115 terms

all vocab and random stuff here and there from chapters 1-6.

inductive reasoning

process of looking for patterns and making conjectures

-based on observations

-based on observations

counter example

an example that shows a conjecture to be false

-specific

-specific

point

has no dimension, but is usually represented by a small dot

line

extends in 1 dimension

-usually represented by a straight line with arrows at each end

-has length

-never ends (infinite)

-usually represented by a straight line with arrows at each end

-has length

-never ends (infinite)

plane

extends in 2 dimensions (has length and width), usually represented by a shape that looks like a wall or a table top

-also infinite

-also infinite

collinear points

2 or more points that lie on the same line

coplanar points

lie on the same plane

line segment

has end points

-like a finite portion of an infinite line

-like a finite portion of an infinite line

ray

has an initial point then in continues infinity

opposite rays

have the same initial point, but go in opposite directions

-when looked at together, make line

-when looked at together, make line

points of intersection

2 or more geometric figures intersect if they have 1 or more points in common

postulate (axiom)

rule that is accepted without proof

distance formula

d = square root of (x2-x1) squared + (y2-y1) squared

angle

2 rays that have the same initial point

2 rays

sides of an angle

vertex

shared initial point

congruent angles

angles that have the same measure

interior of an angle

between the points on each side of the angle

exterior of an angle

points NOT on the sides or in the interior of an angle

acute

less than 90 degrees

right

exactly 90 degrees

obtuse

between 90 and 180 degrees

straight

exactly 180 degrees

adjacent angles

common side and vertex, but no other interior points in common

midpoint of a segment

point in the exact middle of a segment (cuts it in half)

bisect

cut in half

segment bisector

segment, ray, or line that intersects a segment at its midpoint

construction

geometric drawing that uses a limited set of tools (compass and straight edge)

midpoint formula

for end points

-m = (x1 + x2 /2, y1 + y2/2)

-m = (x1 + x2 /2, y1 + y2/2)

angle bisector

ray that divides an angle into 2 adjacent angles that are congruent

vertical angles

two angles whose sides form 2 pairs of opposite rays (2 intersecting lines)

-shared vertex in middle

-shared vertex in middle

linear pair

two adjacent angles whose non common sides are opposite rays

complementary

two angles who measures add to exactly 90 degrees

-can be adjacent or non adjacent

-can be adjacent or non adjacent

supplementary

two angles whose measures add to exactly 180 degrees

-can be adjacent or non adjacent

-all linear pairs are supplementary

-can be adjacent or non adjacent

-all linear pairs are supplementary

conjecture

an unproven statement based on observations

endpoints

part of a line that consist of 2 points

initial point

starting point

conditional statement

type of "logical" statement with 2 parts

hypothesis

setup; comes after "if"

conclusion

result; comes after "then"

converse

formed by switching the hypothesis and conclusion of a conditional statement

negation

writing the negative of a statement (not)

inverse

formed by negating the hypothesis and conclusion of the original conditional statement

contrapositive

formed by negating the hypothesis and conclusion of the converse

definition

uses known words to describe a new word or concept

perpendicular lines

lines that intersect to form right angles

line perpendicular to a plane

that intersects a plane at a point so that it forms a right angle with every line in that plane

biconditional statements

contain the phrase "if and only if" (Or IFF); same as writing a conditional statement and its converse together

theorem

statement that follows as a result of other true statements

-must be proven

-must be proven

two column proof

has numbered statements and reasons that show the logical order of an argument

paragraph proof

proof written in paragraph form

deductive reasoning

uses facts, definitions, and accepted properties in a logical order to write a logical argument

parallel lines

coplanar, but never intersect

skew lines

not coplanar, but never intersect

parallel planes

2 or more planes that never intersect

corresponding angles

occupy same positions on each parallel line

alternate interior angles

between inside the 2 parallel lines, on opposite sides of transversal

alternate exterior angles

between outside the 2 parallel lines, on opposite sides of transversal

consecutive interior angles

between 2 parallel lines on same side of the transversal

flow proof

uses arrows to show the flow of logical argument (reasons are usually written below the statements)

