees

Straight Angle

180 degrees

Reflex Angle

between 180 and 360 degrees

Complete Angle

360 degrees

Congruent Angles

equivalent to each other

Adjacent Angles

angles on the same side of an intersection (linear pair)

Vertical Angles

angles on opposing sides of an intersection (congruent)

Linear Pair

Angles that lie on the same line, aka supplementary

Complementary Angles

sum of angles is 90 degrees

Supplementary Angles

sum of angles is 180 degrees

Equilateral Triangle

triangle with all side lengths equal

Isosceles Triangle

two sides and opposite angles are equal

Scalene Triangle

none of its sides or angles are equal

Right Triangle

contains a right angle

Acute Triangle

all three interior angles are acute

Obtuse Triangle

contains an obtuse angle

Segment Addition Postulate

For line segment AC that contains point B, segments AB + BC = AC

Addition/Subtraction/ Multiplication/Division Properties of Equality

Algebraic foundation of manipulating equations. Applying an operation to both sides of an equation, the two sides remain equal

Substitution Property of Equality

If a=b And a=c Then b=c

Distributive Property of Equality

a(b+c) = ab + ac

Two Point Postulate

Through any two points, there exists exactly one line. A line contains at least two points.

Three Point Postulate

Through any three noncollinear points, there exists exactly one plane.

Plane-Point Postulate

A plane contains at least three noncollinear points.

Plane-Line Postulate

If two points lie in a plane, then the line containing them lies in the plane.

Plane Intersection Postulate

If two planes intersect, then their intersection is a line.

Reflexive Property of Equality

A value is always equal to itself. a=a

Symmetric Property of Equality

If a=b Then b=a

Transitive Property of Equality

If a=b And b=c Then a=c

Reflexive Property of Congruence

Any geometric object is always congruent to itself. X ≅ X

Symmetric Property of Congruence

If one geometric object X ≅ Y, then object Y ≅ X

Transitive Property of Congruence

If geometric object X ≅ Y And Y ≅ Z Then X ≅ Z

Definition of Congruent Segments

Segments are congruent when they can be exactly Superimposed over each other. Turning, flipping, or rotating, an object does not change congruence.

Angle Addition Postulate

If two angles share both a common arm and vertex (adjacent angles), their sum is equal to the resultant angle between the two non-common arms

Vertical Angles Congruence Theorem

When two lines intersect, the angles that are opposite each other (vertical angles) are always equal in measure.

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then those two angles are congruent. If ∠X + ∠Y = 180° And ∠Z + ∠Y = 180° Then ∠X ≅ ∠Z

Right Triangles Congruence Theorem

Two right triangles are congruent if they are congruent in the following ways: Leg-Leg (LL)

Linear Pair Postulate

If two angles are adjacent and their non-common sides form a straight line, then the angles are supplementary

SSS

side, side, and side

SAS

side, (included) angle, and side

ASA

angle, (included) side, and angle

AAS

angle, angle, and (non-included) side

HL

hypotenuse and leg (right triangles only)