Title: Theory / Unimportant / Math / Geometry/ / Moise{ / Absolute
Tags: fromImplication récupéré toBinaryRelation MissingMedia fromMath typToBinary mathToSee closed

Name: A line
Typ: A set of points
Page: 44
Section: 2
Index: I-0

Name: A plane
Typ: A set of points
Page: 44
Section: 2
Index: I-0

Name: \(L\) lies in \(E\)
Definition: \(L\subseteq E\)
Page: 44
Chapter: 2
Variables: \(L\) a line
\(E\) a plane
Index: I-0

Name: \(P\) lies on \(L\)
Definition: \(P\in L\)
Page: 44
Chapter: 2
Variables: \(L\) a line
\(P\) a point
Index: I-0
Name2: \(L\) passes through \(P\)

Name: \(P\) lies in \(E\)
Definition: \(P\in E\)
Page: 44
Chapter: 2
Variables: \(E\) a plane
\(P\) a point
Index: I-0
Name2: \(E\) passes through \(P\)

Name: A figure
Definition: A set of points
Page: 44
Chapter: 2
Index: I-0

Name: A set of colinear points
Definition: A set of points lying on the same line
Page: 44
Chapter: 2
Index: I-0

Name: A set of coplanar points
Definition: A set of points lying in the same plane
Page: 44
Chapter: 2
Index: I-0

Question: The number of lines containing \(P\) and \(Q\)
Answer: is exactly 1.
Page: 44
Chapter: 2
Variables: \(P,Q\) points
Index: I-1

Question: The number of planes containing \(P,Q\) and \(R\)
Answer: is exactly 1.
Page: 44
Chapter: 2
Variables: \(P,Q,R\) points
Index: I-2
Assuming: The points are not colinear

Definition: The only line passing through \(P\) and \(Q\)
Page: 44
Chapter: 2
Variables: \(P,Q\) points
Notation: \(\overleftrightarrow{PQ}\)
Index: I-1

Definition: The only plane passing through \(P,Q\) and \(R\)
Page: 44
Chapter: 2
Variables: \(P,Q,R\) points
Notation: \(\overleftrightarrow{PQR}\)
Index: I-2
Assuming: Not colinear

Smaller0: \(P,Q\in E\)
Greater0: \(\subseteq E\)
Page: 45
Chapter: 2
Variables: \(E\) a plane,
\(P,Q\) points
Index: I-3
Smaller to greater: iff
Prefix Greater: \(\overleftrightarrow{PQ}\)

Question: The intersection of two planes is
Answer: either a line or the empty set.
Page: 45
Chapter: 2
Index: I-4

Question: The number of points of a line is
Answer: at least 2
Page: 45
Chapter: 2
Index: I-5

Question: The number of points of a plane is
Answer: at least 3, non-colinear
Page: 45
Chapter: 2
Index: I-5

Question: The number of points of a geometry
Answer: at least 4, non-coplanar
Page: 45
Chapter: 2
Index: I-5

Question: The intersection of two lines is
Answer: Either a singleton or empty
Page: 45
Chapter: 2
Index: 1
Kind: Theo

Smaller0: \(E\) does not contain \(L\)
Greater0: \(E\cap L\) is a singleton
Page: 45
Chapter: 2
Variables: \(E\) a plane,
\(L\) a line
Kind: Theo
Smaller to greater: iff
Assuming: \(E\) intersects \(L\)

Smaller0: \(P\not\in L\)
Greater0: \(P\cup L\) is contained in exactly one plane
Page: 46
Chapter: 2
Variables: \(P\) a point
\(L\) a line
Index: 3
Kind: Theo
Smaller to greater: iff
Extra: Theo state only one implication

Smaller0: \(L\) intersects \(L'\)
Greater0: \(L\cup L'\) lies in exactly one plane
Page: 46
Chapter: 2
Variables: \(L,L'\) 2 lines
Index: 4
Kind: Theo
Smaller to greater: iff
Extra: The theorem is an implication

Name: A coordinate system for \(L\)
Definition: A function \(f:L\leftrightarrow \mathbb R\) one-to-one such that \(PQ=|f(P)-f(Q)|\) for each \(P,Q\in L\).
Page: 58
Chapter: 2
Variables: \(L\) a line
Kind: Def

Name: The Ruler postulate
Definition: Every line has a coordinate system
Page: 58
Chapter: 2
Kind: Def

Name: The set of coordinate system for \(L\)
Page: 58, 59
Section: 2
Index: 1,2
Variables: \(L\) a line
Kind: Theo
Under: multiplication by -1,
Under2: addition of a constant

Name: A point between \(A\) and \(B\)
Definition: A point \(C\), colinear to \(AB\), such that \(AC+CB=AB\)
Page: 58, 59
Chapter: 2
Variables: \(A,B\) points

Name: \(B\) is between \(A\) and \(C\)
Page: 61, 127
Chapter: 2
Variables: \(A,B,C\) points
Notation: \(A-B-C\)
Kind: Def
Notation2: \(\mathcal B(A,B,C)\)

Definition: \(x<y<z\) or \(x>y>z\)
Page: 61
Chapter: 2
Variables: \(x,y,z\in\mathbb R\)
Notation: \(x-y-z\)
Kind: Def

Smaller0: \(f(A)-f(B)-f(C)\)
Greater0: \(A-B-C\)
Page: 61
Chapter: 2
Variables: \(A,B,C\) colinear points,
\(f\) a ruler for \(\overleftrightarrow {AB}\)
Index: 1
Kind: Lem
Smaller to greater: iff

Question: How many of those points are between the two other ?
Answer: exactly one
Page: 61
Chapter: 2
Variables: \(A,B,C\) three colinear points
Kind: Theo

Question: There exists a sequence \(u_1,u_2,u_3,u_4\) such that
Answer: \(A_{u_1}-A_{u_2}-A_{u_3}-A_{u_4}\)
Page: 62
Chapter: 2
Variables: \(A_1, A_2, A_3, A_4\) four colinear points
Index: B-3
Kind: Theo