slope

-rise/run

-change in y/change in x

-m = y2 - y1 / x2 - x1

-change in y/change in x

-m = y2 - y1 / x2 - x1

perpendicular slopes

have slopes that are negative reciprocals of each other

transversal

line that intersects 2 or more parallel lines

equilateral

all 3 sides are the same length

isosceles triangle

triangle where at least 2 sides are congruent

scalene triangle

triangle where all sides are different length

acute triangle

all 3 angles are acute

equiangular triangle

all 3 angles have the same measure

obtuse triangle

1 angle is obtuse

right triangle

1 angle is 90 degrees

hypotenuse

the side opposite the right angle

adjacent sides

2 sides of the triangle sharing a common vertex

opposite side

the 3rd side of a triangle with adjacent sides

legs

the 2 sides of a right triangle that make the right angle

-perpendicular to each other

-perpendicular to each other

congruent figures

exactly same size and shape

corresponding angles

angles in the same position

corresponding sides

sides in the same position

base angles

2 angles in an isosceles triangle that are adjacent to the base

-non congruent side

-non congruent side

vertex angle

in an isosceles triangle: angle opposite to the base; sides are the 2 congruent legs of the triangle

coordinate proof

involves placing a geometric figure in the coordinate plane

-then we use distance and midpoint formulas long with postulates, theorems, etc, to prove statements

-then we use distance and midpoint formulas long with postulates, theorems, etc, to prove statements

perpendicular bisector

segment, line, or ray, that is perpendicular to a segment at its midpoint

equidistant

the same distance from something

distance from a point to a line

the length of the perpendicular segment from the point to the line

perpendicular bisector of a triangle

a line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side

-every triangle has 3

-all intersect at one point

-every triangle has 3

-all intersect at one point

concurrent lines

3 or more lines that intersect at the same point

point of concurrency

point where lines intersect at same point

circumcenter

point of concurrency of the perpendicular bisectors of a triangle

angle bisectors

bised the angles of a triangle

-every triangle has 3

-are concurrent

-every triangle has 3

-are concurrent

incenter

point of concurrency of angle bisectors

-always inside the triangle

-equidistant from all 3 sides of the triangle

-always inside the triangle

-equidistant from all 3 sides of the triangle

median of a triangle

a segment whose endpoints are vertex of a triangle and the midpoint of the opposite side

-every triangle has 3

-are concurrent

-every triangle has 3

-are concurrent

centroid

point of concurrency of medians of a triangle

-always inside the triangle

-balance point for the triangle

-always inside the triangle

-balance point for the triangle

altitude of a triangle

perpendicular segment from a vertex to the opposite side (or the line containing the opposite side)

-every triangle has 3

-are concurrent

-every triangle has 3

-are concurrent

orthocenter

point of concurrency of altitude of a triangle

-acute triangle: inside

-right triangle: on the triangle, vertex of the right angle

-obtuse triangle: outside

-acute triangle: inside

-right triangle: on the triangle, vertex of the right angle

-obtuse triangle: outside

midsegment of a triangle

segment that connects the midpoints of 2 sides of a triangle

indirect proof

prove a statement is true by first assuming that its opposite is true

-if assumption leads to an impossibility, then the original must be truel

-if assumption leads to an impossibility, then the original must be truel

polygon

a plane figure formed by 3 or more segments so that each segment intersects exactly 2 others at each end point

-no curves, gaps, overlaps, or criss crosses

-no curves, gaps, overlaps, or criss crosses

triangle

3 sides

quadrilateral

4 sides

pentagon

5 sides

hexagon

6 sides

heptagon

7 sides

octagon

8 sides

nonagon

9 sides

decagon

10 sides

dodecagon

12 sides

convex

no line containing a polygon side contains interior points

concave

the extended line of at least 1 side contains points in the interior of the polygon

regular polygon

polygon that is equilateral and equiangular

diagonal of a polygon

segment that joins 2 nonconsecutive vertices

parallelogram

a quadrilateral with both pairs of opposite sides parallel

rhombus

all 4 sides are congruent

-equilateral parallelogram

-equilateral parallelogram

rectangle

all 4 angles are right angles

-equiangular parallelogram

-equiangular parallelogram

square

all 4 angles and all 4 sides are congruent

-equilateral and equiangular parallelogram (REGULAR)

-equilateral and equiangular parallelogram (REGULAR)

trapezoid

quadrilateral that has exactly one pair of opposite sides parallel

kite

quadrilateral with 2 pairs of consecutive, congruent sides, but opposite sides are NOT parallel