Download List of Axioms, Definitions, and Theorems

Axioms, Definitions and Theorems
Undefined Terms: point, line.
Definition (1.1.S1). The universal set is the set of all points.
Axioms of Alignment
Axiom (0). A line is a proper subset of the universal set.
Axiom (1). Each line has at least two points.
Axiom (2). There exists at least two points. Additionally, every set of two (distinct) points is contained
on at least one line.
Theorem (1.1.S1). There exist at least three points and at least three lines.
Metric Geometry: Distance, Segments, Rays
Axiom (3: Metric Axiom). To each pair of points (A, B), distinct or not, there corresponds a real
number AB called the distance from A to B which satisfies the properties:
(a) AB ≥ 0, and AB = 0 if and only if A = B.
(b) AB = BA.
Theorem (1.6.S). There is an α, where α could be a real number or ∞, such that the following are
true:
(a) For all points A, B, AB ≤ α.
(b) If δ < α, then there are points A and B such that AB > δ.
We will usually refer to α as the least upper bound for distance, or simply as α.
Axiom (4). If P and Q are distinct points so that P Q < α, then there is exactly one line which contains
P and Q.
Definition (1.6.1). If 0 < AB < α, then the unique line which contains A and B will be denoted by
←→
AB.
←→
←→
←→
Theorem (1.6.1). If C and D lie on line AB and 0 < CD < α, then AB = CD. If lines ` and m
intersect and distinct points A and B, then AB = α.
Definition (1.7.1). A point B is said to lie between points A and C if:
(a) A, B and C are distinct collinear points.
(b) AB + BC = AC.
We will use (ABC) to indicate the statement “B lies in between A and C.”
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Definition (1.7.2). The notation (ABCD) will be used to indicate that the statements (ABC), (ABD),
(ACD) and (BCD) are all true.
Theorem (1.7.1). If (ABC), (ACD) and (ABD) are true, then (BCD) is also true.
Definition (1.7.3). Let A and B be distinct points such that AB < α. The segment joining A and B
is the set of all points P such that P = A, P = B or (AP B) is true. A and B are called the endpoints
of the segment, and the other points are called interior points. The ray joining A and B is usually
indicated by AB. Let A and B be distinct points such that AB < α. The ray originating at A and
passing through B is the set of all points P such that P = A, P = B, (AP B) is true or (ABP ) is true.
A and any point A∗ such that AA∗ = α is called an endpoint of the ray, while all other points are called
−→
interior points. The ray originating at A and passing through B is usually indicated by AB.
Axiom (5). If A, B, C and D are any four distinct collinear points such that (ABC) is true, then at
least one of the following is true: (DAB), (ADB), (BDC) or (BCD).
Theorem (1.7.S). If (ABC) is true, then there is exactly one line ` which contains A, B and C.
Definition (1.7.3). The rays h and k are called opposite rays if they share at least one endpoint and
their union is a line.
Theorem (1.7.3). If (ABC) is true and ` is the line which contains A, B and C, then ` is the union
−→
−−→
of the opposite rays BA and BC.
Coordinates for Rays and Lines
Axiom (6:Ruler Postulate for Rays). There is a one-to-one onto correspondence between the points of
any ray h and the nonnegative real numbers in the interval 0 ≤ x ≤ α, called coordinates. We will
use the notation A[a] to indicate that the geometric point A corresponds to the real number a. The
coordinates satisfy the following properties:
• The point O[0] (O with coordinate 0) is an endpoint of the ray (so that all other points have
positive coordinates).
• If A[a] and B[b] are any two points of h, then:
AB = |a − b|
Theorem (1.8.1). Suppose that A, B and C are three points on the ray h with coordinates a, b and c
respectively. Then (ABC) holds exactly when one of a < b < c or c < b < a is true.
Theorem (1.8.S1). Let A and B be points on a ray, and d a number so that AB + d ≤ α. Then there
is a point P on the ray so that (ABP ) and BP = d.
Theorem (1.8.S2). Suppose that A, B, C and D are any four distinct points so that (ABC) and (ACD)
hold. Then (ABCD) holds.
Theorem (1.8.2: Part One). Suppose that A, B and C are distinct points on the same line and that
the least upper bound for distance is α = ∞. Then exactly one of (BAC), (ABC) or (ACB) is true.
Theorem (1.8.S3). If α < ∞, and A is a point on the line `, then there is a unique point A∗ on ` so
that AA∗ = α.
2
Definition (1.8.1). Let A be a point on `. Then the unique point A∗ on ` so that AA∗ is called the
polar opposite of A on ` (or the antipodal point of A on `, or the extremal opposite of A on `).
