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# Geometry Ch. 1 and 2

point a location. it has neither shape nor size.
line made up of points and has no thickness or width. exactly one through any two points.
plane a flat surface made up of points that extends infinitely in all directions. there is exactly one through any three points not on the same line.
collinear points that lie on the same line.
coplanar points that lie on the same plane.
intersection two or more geometric figures is the set of points they have in common. two lines = point 2 planes = line
space defined as boundless, three dimensional set of all points
line segment can be measured because it has two endpoints
Betweenness of points for any two points A and B on a line, there is another point C between A and B if and only if A, B, and C are collinear and AC+CB=AB
Congruent Segments segments that have the same measure
Constructions are methods of creating these figures without the benefit of measuring tools
distance between two points is the length of the segment with those points as its endpoints √(x2-x1)² + (y2-y1)²
midpoint is the point halfway between the endpoints of the segment m= (x1 + x2)/2
segment bisector any segment, line, or plane that intersects a segment at its midpoint
ray a part of a line. it has one endpoint and extends indefinitely in one direction
opposite rays if you choose a point on a line, the point determines exactly two rays called this. since both rays share a common enpoint, ______ are collinear
angle formed by two noncollinear rays that have a common endpoint.
sides the rays are called this of the angle
vertex the common endpoint
interior inside of angle
exterior outside of angle
degrees angles are measured in units
right angle measure = 90
acute angle measure = less than 90
obtuse angle measure = more than 90
angle bisector a ray that divides an angle into two congruent angles
adjacent angles two angles that lie in the same plane and have a common vertex and a common side but no common interior points
linear pair a pair of adjacent angles with noncommon sides that are opposite rays
vertical angles two nonadjacent angles formed by two intersecting lines
complementary angles two angles with measures that have a sum of 90
supplementary angles are two angles with measures that have a sum of 180
perpendicular lines, segments, or rays that form right angles
polygon a closed figure by a finite number of coplanar segments called sides
vertex of the polygon vertex of each angle
concave some of the lines pass through the interior
convex no points of the lines are in the interior
n~gon a polygon with n sides
regular polygon a convex polygon that is both equilateral and equiangular
equilateral polygon a polygon in which all sides are congruent
equiangular polygon a polygon in which all angles are congruent
perimeter the sum of the lengths of the sides of the polygon
circumference of circle is the distance around the circle
area the number of square units needed to cover a surface
polyhedron a solid with all flat surfaces that enclose a single region of space
face each flat surface of a polygon
edges the line segments where the faces intersect
vertex the point where three or more edges intersect
prism a polyhedron with two parallel congruent faces called bases connected by parallelogram faces
base of a polyhedron the two parallel congruent faces of a polyhedron
pyramid a polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex
cylinder a solid with congruent parallel circular bases connected by curved surfaces
cone a solid with a circular base connected by a curved surface to a single vertex
regular polyhedron if all of its faces are regular congruent polygons and all of the edges are congruent
platonic solids exactly five types of regular polyhedrons
surface area two- dimensional measurement of the surface of a solid figure
volume the measure of the amount of space enclosed by a solid figure
inductive reasoning reasoning that uses a number of specific examples to arrive at a conclusion
conjecture a concluding statement reached using inductive reasoning
counterexample false example, called this, and it can be a number, drawing, or statement.
statement a sentence that is either true or false
truth value a statement is either t or f represented using p and q
negation has the opposite meaning as well as an opposite truth value. not p or ~p
conjunction a compound statement using the word and
truth table a convenient method for organizing truth values of statements
disjunction a compound statement using the word or
pyramid a polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex
cylinder a solid with congruent parallel circular bases connected by curved surfaces
cone a solid with a circular base connected by a curved surface to a single vertex
sphere a set of point in the space that are the same distance from a given point. no faces, edges, or vertices
regular polyhedron if all of its faces are regular congruent polygons and all of the edges are congruent
platonic solids exactly five types of regular polyhedrons
surface area two- dimensional measurement of the surface of a solid figure
volume the measure of the amount of space enclosed by a solid figure
inductive reasoning reasoning that uses a number of specific examples to arrive at a conclusion
conjecture a concluding statement reached using inductive reasoning
counterexample false example, called this, and it can be a number, drawing, or statement.
