Geometry Definitions Flashcards
(116 cards)
1
Q
Point
A
That which has no part.
2
Q
Line
A
Limitless, breathless length.
3
Q
Collinear
A
A set of points are collinear if they all lie on the same line.
4
Q
Intersect or Intersection
A
Two or more figures intersect if at least one point is on them all. The intersection of two or more figures is the set of points on them all.
5
Q
Segment
A
A segment is two points on a line and all points on that line between them.
6
Q
Ray
A
Choose a point on a line. Add to it all points on that line on one of the two sides of the point you choose. The result is a ray.
7
Q
Between
A
Point B is between points A and C when B lies in the interior of segment AC.
8
Q
Midpoint
A
The midpoint of a line segment is the point in its interior that divides it into a pair of equal sub-segments. In symbols: if C is the midpoint of segment AB, then C lies between A and B and AC=BC.
9
Q
Angle
A
An angle is a pair of rays that share an endpoint in common.
10
Q
Angle Side
A
An angle is composed of a pair of coterminal rays. Each of those two rays is a side of the angle.
11
Q
Angle Vertex
A
The vertex of an angle is the point of intersection of its two sides.
12
Q
Degree
A
When a ray completes one complete rotation about its endpoint, it sweeps out 360 degrees. A degree is thus 1/360th of one complete rotation of a ray about its endpoint.
13
Q
Angle Measure
A
The measure of an angle is the number of degrees through which one side must rotate to coincide with the other. Two measures are possible for all angles that don’t measure 180 degrees. We choose the smaller unless otherwise instructed.
14
Q
Angle Bisector
A
An angle bisector is a ray from an angle’s vertex through its interior that divides it into a pair of equal sub-angles. In symbols: if ray BP is the bisector of angle ABC, then BP passes through the interior of angles ABC, and the measure of angle ABP = the measure of angle CBP.
15
Q
Right Angle
A
An angle is right when it measures 90 degrees.
16
Q
Straight Angle
A
An angle is straight when its two sides form a line. (Note that the definition does not say what straight angles measure. Yes, they measure 180 degrees. But that’s a postulate not a definition).
17
Q
Opposite Rays
A
Two rays are opposite when they are coterminal and together they from a line.
18
Q
Acute Angle
A
An angle is acute when it measures less than 90 degrees.
19
Q
Obtuse Angle
A
An angle is obtuse when its measure is greater than 90 degrees and less than 180 degrees.
20
Q
Reflex Angle
A
An angle is reflex when it measures greater than 180 degrees.
21
Q
Adjacent Angles
A
Two angles are adjacent if they have a common side and that common side passes through the interior of the angle formed by the non-common sides.
22
Q
Vertical Angles
A
Opposite angle pairs formed when two lines intersect. Equivalently, when four coterminal rays form two lines, opposite pairs are vertical.
23
Q
Linear Pair
A
Two angles form a linear pair when they are adjacent and their non-common sides form a line.
24
Q
Complementary
A
Two angles are complementary when the sum of their measures is 90 degrees.
25
Supplementary
Two angles are supplementary when the sum of their measures in 180 degrees.
26
Perpendicular
Two lines are perpendicular when they from a right angle. (Note: the definition does not say that perpendicular lines form four right angles. That's true but it's not part of the definitions. It is a theorem!)
27
Polygon
A polygon is composed of sides. Each side is a line segment. Each side intersects precisely two others, one at one of its endpoints and a second at its other endpoint.
28
Polygon Side
The sides of a polygon are the line segments of which it is composed.
29
Polygon Vertex
The vertices of a polygon are the points at which its sides intersect.
30
Diagonal
A diagonal is a segment that joins a pair of non-adjacent vertices in a polygon.
31
Concave
A polygon is concave if at least one segment that joins points in its interior passes outside the polygon.
32
Convex
A polygon is convex if it is not concave (no segments that join points in its interior pass outside the polygon).
33
Equilateral
A polygon is equilateral if all its sides have the same length.
34
Equiangular
A polygon is equiangular if all its angles have the same measure.
35
Regular
A polygon is regular if it is both equilateral and equiangular.
36
Irregular
A polygon is irregular if it is not regular (the sides and angles do not measure the same)
37
Postulate
An unproven self-evident assumption.
38
Greater Than and Less Than
We say that p>q when there exists a positive quantity t such that p=q+t. We say that p
39
Inference
An inference links a pair of statements together in such a way that if the first is true, then the second must be true as well.