Theorem (1.8.S4). Let A∗ be the polar opposite of A on the line `, and B be any other point on `.
Then (ABA∗ ).
Theorem (1.8.2: Part Two). Suppose that α < ∞, and A, B, C are three distinct points on the line `.
Let A∗ be the polar opposite of A on `. Then one of the following is true (BAC), (ABC), (ACB) or
(BA∗ C). Furthermore if BC 6= α, then only one case is true.
Theorem (1.8.3). Let A be any point, and B, C points so that AB < α and so that C is an interior
−→
−→ −→
point of AB. Then AB = AC.
−→ −→
Theorem (1.8.4). Suppose that (P AQ) and ` = AP ∪ AQ with P Q < α. Let B be an interior point of
−→
−→
AP and C be an interior point of AQ. Then:
• If AB + AC ≤ α, (BAC) holds
• if AB + AC > α, (BA∗ C) holds where C is the polar opposite of A on `.
Theorem (1.8.5: Ruler Theorem for Lines). Let ` be a line and A, B be points on the line. Then it
is possible to put the points on ` in correspondence as coordinates with the real numbers in the interval
−α < x ≤ α, so that A has the coordinate 0, B has a positive coordinate, and if C and D have
coordinates c and d respectively, then:
• If |c − d| ≤ α, CD = |c − d|.
• If |c − d| > α, CD = 2α − |c − d|.
Congruence and Construction of Segments
Definition (1.10.1). The segments AB and CD are called congruent if AB = CD.
Theorem (1.10.1: Segment Construction Theorem). Let AB and CD be segments, and AB < CD.
Then there is a unique interior point E or CD such that AB ∼
= CE (that is AB = CE).
Definition (1.10.2). The point P is called the midpoint of AB if (AP B) and AP = P B.
Theorem (1.10.3: Segment Doubling Theorem). Let AB be any segment such that AB < α/2. Then
−→
there exists a unique point on AB such that B is the midpoint of AC (and so the length of AC is twice
that of AB).
Theorem (1.10.4: Segment Extension Theorem). Let AB be a segment and c a real number so that
AB < c < α. Then there is a unique point C such that AC is an extension of AB and AC = d.
Theorem (1.10.5: Uniqueness of Opposite Rays). Let r be any ray. Then r has a unique opposite ray.
(That is, there is exactly one ray s such that r ∩ s is one or two points and r ∪ s is a line).
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Angle Measures
Definition (2.1). An angle is the set of points belonging to each of two rays having a common endpoint.
The two rays are called the sides, and the common endpoint is called the vertex. If the sides are the
same, the angle is called degenerate, and if the sides are opposite the angle is called straight.
Axiom (7). To each angle there corresponds a real number called its measure. If our angle is written
in the form hk, we will refer to its measure as hk, and if it is written in the form ∠ABC, we will refer
to its measure as m∠ABC. For each angle hk, the measure satisfies the following properties:
• hk ≥ 0, with equality if and only if h = k.
• hk = kh.
Axiom (8). All straight angles have the same measure, and every angle has a measure less than or
equal to that of a straight angle.
Definition (2.1.S). The measure of a straight angle will be called β.
Theorem (2.1.1). The measure β is a maximum for all angle measures. That is, for any angle hk, we
have:
0 ≤ hk ≤ β
Definition (2.1.2). The ray u is said to lie in between the rays h and k if:
1. h, k and u are distinct rays with a common endpoint. (We will often say that the rays are
concurrent when this occurs).
2. hu + uk = hk.
If u is between h and k, we will write (huk).
Theorem (2.1.S1). If (huk) is true then (kuh) is also true.
Definition (2.1.3). If h, k, u and v are rays, then the statement (hkuv) is true exactly when the statements (hku), (hkv), (huv) and (kuv) are true.
Plane Separation Interlude
Definition (2.2.1). A set S is called convex, if for each pair of points A and B in S so that AB < α,
the line segment AB ⊂ S.
Theorem (2.2.1). The intersection of two convex sets is convex.
Axiom (9: Plane-Separation Postulate). There corresponds to each line ` two regions H1 and H2 such
that:
1. The sets `, H1 and H2 are disjoint, and their union is the universal set.
2. H1 and H2 are nonempty convex sets.
3. If A ∈ H1 , B ∈ H2 and s is a segment containing A and B, then s intersects `.
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Theorem (2.3.1). If A lies on line ` and B lies in one of the half planes H1 determined by ` so that
AB < α, then the interior of the segment AB lies in H1 .