Question: there exists \(C\) and \(D\) such that
Answer: \(A-D-B-C\)
Page: 63
Chapter: 2
Variables: \(A,B\) two points
Index: B-4
Kind: Theo

Smaller0: \(A-B-C\)
Greater0: \(C-B-A\)
Page: 64
Chapter: 2
Variables: \(A,B,C\) three points
Index: B-1
Smaller to greater: iff

Name: The angle whose rays are \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\)
Page: 65
Chapter: 3
Section: 5
Variables: \(A,B,C\) three points,
Notation: \(\angle BAC\)
Notation2: \(\angle A\), assuming \(B,C\) are clear from the context

Name: The sides of \(\angle BAC\)
Definition: \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\)
Page: 65
Chapter: 3
Section: 5

Name: The vertex(ices) of \(\angle BAC\)
Definition: \(A\)
Page: 65
Chapter: 3
Section: 5

Name: The triangle with ends \(A\), \(B\) and \(C\)
Definition: \(\overline{AB}\cup\overline{AC}\cup\overline{CB}\) for three non colinear points \(A,B,C\)
Page: 65
Chapter: 3
Section: 5
Notation: \(\triangle ABC\)

Name: An angle
Definition: The union of two rays having the same end
Page: 65
Chapter: 3
Section: 5
Denoted by: \(\angle BAC\)

Name: The ray from \(A\) through \(B\)
Definition: The union of the segment \(\overline{AB}\) and of the points \(C\) such that \(B\) is between \(A\) and \(B\)
Page: 65
Chapter: 3
Section: 5
Variables: \(A,B\) two points
Notation: \(\overrightarrow{AB}\)

Name: The segment between \(A\) and \(B\)
Definition: The set of points \(A,B\) or are between \(A\) and \(B\),
Page: 65
Chapter: 3
Section: 5
Variables: \(A,B\) two points
Notation: \(\overline{AB}\)

Definition: The triangle \(\overline{AB}\cup\overline{AC}\cup\overline{CB}\)
Page: 65
Chapter: 3
Section: 5
Variables: \(A,B,C\) three points
Notation: \(\triangle ABC\)
Assuming: \(A,B,C\) not colinear

Name: The vertices of \(T\)
Definition: \(A\), \(B\) and \(C\)
Page: 65
Chapter: 3
Section: 5
Name2: A vertex of \(T\)
Assuming: \(A,B,C\) not colinear

Name: The sides of \(\triangle ABC\)
Definition: \(\overline{AB}\), \(\overline{BC}\) and \(\overline{AC}\)
Page: 65
Chapter: 3
Section: 5
Assuming: \(A,B,C\) not colinear

Name: The angles of \(\triangle ABC\)
Definition: \(\angle{ABC}\), \(\angle{BCA}\) and \(\angle{CBA}\)
Page: 66
Chapter: 3
Section: 5
Assuming: \(A,B,C\) not colinear

Smaller0: \(\overline{AB}\)
Greater0: \(\overline{BA}\)
Page: 66
Chapter: 3
Section: 5
Variables: \(A,B\) two points,
Index: 1
Kind: Theo
Smaller to greater: =

Smaller0: \(C\in\overrightarrow{AB}\)
Greater0: \(\overrightarrow{AB}=\overrightarrow{AC}\)
Page: 66
Chapter: 3
Section: 5
Index: 2
Kind: Theo

Smaller0: \(B'\in\overrightarrow{AB}\) and \(C'\in\overrightarrow{AC}\)
Greater0: \(\angle{BAC}=\triangle{B'AC'}\)
Page: 66
Chapter: 3
Section: 5
Kind: Theo

Smaller0: \(\overline{AB}=\overline{CD}\)
Greater0: \(\{A,B\}=\{C,D\}\)
Page: 66
Chapter: 3
Section: 5
Variables: \(A,B,C,D\) points,
Index: 4
Kind: Theo

Smaller0: \(\triangle{ABC}=\triangle{DEF}\)
Greater0: \(\{A,B,C\}=\{D,E,F\}\)
Page: 66
Chapter: 3
Section: 5
Variables: \(A,B,C,D,E,F\) points,
Kind: Theo
Smaller to greater: iff

Page: 69
Definition: Segments of the same length
Chapter: 3
Section: 5
Notation: \(\cong\)
Name: Congruent segments
Kind: Def, Theo
Index: ,C-1
Context: Segments
Equivalence: Equivalence relation

Question: The number of point \(E\in \overrightarrow{CD}\) such that \(\overline{AB}\cong\overline{CE}\) is
Answer: exactly one
Page: 69
Chapter: 3
Section: 5
Variables: \(A,B,C,D\) points
Index: C-2
Kind: Theo
Name: Segment Construction Theorem

Smaller0: \(A-B-C\) and \(A'-B'-C'\)
Greater0: \(\overline AC \cong\overline{A'C'}\)
Page: 70
Chapter: 3
Section: 5
Index: C-3
Kind: Theo
Smaller1: \(\overline AB \cong\overline{A'B'}\) and \(\overline BC \cong\overline{B'C'}\)
Name: The segment-addition theorem

Smaller0: \(A-B-C\) and \(A'-B'-C'\)
Greater0: \(\overline BC \cong\overline{B'C'}\)
Page: 70
Chapter: 3
Section: 5
Index: C-4
Kind: Theo
Smaller1: \(\overline AB \cong\overline{A'B'}\) and \(\overline AC \cong\overline{A'C'}\)
Name: The segment-substraction theorem

Name: A midpoint of \(\overline{AC}\)
Definition: A point \(C\) such that \(A-C-B\) and \(\overline{AC}\cong\overline{CB}\)
Page: 71
Chapter: 3
Section: 5
Variables: \(A,B\) points
Cardinal: Exactly one