40
Theorem
A statement that has been proven true. (Not "can be" but "has been")
41
Proof
A sequence of statements that begins with the given and ends with the conclusion. If we place all givens at the start, each statement after the given follows from a prior statement or statements and is justified by some definition, postulate or previous Theron.
42
Lemma
A helper theorem, that is, a theorem whose purpose is to simplify the proof of a later, typically more significant theorem,
43
Corollary
A quick and easy consequence of a theorem already proven.
44
Counterexample
A counterexample is a specific example that contradicts a universal statement.
45
Conditional Statement
A conditional statement is one that can be written in "if...,then..." form.
46
Given and Conclusion
In a conditional statement, the sub-statement that follows the 'if' is the given and the statement that follows the 'then' is the conclusion.
47
Related Conditionals
A conditional has three related conditionals. They are it's converse, it's inverse and it's contrapositive.
48
Converse
To form the converse of a conditional, exchange hypothesis and conclusion. The the converse of conditional "If G, then C" is "If C then G."
49
Inverse
To form the inverse of a conditional, negate both hypothesis and conclusion. Thus the inverse of the conditional "If G, then C" is "If not G, then not C".
50
Contrapositive
To form the contrapositive of a conditional, negate both hypothesis and conclusion and exchange them. Thus the contrapositive of the conditional "If G, then C" is "If not C, then not G". Alternative definition: the contrapositive of a conditional is the inverse of its converse (equivalently, the converse of it's inverse".
51
Congruent Polygons
Two Polygons are congruent if sides and angles which correspond are equal in measure. This definition is often abbreviated as "CPCPE", short for "Corresponding Parts of Congruent Polygons are Equal".
52
Isosceles Triangle
A triangle is isosceles if it has at least two sides equal.
53
Legs, Base, Base Angles, Vertex Angle
In an isosceles triangle, if a pair of sides are equal in length we call them legs. (Notice that in an equilateral triangle, we may call any two sides legs.) The base is the third side. The base angles are the angles opposite the legs. The vertex angle is the angle opposite the base.
54
Construction
An addition made to a given diagram justified by a postulate or definition that asserts its existence.
55
Exterior Angle
The angle formed when a side of a polygon is extended through a vertex. An exterior angle forms a linear pair with the polygon angle to which it is adjacent.
56
Remote Angles
A triangle exterior angle has two remotes. These are the angles of the triangle with which it is not adjacent.
57
Parallel Lines
Lines are parallel when they are coplanar and do not intersect at just one point.
58
Parallel Segments, Parallel Rays
Two segments (or two rays) are parallel if the lines through them are parallel.
59
Skew Lines
Two lines are skew if they are non-coplanar and do not intersect.
60
Transversal
A transversal is a line that intersects each of two other coplanar lines at different points.
61
Greater Than
One quantity is greater than another when it is that other plus some positive quantity. In symbols: for any quantities p, q and a, p=q+a just if p>q.
62
Less Than
One quantity is less than another when it is that other minus some positive quantity. In symbols: for any quantities p, q, and a, p=q-a just if p
63
Anisosceles
A triangle is anisosceles when it has a pair of unequal sides.
64
Circle
A circle is a set of points on a given plane a given distance from a given point on that plane.
65
Radius
A radius is a segment that extends from the center of a circle to a point on the circle.
66
Chord
A chord is a segment that joins two points on a circle.
67
Diameter
A chord that passes through the circle's center.
68
Triangle Altitude
An altitude of a triangle is a segment from a vertex to the line through the opposite side, perpendicular to that line.
69
Distance
The distance between two figures is the length of the shortest segment that joins a point on one to a point on the other.
70
Parallelogram
a quadrilateral with both pairs of opposite sides parallel.
71
Rectangle
A quadrilateral with four right angles.
72
Rhombus
A quadrilateral with four equal sides.
73
Square
A rectangular rhombus (four right angles and equal sides).
74
Trapezoid
A quadrilateral with precisely one pair of parallel sides.
75
Trapezoid Bases
The bases of a trapezoid are its parallel sides.
76
Trapezoid Legs
The legs of a trapezoid are its non-parallel sides.
77
Isosceles Trapezoid
A trapezoid whose legs are equal in length.
78
Kite
A quadrilateral with two pairs of equal adjacent sides but not all sides are equal.
79
Ratio
A comparison of two quantities by means of division.
80
Proportion
An equation of ratios.