Theorem (2.3.C). If A lies on `, B lies in H1 and (ABC), where AC < α, then C lies in H1 .
Theorem (2.3.2). If A lies on line ` and B lies in one of the half planes H1 or H2 determined by ` so
−→
that AB < α, then every interior point of the ray AB will lie in H1 .
Theorem (2.3.3). Let B and F lie on opposite sides of a line ` and let A and G be any two distinct
−→
points on ` such that AG < α. Then any segment GB and any ray AF are disjoint sets.
Theorem (2.3.4). Two points A and B not on line ` lie on the same side of ` iff there exists no point
X on ` such that (AXB).
Theorem (2.3.5: “Postulate” of Pasch). Suppose thatA, B and C are three distinct, but possibly
collinear or not collinear, points, that D is a fourth point such that (ADB) holds, and that ` is a
line which intersects D but not A, B or C. Then exactly one of the following is true: either ` contains
a point E such that (AEC) holds, or ` contains a point F such that (BF C) holds.
Angle Addition
Definition (2.4.1). The interior of the nondegenerate, nonstraight angle ∠ABC is the intersection of
←→
←→
the A side of BC and the B side of AB. The interior of a degenerate angle is the empty set and the
interior a straight angle is the whole plane.
Axiom (10: Angle Addition Postulate). If point D lies in the interior of ∠ABC, or is an interior point
of one of the sides, then m∠ABD + m∠DBC = m∠ABC.
Theorem (2.4.1: Linear Pair Theorem). If rays h and h0 are opposite rays, and k is any other ray
which originates from a common endpoint of h and h0 then (hkh0 ) and therefore hk + kh0 = β.
Definition (2.4.2). An angle is called right if its measure is equal to half that of a straight angle. An
acute angle is an angle with measure less than that of a right angle, and an obtuse angle is an angle
with measure of less than a right angle.
Two angles are called supplementary if the sum of their measures is equal to the measure of a straight
angle. Two angles are called complementary if the sum of their measures is equal to that of a right angle.
Two angles are called adjacent if they share a common side and have disjoint, nonempty interiors. Two
angles are called a vertical pair if the sides of one are opposite rays to the sides of the other.
Theorem (2.4.S1). Suppose that angles ab and cd are supplementary, and that cd and ef are also
supplementary. Then ab and ef have equal measures.
Theorem (2.4.2: Vertical Pair Theorem). Vertical angles have equal measures.
Theorem (2.4.S2). If A, B, C, D are distinct points such that m∠ABD + m∠DBC = m∠ABC, then
either D ∈ Int ∠ABC or D is an interior point of one of the sides of ∠ABC.
Theorem (2.4.S3). Suppose that ` is a line and r is a ray not contained in ` with endpoint on `. Then
any interior point of the opposite ray r0 to r is on the opposite side of ` from any interior point of r.
Theorem (2.4.S4). Suppose that ` is a line and that r is a ray not contained in ` with endpoint on `.
Then every endpoint of r is on `.
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Theorem (2.4.3). Suppose that u, v, h are concurrent rays, that ` is the line containing h and that the
interiors of u and v lie on the same side of `. Then (hvu) or (huv).
Definition (2.4.2). Two lines ` and m are said to be perpendicular, if they intersect and they contain
a right angle. In this case we write ` ⊥ m. If h and k are each a line segment, line or ray (where each
could be different types of shape), they are said to be perpendicular when the lines containing them are
perpendicular.
Theorem (2.4.4: Criterion for Perpendicularity). Suppose that h, h0 , k are concurrent angles with h
opposite to h0 . Then h ⊥ k iff hk = h0 k.
Theorem (2.4.5: Crossbar Theorem). Let ∠BAC be a nondegenerate, nonstraight angle, where B and
C are interior points of the sides, and let D be an interior point of ∠BAC with AD < α. Then the ray
−−→
AD intersects a segment s with endpoints B and C at an interior point.
←→
Axiom (11: Angle Construction Postulate). Given any line AB and half plane H determined by that
line, for every real number r between 0 and 180 (including 0 and 180), there is exactly one ray AP in
H such that m∠P AB = r.
Theorem (2.5.1: Construction of Perpendicular Lines). Given a point A on the line `, there exists
exactly one line m which is perpendicular to ` and which contains A.
Theorem (2.5.2: Angle Construction Theorem). Let two nondegenerate, nonstraight angles ∠ABC and
−−→
−−→
∠DEF be given, such that m∠ABC < m∠DEF . There exists a unique ray EG lying between rays ED
−→
and EF so that ∠GEF ∼
= ∠ABC.