Name: A convex set (of points)
Definition: A set \(X\) (of points) such that, for each \(A,B\in X\), \(\overline{AB}\subseteq X\)
Page: 72, 73
Chapter: 4
Section: 1
Example: segments, lines,
Example2: triangle
Example3: a singleton, the entire set of points
Example4: half-planes, planes

Name: A half plane
Definition: A set containing every points \(P\in E\) such that \(\overline{PP'}\) does not intersect \(L\), for \(P'\in E\), \(L\subseteq E\), points, line and plane
Page: 73
Chapter: 4
Section: 1

Question: \(E\setminus L\) is the union of
Answer: Two disjoint convex sets such that, for \(P,Q\) in those two sets, then \(\overline{PQ}\) intersects \(L\)
Page: 74
Chapter: 4
Section: 1
Variables: \(L\subseteq E\) a line and a plane,
Index: PS-1
Name: Plane-Separation postulate

Name: The edge of \(H\)
Definition: A line \(L\) such as in the definition of half planes
Page: 74
Chapter: 4
Section: 1
Variables: \(H\) an half-plane

Smaller0: \(L\) contains a point between \(A\) and \(C\)
Greater0: \(L\) intersects \(\overline AB\) or \(\overline BC\)
Page: 74
Chapter: 4
Section: 1
Variables: \(\triangle ABC\)
Kind: Theo
Smaller1: \(L\) coplanar

Name: The half planes of \(L\)
Definition: The two subsets of \(E\) as in the definition of half plane
Page: 74
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
Index: 2
Name2: The sides of \(L\)

Name: Two points on opposite sides of \(L\)
Definition: Two points belonging each to a distinct sides of \(L\)
Page: 74
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
Index: 2

Smaller0: \(P_i\) is on the opposite side of \(Q\)
Greater0: \(P_1,P_2\) are on the same side
Page: 76
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
\(P_1,P_2,Q\in E\setminus E\)
Index: 2
Smaller to greater: iff
Extra: Theorem is an implication

Smaller0: \(P,R\) are on the same side of \(L\) and \(P,Q\) are on opposite side
Greater0: \(Q,R\) are on opposite side
Page: 76
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
\(P,Q,R\in E\setminus E\)
Index: 2

Question: The number of sides crossed by \(\overrightarrow{AB}\)
Answer: one
Page: 76
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
\(A\in L\), \(B\in E\setminus L\)
Index: 3
Kind: Theo

Question: The number of sides crossed by \(\overline{AB}\)
Answer: one
Page: 76
Chapter: 4
Section: 1
Variables: \(P\) a plane
\(L\subseteq E\) a line
\(A\in L\), \(B\in E\setminus L\)
Kind: Theo

Name: The interior of \(\angle BAC\)
Definition: The intersection of (the half-planes \(A_iB\) containing \(A_{3-i}\))
Page: 76
Chapter: 4
Section: 1

Question: Its interior of \(\angle BAC\)
Answer: \(\overline{BC}\setminus\{B, C\}\)
Page: 78
Chapter: 4
Section: 1
Index: 5
Kind: Theo

Smaller0: \(F\) is in the interior of \(\angle BAC\)
Greater0: \(\overrightarrow{BF}\setminus\{F\}\) is included in the interior of \(\angle BAC\)
Page: 78
Chapter: 4
Section: 1
Variables: \(F\) a point
Index: 6
Kind: Theo
Smaller to greater: iff
Extra: Theorem is an implication

Name: The interior of \(\triangle ABC\)
Definition: The intersection of the sides of the triangle containing the third points
Page: 80
Chapter: 4
Section: 1
Index: , 8
Kind: , Theo
Definition2: The intersections of the interiors of the angles of the triangles
Typ: A convex set

Smaller0: \(B,G\) lies on different sides of \(L\)
Greater0: \(\overline{FB}\) does not intersect \(\overrightarrow{AG}\)
Page: 81
Chapter: 4
Section: 3
Index: 1
Kind: Theo

Smaller0: \(D\) and \(B\) are on the same side of \(\overleftrightarrow{FC}\)
Greater0: \(\overrightarrow{AD}\) intersects either \(\overline{FB}\) or \(\overline{BC}\)
Page: 81
Chapter: 4
Section: 3
Index: 2
Kind: Theo
Smaller1: \(F-A-C\)
Extra: G

Smaller0: \(D\) is in the interior of the triangle
Greater0: \(\overrightarrow{AD}\) intersects \(\overline{BC}\setminus\{B,C\}\)
Page: 81
Chapter: 4
Section: 3
Kind: Theo
Name: The Crossbar Theorem

Name: A quadrilateral
Definition: coplanar points
Page: 83
Chapter: 4
Definition2: not colinear
Conjdef: with
Definition3: sides intersects only at their end
Denoted by: \(\square ABCD\)
DefType: \(AB\cup BC \cup CD \cup DA\)

Name: The angles of \(\square ABCD\)
Definition: \(\angle ABC\), \(\angle BCD\), \(\angle CDA\), \(\angle DAB\)
Page: 83
Chapter: 4

Name: The sides of \(\square ABCD\)
Definition: \(\overline {AB}\), \(\overline {BC}\), \(\overline {CD}\), \(\overline {DA}\)
Page: 84
Chapter: 4

Name: A pair of adjacent sides of \(\square ABCD\)
Definition: A pair of sides of \(\square ABCD\) with an intersecting point
Page: 84
Chapter: 4

Name: A pair of opposite sides of \(\square ABCD\)
Definition: A pair of sides of \(\square ABCD\) with no intersection
Page: 84
Chapter: 4

Name: A pair of adjacent angles
Definition: Two angles of \(\square A_0A_1A_2A_3\) whose intersection is a side
Page: 84
Chapter: 4

Name: A pair of opposite angles
Definition: Two angles of \(\square A_0A_1A_2A_3\) whose intersection is empty
Page: 84
Chapter: 4