statement a sentence that is either true or false
truth value a statement is either t or f represented using p and q
negation has the opposite meaning as well as an opposite truth value. not p or ~p
compound statement the statement formed from two or more simple statements using connective words like "and" or "or."
conjunction a compound statement using the word "and"
disjunction a compound statement using the word "or"
truth table a convenient method for organizing truth values of statements
if-then statements if p, then q
hypothesis conditional statement is the phrase immediately following the word if... p
conclusion conditional statement is the phrase immediately following the word then... q
related conditionals there are other statements that are based on a given conditional
converse exchanging the hypothesis and conclusion of the conditional..... q-p
inverse formed by negating both the hypothesis and conclusion of the conditional...... ~p - ~q
contrapositive formed by negating both the hypothesis and the conclusion of the converse of the conditional..... ~q - ~p
logically equivalent statements with the same truth values
deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions from given statements
valid method of proving a conjecture
law of detachment one valid form of deductive reasoning if p - q is a true statement and p is true then q is true
postulate or axiom is a statement that is accepted as true without proof
postulate 2.1 through any two points, there is exactly one line
postulate 2.2 through any three noncollinear points, there is exactly one plane
postulate 2.3 a line contains at least two points
postulate 2.4 a plane contains at least three noncollinear points
postulate 2.5 if two points lie in a plane, then the entire line containing those points lies in that plane
postulate 2.6 if two line intersect, then their intersection is exactly one point
postulate 2.7 if two planes intersect then their intersection is a line
proof a logical argument in which each statement you make is supported by a statement this is accepted as true
deductive argument a proof formed by a group of algebraic steps used to solve a problem
paragraph proof an informal proof written in the form of a paragraph that explains why a conjecture for a given situation is true
informal proofs a paragraph proof
midpoint theorem if M is the midpoint of segment AB, then seg. AM is congruent to seg. MB. A_______M_______B
addition property of equality if a = b then a + c = b + c
subtraction property of equality if a = c then a - c = b - c
multiplication prop of equality if a = b then a*c=b*c
division prop of equality if a = b and c ≠ 0 then a /c = b/c
reflexive prop of equality a = a
symmetric prop of equality if a = b then b = a
transitive prop of equality if a = b and b = c then a = c
substitution prop of equality if a = c then a may b replaced by b in any equation of expression
distributive prop a(b + c) = ab + ac
algebraic proof is a proof that is made up of a series of algebraic statement
two-column proof or formal proof contains statements and reasons organized in two columns
ruler postulate the points on any line or line segment can be put into one-to-one correspondence with real numbers
segment addition postulate if A, B, and C are collinear, the point B is between A and C if and only if AB+BC=AC
Reflexive prop of congruence seg. AB ≅ seg. AB
symmetric prop of congruence if seg. AB ≅ seg. CD, the seg. CD ≅ seg. AB
Transitive prop of congruence if seg. AB ≅ seg. CD and seg. CD ≅ seg. EF then seg. AB ≅ seg. EF
Protractor Postulate given any angle, the measure can be put into one- to- one correspondence with real numbers between 0 and 180
angle addition postulate states that if D is in the interior of angle ABC, then the measure of angle ABD + the measure of angle DBC = the measure of angle ABC.
Supplement Theorem if two angles form a linear pair, then they are supplementary angles
Complement Theorem if the noncommon sides of two adjacent angles form a right angle then the angles are complentary angles
Reflexive prop of congruence ∠1 ≅ ∠1
Symmetric prop of congruence if ∠1 ≅ ∠2 then ∠2 ≅ ∠1
transitive prop of congruence if ∠1 ≅ ∠2 and ∠2 ≅ ∠3 then ∠1 ≅ ∠3
Congruent Supplements Theorem Angles supplementary to the same angle or to congruent angles are congruent
Vertical Angles Theorem if two angles are vertical angles then they are congruent
Right angle theorem 2.9 perpendicular lines intersect to form four right angles
Right angle theorem 2.10 all right angles are congruent
Right Angle theorem 2.11 perpendicular lines form congruent adjacent angles
Right Angle theorem 2.12 if two angles are congruent and supplementary then each angle is a right angle
Right Angle theorem 2.13 if two congruent angles form a linear pair, then they are right angles
Created by: 15jsmith
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