81
Similar
Two polygons are similar if sides which correspond are proportional and angels which correspond are equal.
82
Scale Factor
The factor by which we multiply the side lengths of one polygon to yield the side lengths of a second similar polygon.
83
Midsegment of a Triangle
A segment that joins the midpoints of two sides of a triangle.
84
Median of a Triangle
A segment that joins a vertex of a triangle to the midpoint of the opposite side.
85
Geometric Mean
The geometric mean of two (positive) quantities a and b is x such that a:x::x:b. Alternative definition: the geometric mean of two (positive) quantities is the square root of their product.
86
Opposite and Adjacent Leg of a Triangle
The leg adjacent to an acute angle of a right triangle is the leg that forms one of the angle's sides. The leg opposite an acute angle of a right triangle is the leg that does not form a side of that angle, I.e. the opposite leg is the leg that is not adjacent.
87
Remote Angles of a Triangle
A triangle exterior angle has two remotes. These are the angles of the triangle with which it is not adjacent.
88
Parallel Lines
Lines are parallel when they are coplanar and do not intersect at just one point.
89
Parallel Segments, Parallel Rays
Two segments (or two rays) are parallel if the lines through them are parallel.
90
The Trigonometric Functions
Let A be an acute angle of a right triangle. (I) The sine of A is the ratio of the leg opposite A to the hypotenuse. (ii) The cosine of A is the ratio of the leg adjacent to A to the hypotenuse. (III) The tangent of A is the ratio of the opposite leg to the adjacent leg.
91
The Inverse Trigonometric Functions
The Inverse sine of r is that angle measure A such that Sin A = r. Inverse cosine and inverse tangent are similar.
92
Disc
The union of a circle and its interior.
93
Arc
If we choose two points on a circle and a direction of travel from one to the other, an arc is those two points and all points on the circle encountered as we travel in our chosen direction from first to second.
94
Circumference
The total distance around a circle.
95
Pi
The ratio of a circle's circumference to the length of its diameter.
96
Central Angle
An angle whose vertex is the center of a given circle.
97
Intercepted Arc (or Arc Cut)
The arc that lies in the anterior of an angle.
98
Arc Measure
The measure of an arc equals the measure of the central angle that cuts it.
99
Semicircle
A half-circle.
100
Inscribed Angle
An angle whose sides are chords of a given circle.
101
Inscribed Polygon
A polygon whose sides are chords of a given circle.
102
Tangent Line, Tangent Segment and Point of Tangency
A line is tangent to a circle when the two are coplanar and intersect at precisely one point. We call their intersection the point of tangency. A segment is tangent to a circle when it intersects the circle and its extension is a line tangent to the circle.
103
Circumscribed Polygon
A polygon is circumscribed about a circle if each of its sides is tangent to the circle.
104
Area
The area of a figure is the number of unit squares needed to cover it without gap or overlap.
105
Center of a Regular Polygon
The point in the interior of a regular polygon that is equidistant from its sides. (As before, this definition does not guarantee the existence of such a point. How then do we know it exists? We prove it!)
106
Apothem
An apothem of a regular polygon is a segment from center to side perpendicular to the side.
107
Circumradius
A circumradius of a regular polygon is a segment from center to vertex.
108
Cyclic
A regular polygon is cyclic if a circle can be constructed that passes through each of its vertices.
109
Circle Sector
A portion of a disc bounded by two radii and an arc.
110
Circle Segment
A portion of a disc bounded by an arc and a chord.
111
Point on Concurrency
A point at which three or more lines intersect.
112
Circumcenter
The point of concurrency of the three perpendicular bisectors of the sides of a triangle. (Note that the definition does not guarantee the existence of such a point. It only tells us what to call it if in fact it exists. How do we know it exists? We prove it!)
113
Incenter
The point of concurrency of a triangle's three angle bisectors. (Note that the definition does not guarantee the existence of such a point. It only tells us what to call it if in fact it exists. How do we know it exists? We prove it!)
114
Centroid
The point of concurrency of a triangle's three medians (Note that the definition does not guarantee the existence of such a point. It only tells us what to call it if in fact it exists. How do we know it exists? We prove it!)
115
Orthocenter
The point of concurrency of a triangle's three altitudes. (Note that the definition does not guarantee the existence of such a point, It only tells us what to call it if in fact it exists. How do we know it exists? We prove it!)
116
Cevian
A segment whose endpoints are a vertex of a triangle and a point in the interior of the opposite side.
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