Name: The diagonals of \(\square ABCD\)
Definition: \(\overline {AC}\) and \(\overline{BD}\)
Page: 84
Chapter: 4

Name: A convex quadrilateral
Definition: A quadrilateral such that each of its sides lies in one of the half planes determined by the opposite side.
Page: 84
Chapter: 4
Definition2: The interior of \(ABCD\) is convex
Definition3: The diagonals \(\overline{AC}\) and \(\overline{BD}\) intersects

Smaller0: all points of \(\overline{AB}\) are on the same side of \(\overleftrightarrow{CD}\)
Greater0: \(A\) and \(B\) are on the same side of \(\overleftrightarrow{CD}\) (addition modulo 4)
Page: 84
Chapter: 4
Smaller to greater: iff

Smaller0: the interior of \(Q\) is convex
Greater0: \(Q\) is convex
Page: 84
Chapter: 4
Smaller to greater: iff

Smaller0: \(Q\) is convex
Greater0: intersects
Page: 84
Chapter: 4
Index: 1
Kind: Theo
Prefix Greater: its diagonal

Question: The set of all points that do not lie in \(E\) is
Answer: two disjoint convex sets such that, for \(P,Q\) in those two sets, then \(\overline{PQ}\) intersects \(E\)
Page: 85
Chapter: 4
Section: 5
Variables: \(E\) a plane,
Index: PS-1
Name: Space-Separation postulate

Name: A dihedral angle
Definition: \(H_1\cup H_2\cup L\) where \(H_1,H_2\) are two non-coplanar half planes with side \(L\)
Page: 86
Chapter: 4
Section: 5

Name: The edge of \(D\)
Definition: \(L\)
Page: 86
Chapter: 4
Section: 5
Variables: D=

Name: The sides of \(D\)
Definition: \(H_1\) and \(H_2\)
Page: 86
Chapter: 4
Section: 5
Variables: D=

Smaller0: \(E\) does not contains \(H\)
Greater0: \(H\) lies in a single side of \(E\)
Page: 86
Chapter: 4
Section: 5
Variables: \(H\) a half plane with edge \(L\),
\(E\) a plane
Kind: Theo

Name: The interior of \(D\)
Definition: The intersections of the sides of the plane of \(H_1\) containing \(H_2\) and of the side of the plane of \(H_2\) containing \(H_1\)
Page: 86
Chapter: 4
Section: 5
Variables: D=
Index: , 6
Kind: , Theo
Typ: A convex set

Smaller0: \(P_1\in H_1\), \(P_2\in H_2\)
Greater0: in the interior of the dihedral angle
Page: 87
Chapter: 4
Section: 5
Variables: D=
\(P_1,P_2\) points
Index: 7
Kind: Theo
Prefix Greater: the points between the  \(P_1\) and \(P_2\) are

Name: note
Definition: missing
Page: 138, 144, 145, 215, 216, 218-219, 221, 222, 229,230, 416, 454
Chapter: 4
Section: 5

Name: The set of angles
Page: 93
Chapter: 5
Context: Geometry
Notation: \(\mathcal A\)

Name: The set of lines
Page: 93 ?
Chapter: 5
Notation: \(\mathcal L\)

Name: The set of points
Page: 93 ?
Chapter: 5
Notation: \(S\)
Name2: The space

Name: The set of plane
Page: 93 ?
Chapter: 5
Notation: \(\mathcal P\)

Name: An angle measure
Typ: A function \(\mathcal A\to[0,180]\)
Page: 93
Chapter: 5
Denoted by: \(m\)
Index: M-1,M-2
Kind: Postulate

Question: The number of rays \(\overrightarrow{AP}\) with \(P\in H\) such that \(m\angle BAP=r\) is
Answer: exactly one
Page: 95
Chapter: 5
Variables: \(A,B\) points,
\(H\) an half plane determined by \(\overleftrightarrow{AB}\),
\(r\in[0,180]\)
Index: M-3
Kind: Postulate
Name: Angle-Construction Theorem

Smaller0: \(D\) is in the interior of \(\angle BAC\)
Greater0: \(m\angle BAC=m\angle BAD+m\angle DAC\)
Page: 96
Chapter: 5
Variables: \(D\)
Index: M-4
Kind: Postulate
Smaller to greater: iff
Extra: Postulate is just an implication

Name: Opposites rays
Definition: A pair \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\) where \(A-B-C\).
Page: 76
Chapter: 4
Section: 2

Name: A linear pair of angles
Definition: A pair \((\angle BAD, \angle DAC)\) where \(B-A-C\).
Page: 96
Chapter: 5

Name: Supplementary angles
Definition: A pair of angle whose sum of degree is 180
Page: 96
Chapter: 5

Smaller0: \(\angle BAD\) and \(\angle DAC\) forms a linear pair
Greater0: \(\angle BAD\) and \(\angle DAC\) are supplementary
Page: 96
Index: M-5
Kind: Postulate
Name: The Supplement postulate

Page: 97
Definition: The relation between angles with equal measure
Chapter: 5
Notation: \(\cong\) for angles
Name: Congruence for angles
Kind: Def, Theo
Index: , 1
Equivalence: Equivalent relation

Name: A right angle
Definition: An angle which is congruent to another angle in the same linear pair.
Page: 97
Chapter: 5
Kind: Def
Definition2: An angle whose degree is 90

Question: The number of ray \(\overrightarrow{B'A'}\) with \(A'\in H\) such that \(\angle ABC=\angle A'B'C'\)
Answer: is one.
Page: 97
Variables: \(\angle ABC\),
\(\overrightarrow{B'C'}\)
\(H\) a half plane with edge \(\overleftrightarrow{B'C'}\)
Index: 2
Kind: Theo

Smaller0: \(\angle BAD\cong\angle B'A'D'\) and \(\angle CAD\cong\angle C'A'D'\)
Greater0: \(\angle BAC\cong\angle B'A'C'\)
Page: 97
Index: 3
Kind: Theo
Smaller1: \(D\) in the interior of \(\angle BAC\)
\(D'\) in the interior of \(\angle B'A'C'\)
Name: Angle addition theorem

Smaller0: \(\angle BAD\cong\angle B'A'D'\) and \(\angle BAC\cong\angle B'A'C'\)
Greater0: \(\angle CAD\cong\angle C'A'D'\)
Page: 98
Index: 4
Kind: Theo
Smaller1: \(D\) in the interior of \(\angle BAC\)
\(D'\) in the interior of \(\angle B'A'C'\)
Name: Angle substraction theorem

Page: 98
Definition: The relation between rays/lines/segment whose union is a right angle.
Chapter: 5
Notation: \(\bot\)
Name: Perpendicular

Name: An acute angle
Definition: An angle whose measure is less than \(90\)
Page: 98
Chapter: 5

Name: An obtuse angle
Definition: An angle whose measure is greater than \(90\)
Page: 98
Chapter: 5

Page: 98
Definition: The smaller relation for the measure of angle
Chapter: 5
Notation: \(<\)
Name: Smaller relation for angle

Name: Angles in vertical pair
Definition: Angles whose sides form pairs of opposite rays.
Page: 98
Chapter: 5

Smaller0: \(\angle BAC\) and \(\angle B'AC'\) forms a vertical pair
Greater0: \(\angle BAC\) and \(\angle B'AC'\) are congruent
Page: 98
Index: 5
Kind: Theo
Name: The Vertical Angle Theorem

Smaller0: \(L\) and \(L'\) forms one right angle
Greater0: \(L\) and \(L'\) forms four right angle
Page: 98
Variables: \(L,L'\in\mathcal L\) intersection
Index: 6
Kind: Theo
Smaller to greater: iff
Extra: Theorem 6 is an implicaton

Page: 101
Definition: A bijective relation between two elements of \(S^3\)
Chapter: 6
Section: 1
Name: A correspondance on \(S^3\)

Page: 101
Definition: The relation between triangles whose vertices can be reordered in a congruent way
Chapter: 6
Section: 1
Name: A congruence of triangle

Page: 102
Definition: The relation on 3-tuple of vertices \(A^i_1A^i_2A^i_3\) such that \(\angle A^1_j\cong A^2_j\) and \(\overline{A^1_{j}A^1_{j'}}\cong \overline{A^2_{j}A^2_{j'}}\)
Chapter: 6
Section: 1
Notation: \(\leftrightarrow\)
Name: A congruence
Context: \(S^3\)

Smaller0: \(\overline AB\cong\overline DE\)
Greater0: the \(\triangle ABC\) and \(\triangle DEF\) are congruent
Page: 103
Chapter: 6
Section: 1
Variables: \(\leftrightarrow\) a congruence between those triangles
Index: SAS
Kind: Postulate
Smaller1: \(\overline BC\cong\overline EF\)
Name: Side Angle Side (postulate)
Smaller2: \(\angle ABC\cong\angle DEF\)

Name: An isoceles triangle
Definition: A triangle whose at least two sides are congruent.
Page: 104
Chapter: 6
Section: 2

Name: An equilateral triangle
Definition: A triangle whose three sides are congruent
Page: 104
Chapter: 6
Section: 2

Name: A scalene triangle
Definition: A triangle which is note isoceles
Page: 104
Chapter: 6
Section: 2

Smaller0: \(\overline {AB}\cong\overline{BC}\)
Greater0: \(\angle {CAB}\cong\angle{BCA}\)
Page: 104, 107
Chapter: 6
Section: 2
Index: 1, 2-1
Kind: Theo, Cor
Smaller to greater: iff
Name: Isoceles Triangle Theorem

Name: The base angle of \(T\)
Definition: The angles \(\angle BAC\) and \(\angle ACB\) opposite the two congruent sides.
Page: 104
Chapter: 6
Section: 2
Variables: \(T\)=

Definition: The conclusion of the illustrated theorem
Page: 104
Chapter: 6
Section: 2
Notation: ! in figures

Definition: Notation stating that multiple angles, segments... are congruent
Page: 104
Chapter: 6
Section: 2
Notation: Some symbol appearing many times in a figure, annotating angles, segments...

Name: An equiangular triangle
Definition: A triangle whose angles are congruent
Page: 106
Chapter: 6
Section: 2
Extra: similar?

Smaller0: \(\triangle ABC\) is equiangular
Greater0: \(\triangle ABC\) is equilateral
Page: 106, 107
Chapter: 6
Section: 2
Index: 1, 2-2
Kind: Theo, Cor
Smaller to greater: iff

Smaller0: \(\angle A\cong\angle D\)
Greater0: The \(\triangle ABC\cong\triangle DEF\)
Page: 106
Chapter: 6
Section: 2
Variables: \(\leftrightarrow\) a congruence between those triangles
Kind: Theo
Smaller1: \(\angle C\cong\angle F\)
Name: Angle Side Angle (postulate)
Smaller2: \(\overline{AC}\cong\overline{DF}\)
Abbreviation: ASA theorem

Smaller0: \(\overline{AB}\cong\overline{DE}\) and \(\overline{BC}\cong\overline{EF}\) and \(\overline{CA}\cong\overline{FD}\)
Greater0: \(\triangle ABC\cong\triangle DEF\)
Page: 107
Chapter: 6
Section: 2
Index: 3
Kind: Theo
Smaller to greater: iff
Name: Side Side Side (Theorem)
Abbreviation: SSS theorem

Name: The bisector of \(\angle ABC\)
Definition: A ray \(\overrightarrow{AD}\) such that \(D\) is in the interior of \(\angle ABC\) and \(\angle ABD\cong\angle DBC\)
Page: 109
Chapter: 6
Section: 2
Cardinal: 1

Question: The number of perpendicular to \(L\) passing through \(P\) is
Answer: exactly one
Page: 114, 118
Chapter: 6, 7
Section: 5
Variables: \(L\in\mathcal L\),
\(P\not\in L\)
Index: 1, 1-1
Kind: Theo, Cor

Definition: \(m\angle ABC<m\angle DEF\)
Page: 116
Chapter: 7
Notation: \(\angle ABC<\angle DEF\)

Page: 116
Definition: There exists a point \(E\) between \(C\) and \(D\), such that \(\overline{AB}\cong\overline{CE}\)
Chapter: 7
Variables: \(A,B,C,D\) points
Notation: \(\overline{AB}<\overline {CD}\)
Transitive: yes

Name: An exterior angle of \(\triangle ABC\)
Definition: Some \(\angle BCD\), where \(A-C-D\).
Page: 117
Chapter: 7

Question: The number of exterior angle of \(T\) is
Answer: 6
Page: 117
Chapter: 7
Variables: \(T\) =

Name: The remote interior angle of the exterior angles \(\angle BCD\)
Definition: \(\angle A\) and \(\angle C\)
Page: 117
Chapter: 7

Smaller0: \(\overline{AB}>\overline{AC}\)
Greater0: \(\angle C>\angle B\)
Page: 119
Chapter: 7
Index: 2, 3
Kind: Theo
Smaller to greater: iff

Question: The shortest segment joining \(P\) to \(L\) is
Answer: \(\overline{PQ}\) such that \(\overline{PQ}\bot L\)
Page: 120
Chapter: 7
Variables: \(L\in\mathcal L\),
\(P\)

Smaller0: \(AB+BC\)
Greater0: \(AC\)
Page: 120
Chapter: 7
Variables: \(\triangle ABC\)
Index: 5
Kind: Theo
Smaller to greater: >
Name: Triangular inequality
Greater to smaller: <

Smaller0: \(\overline{AB}\cong\overline{DE}\)
Greater0: \(\overline{BC}>\overline{EF}\)
Page: 121
Chapter: 7
Index: 6
Kind: Theo
Assuming: \(A^1_jB^1 \cong A^2_jB^2\)
Smaller1: \(\overline{AC}\cong\overline{DF}\)
Name: The Hinge Theorem
Smaller2: \(\angle A>\angle D\)

Smaller0: \(\angle ACB\cong\angle EFD\)
Greater0: \(\triangle ABC\) and \(\triangle DEF\) are congruents
Page: 123
Chapter: 7
Kind: Theo
Smaller1: \(\angle BAC\cong\angle DEF\)
Name: The SAA Theorem
Smaller2: \(\overline{AB}\cong\overline{DE}\)

Name: A right triangle
Definition: A triangle with a right angle
Page: 123
Chapter: 7
Kind: Def

Name: The hypothenuse of a triangle
Definition: The side opposed to the right angle
\(BC\)
Page: 123
Chapter: 7
Kind: Def

Name: The leg(s) of \(T\)
Definition: The sides which of \(T\) which is not the hypothenuse. I.e. \(AC\) and \(AB\)
Page: 123
Chapter: 7
Variables: \(T=\)
Kind: Def

Page: 127
Definition: A given relation on triples of points
Chapter: 8
Section: 1
Notation: \(\mathcal B\)
Name: Betweenness
Context: geometry

Name: Betweenness Postulates
Definition: \(A-B-C\) implies \(C-B-A\)
Page: 127
Chapter: 8
Section: 1
Definition2: for \(L\in\mathcal L\), \(A,B,C\in L\), exactly one of \(A-B-C\), \(B-C-A\) or \(C-A-B\) holds
Definition3: Any four points of a line can be named in an order \(A, B, C, D\), in such a way that \(A-B-C-D\).
Definition4: For each \(A,B\), there exists \(C,D\) such that \(A-C-B-D\),
Definition5: \(A_1-A_2-A_3\) implies there exists \(L\in\mathcal L\) such that \(A_i\in\mathcal L\) and \(A_i\ne A_j\)

Name: Congruence Postulates for Angles
Definition: \(\cong\) is an equivalence relation over angles
Page: 127
Chapter: 8
Section: 1
Definition2: Angle-construction postulate
Definition3: Angle addition/substraction Postulate
Definition4: SAS

Page: 128
Definition: There is a point \(B'\) between \(C\) and \(D\) such that \(\overline{AB}\cong\overline{CB'}\).
Chapter: 8
Section: 2
Variables: \(S\) a non-metric space,Let \(A,B,C,D\in S\)
Notation: \(\overline{AB}<\overline{CD}\)
Transitive: Order relation compatible with \(\cong\)

Page: 131
Definition: There is a point \(G\) in the interior of \(\angle DEF\) such that \(\overline{ABC}\cong\overline{GEF}\).
Chapter: 8
Section: 2
Variables: \(S\) a non-metric space,Let \(A,B,C,D,E,F\in S\)
Notation: \(\angle{ABC}<\angle{DEF}\)
Transitive: Order relation compatible with \(\cong\)

Smaller0: \(\angle BAC\) and \(\angle CAD\) form a linear pair, \(\angle B'A'C'\) and \(\angle C'A'D'\) form a linear pair,
Greater0: \(\angle BAC\cong\angle B'A'C'\)
Page: 132
Chapter: 8
Section: 2
Variables: \(S\) a non-metric space,Let \(\angle B^iA^iC^i_j\) forms a linear pair for \(j\in\{1,2\}\),
Index: 1
Kind: Theo
Smaller1: \(\angle CAD\cong\angle C'A'D'\)

Smaller0: \(\alpha\) is congruent to a right angle
Greater0: \(\alpha\) is a right angle
Page: 132
Chapter: 8
Section: 2
Variables: \(S\) a non-metric space,
\(\alpha\) an angle
Index: 1
Kind: Theo
Smaller to greater: iff

Smaller0: \(\alpha\) and \(\alpha'\) are right angles
Greater0: \(\alpha\) and \(\alpha'\) are congruents
Page: 133
Chapter: 8
Section: 2
Variables: \(S\) a non-metric space,
\(\alpha,\alpha'\) angles
Index: 3
Kind: Theo

Page: 136
Definition: The set of all segments congruent to \(\overline{AB}\)
Chapter: 8
Section: 2
Variables: \(A,B\in S\)
Notation: \([\overline{AB}]\)
Name: The congruence class of \(\overline{AB}\)
Context: Geometry

Name: Orthogonal circles
Page: 141
Chapter: 9

Name: A Saccheri quadilaterals
Definition: A quadrilateral \(\square ABCD\) with \(\angle BAD\), \(\angle CDA\) right and \(AB=CD\).
Page: 152
Chapter: 10
Section: 3

Name: The lower base of \(S\)
Definition: \(\overline{AD}\)
Page: 152
Chapter: 10
Section: 3
Variables: \(S=\) a Saccheri quadilaterals

Name: The upper base of \(S\)
Definition: \(\overline{BC}\)
Page: 152
Chapter: 10
Section: 3
Variables: \(S=\) a Saccheri quadilaterals

Name: The lower base angles of \(S\)
Definition: \(\angle BAD,\angle CDA\)
Page: 152
Chapter: 10
Section: 3
Variables: \(S=\) a Saccheri quadilaterals

Name: The upper base angles of \(S\)
Definition: \(\angle ABC,\angle DCB\)
Page: 152
Chapter: 10
Section: 3
Variables: \(S=\) a Saccheri quadilaterals

Question: Its diagonals of \(S\) are
Answer: congruent
Page: 152
Chapter: 10
Section: 3
Variables: \(S=\) a Saccheri quadilaterals
Index: 1
Kind: Theorem

Page: 148
Definition: Lines, lying in the same plan, not intersecting
Chapter: 10
Section: 1
Name: Parallel lines
Equivalence: An equivalence relation

Page: 148
Definition: Segment whose lines are parallel
Chapter: 10
Section: 1
Name: Parallel segment
Transitive: An equivalence relation

Question: The number of lines, through \(P\), parallel to \(L\) is
Answer: at least one
Page: 149
Chapter: 10
Section: 1
Variables: \(L\in\mathcal L\),
\(P\in S\setminus L\)
Index: 2
Kind: Theo

Name: A transversal to \(L_1,L_2\)
Definition: A line, in the plane of the \(L_i\)'s cutting the \(L_i\) in distinct points.
Page: 149
Chapter: 10
Section: 1
Variables: \(L_1,L_2\)
Context: Plane geometry

Name: Alternate interior angles
Definition: A pair of angles \(A_1B_1B_2\) and \(B_1B_2A_2\), where \(\overleftrightarrow{B_1B_2}\) separates \(\overline{A_1A_2}\).
Page: 150
Chapter: 10
Section: 1

Page: 150
Definition: The relation between an angle and the corresponding angle of its vertical angle
Chapter: 10
Section: 1
Name: Corresponding angles

Smaller0: \(A_1A_n\)
Greater0: \(\sum A_iA_{i+1}\)
Page: 151
Chapter: 10
Section: 2
Variables: \((A_i)_{i\in[1,n]}\in S\)
Kind: Theo
Smaller to greater: \(\le\)
Name: The polygonal inequality
Greater to smaller: \(\ge\)

Question: \(m\angle B+m\angle D\)
Answer: \(\le90\)
Page: 155
Chapter: 10
Section: 4
Variables: \(\triangle ABD\) right at \(A\)
Index: 2
Kind: Theo

Question: \(m\angle B\)
Answer: \(<90\)
Page: 155
Chapter: 10
Section: 4
Kind: Theo

Question: Its longer side
Answer: is its hypothenuse
Page: 156
Chapter: 10
Section: 4
Index: 5
Kind: Theo

Smaller0: \(R\cong R '\)
Greater0: \(R\) and \(R'\) have the same measure
Page: 185
Chapter: 13
Section: 1
Variables: \(R,R'\in\mathcal R\)
Name: The Congruence Postulate

Smaller0: \(R\cap R '\) is a union of edges and vertices
Greater0: the area of their union is the sum of their area
Page: 185
Chapter: 13
Section: 1
Variables: \(R,R'\in\mathcal R\)
Name: The Additivity Postulate
Counterexample: ?

Name: An equidistant points from \(P\) and \(Q\)
Definition: A point \(A\) such that \(AP=AQ\)
Page: 212
Chapter: 15
Section: 1
Variables: \(P,Q\) points

Name: \(L\) is perpendicular to \(E\)
Definition: \(L\) intersects \(E\) and is perpendicular to every/two lines of \(E\) intersecting \(L\)
Page: 212, 213
Chapter: 15
Section: 1
Variables: \(L\in\mathcal L\) a line,
\(E\in\mathcal S\) a space
Index: 3
Kind: Theo

Question: The number of perpendicular to \(E\) through \(A\) is
Answer: exactly one
Page: 217
Chapter: 15
Section: 1
Variables: \(A\) a point and
\(E\) a plane
Index: 13-17
Kind: Theo

Name: The circle with center \(P\) and radius \(r\)
Definition: The set of points \(Q\in E\) such that \(\overline{PQ}\cong r\)
Page: 224
Chapter: 16
Section: 1
Variables: \(E\) a plane,
\(P\in E\) a point,
\(r>0\)

Name: Concentric cercles
Definition: Two cercles with the same center
Page: 224
Chapter: 16
Section: 1

Name: A center of \(C\)
Definition: A point \(P\) such that for all \(Q,Q'\in C\), \(\overline{PQ}\cong \overline{PQ'}\)
Page: 224
Chapter: 16
Section: 1
Variables: \(C\) a circle

Name: A radius of \(C\)
Definition: A segment \(\overline{PQ}\) for \(P\) a center and \(Q\in C\)
Page: 224
Chapter: 16
Section: 1
Variables: \(C\) a circle

Name: The outer end of \(\overline{PQ}\)
Definition: \(Q\)
Page: 224
Chapter: 16
Section: 1
Context: \(\overline{PQ}\)

Name: A chord
Definition: \(\overline{QR}\) for \(Q,R\in C\)
Page: 224
Chapter: 16
Section: 1
Variables: \(C\) a circle,

Name: A diameter
Definition: A chord containing a center
Page: 224
Chapter: 16
Section: 1
Variables: \(C\) a circle,

Name: The interior of the circle with center \(P\) and radius \(\overline{AB}\)
Definition: The set of points \(Q\) such that \(\overline{PQ}<\overline{AB}\)
Page: 224
Chapter: 16
Section: 1
Variables: \(P\) a point,
\(\overline{AB}\),

Name: The exterior of the circle with center \(P\) and radius \(\overline{AB}\)
Definition: The set of points \(Q\) such that \(\overline{PQ}>\overline{AB}\)
Page: 224
Chapter: 16
Section: 1
Variables: \(P\) a point,
\(\overline{AB}\),

Name: A tangent to \(C\)
Definition: A line whose intersection with \(C\) is a singleton
Page: 225
Chapter: 16
Section: 2
Variables: \(C\) a circle

Name: The point on tangency of \(L\) and \(C\)
Definition: The element of \(L\cap C\)
Page: 225
Chapter: 16
Section: 2
Variables: \(C\) a circle,
\(L\) a tangent to \(C\)
Name2: The point of contact of \(L\) and \(C\)

Q: -[...],
-[...],
-[...]
A: \(L\) is a perpendicular bissector of \(K\); \(L\) containg \(P\) and the center of \(K\); \(L\) is perpendicular to \(K\) passing through \(P\)
Page: 227
Chapter: 16
Section: 2
Variables: \(C\) a circle,
\(P\) its center
\(K\) a chord,
\(L\) a line,
Index: 3-5
Kind: Theo

Name: Congruent circles
Definition: Circles with the same radius
Page: 227
Chapter: 16
Section: 2

Name: Equidistant chords from the center
Definition: Chords whose distance from the center are the same
Page: 227
Chapter: 16
Section: 2
Variables: \(C\) a circle,
\(P\) its center

Name: A central angle of \(C\)
Definition: An angle whose vertex is the center of \(C\)
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle

Name: The minor arc of
Definition: The intersection of the interiors of \(\angle APB\) and of \(C\)
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,
\(A,B\in C\)

Name: The major arc of
Definition: The intersection of the exterior of \(\angle APB\) and of the interior of \(C\)
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,
\(A,B\in C\)

Name: endpoint of the arc \(\angle APB\)
Definition: \(A\) and \(B\)
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,

Name: A semicircle of \(C\)
Definition: An arc of \(A,B\) with \(\overline{AB}\) a diameter
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,

Definition: The semicircle of \(C\) with ends \(A,B\) containing \(X\)
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,
\(A,X,B\in C\)

Name: The degree measure of
Definition: if is minor, the measure of the corresponding central arc,
if is a semicircle, it is 180
if is a major arc, the measure of the corresponding minor arc
Page: 231
Chapter: 16
Section: 4
Variables: \(C\) a circle,
\(A,X,B\in C\)
Notation: \(m\)

Name: An angle inscribed in
Definition: An angle \(\angle ABC\)
Page: 233
Chapter: 16
Section: 4
Variables: \(C\) a circle,
 an arc

Name: An angle intercepts
Definition: And arc such that:
-its endpoints lie on the angle
-each side of the angle contains at least one endpoint of the arc
-except for its end points, the arc lies in the interior of the angle
Page: 234
Chapter: 16
Section: 4
Variables: \(C\) a circle,
 an arc

Question: An angle inscribed in a semicircle is
Answer: a right angle
Page: 236
Chapter: 16
Section: 4
Variables: \(C\) a circle,
an arc
Index: 3
Kind: Theo

Question: Two angles inscribed in the same arc are
Answer: congruent
Page: 236
Chapter: 16
Section: 4
Variables: \(C\) a circle,
an arc
Kind: Theo

Question: the measure of the angle is
Answer: half the measure of its intercepted arc
Page: 237
Chapter: 16
Section: 4
Variables: \(C\) a circle,
\(S\) a secant ray,
\(T\) a tangent ray with its vertex on the circle
Index: 8
Kind: Theo

Question: \(C\cap C'\) is
Answer: either \(C\) or a set of cardinality at most 2
Page: 237
Chapter: 16
Section: 4
Variables: \(C,C'\) circles,
Index: 9
Kind: Theo

Smaller0: \(c,r_1,r_2\) satisfies the triangular inequality theorem
Greater0: \(C_1\cap C_2\) contains exactly two points
Page: 242
Chapter: 16
Section: 5
Variables: \(C_i\) circles of radius \(r_i\)
\(c\) the distance between their centers
Name: The two-circle postulate

Name: Congruence Postulates for Segments
Definition: \(\cong\) is an equivalence relation over segments
Page: 127
Chapter: 8
Section: 1
Definition2: Segment construction postulate
Definition3: segment addition/substraction Postulate
Definition4: each segment have exactly one midpoint

Name: The length of \(\overline AB\)
Definition: The distance between \(A\) and \(B\)
Page: 68
Chapter: 3
Section: 5
Variables: A, B points

Smaller0: \(B-F-C\)
Greater0: \(G\) belongs to \(\angle BCD\)
Page: 79
Chapter: 4
Section: 1
Index: 7
Kind: Theo
Smaller1: \(A-C-D\)
Smaller2: \(A-F-G\)