Definition (2.5.1). If h, u k are concurrent rays so that (huk) and hu = uk, then u is called the bisector
of hk.
Theorem (2.5.2: Angle Bisector Theorem). Every nondegenerate, nonstraight angle has a unique bisector.
Theorem (2.5.3: Circular Protractor Theorem). There exists a one-to-one correspondence between the
set of all rays that are concurrent at a given point O and the set of real numbers θ, so that −180 < θ ≤
180, called their coordinates, such that:
(a) If h and k are two rays concurrent at O, then h can be given the coordinate 0 and k can be given a
positive coordinate.
(b) If r and s are rays with coordinates θ and φ respectively, then:
• If |θ − φ| ≤ β, hk = |θ − φ|.
• If |θ − φ| > β, hk = 2β − |θ − φ|.
Antipodal Points in Spherical Geometry
Theorem (2.6.1). Suppose α < ∞. Let A be a point on line ` and A∗ the polar opposite of A on ` (that
is, the unique point on ` so that AA∗ = α). Then every line which contains A also contains A∗ .
Theorem (2.6.A1). Let α < ∞. Then given a point A, there is a unique point A∗ such that AA∗ = α.
6
Definition (2.6.1). If A and A∗ are two points such that AA∗ = α, then the points will be called
antipodes, antipodal points, extreme points, or extremal opposites without any qualifications.
Theorem (2.6.A2). If α < ∞, then every line passing through a point A also passes through its antipode
A∗ .
Theorem (2.6.B). If α < ∞ and ` and m intersect at point A, then they also intersect at the antipodal
point A∗ .
Theorem (2.6.C). If A and A∗ are two antipodal points, and X is any other point, then (AXA∗ ) holds.
Theorem (2.6.D). Suppose that α < ∞, that ` is a line and that A and B are points on the same side
of `. Then AB < α.
Triangles
Theorem (2.6.2: Existence of Triangles). Let A, B and C be three noncollinear points. Then AB, AC
and BC are all less than α, so that the segments AB, AC, BC are well defined. Furthermore, each of
the angles ∠BAC, ∠ABC, ∠BCA are neither degenerate nor straight.
Definition. Let A, B, C be three distinct, noncollinear points. Then the triangle 4ABC is the set
AB ∪ BC ∪ AC. The sides of the triangle are the segments AB, BC and AC. The angles of the triangle
are the angles ∠CAB, ∠ABC, ∠BCA. These are often called the angles at A, B and C respectively.
Definition. The length of a segment is the distance between its endpoints. Two segments are said to
be congruent if they have the same length.
Two angles are said to be congruent if they have the same measure.
When we say that 4ABC ∼
= 4DEF , we mean that corresponding sides and angles are congruent. That
is, this statement will be shorthand for the following six statements:
AB ∼
= DE, BC ∼
= EF , AC ∼
= DF
∠BAC ∼
= ∠F DE, ∠ABC ∼
= ∠DEF, ∠BCA ∼
= ∠EF D
Basic Congruence Results
Axiom (13: SAS Postulate). If 4ABC and 4DEF are triangles so that AB ∼
= DE, BC ∼
= EF and
∠ABC ∼
= ∠DEF , then 4ABC ∼
= 4DEF .
Theorem (3.3.1: ASA Theorem). Suppose 4ABC and 4DEF are triangles so that ∠BAC ∼
= ∠EDF ,
∼
∼
∼
∠BCA = ∠EF D and AC = DF . Then 4ABC = 4DEF .
Definition (3.3.S1). In triangle 4ABC the angle ∠BAC is opposite to BC, ∠ABC is opposite to AC
and ∠ACB is opposite to AB.
Definition (3.3.1: Isosceles Triangles). A triangle is called isosceles if it has two congruent sides which
are congruent and which have congruent opposite angles. The congruent sides are called the legs, the
remaining side the base, the congruent angles the base angles and the remaining angle the vertex angle.
Theorem (3.3.2: Isosceles Triangle Theorem). In a triangle 4ABC, AB ∼
= BC iff ∠BAC ∼
= ∠BCA.
As a consequence, it is sufficient to check that a triangle has two congruent sides or two congruent
angles to see if it is isosceles.
7
Definition (3.3.S2). The interior of a triangle is the intersection of the interiors of the three angles
which it contains. That is:
Int 4ABC = Int ∠ABC ∩ Int ∠BCA ∩ Int ∠CAB
Theorem (3.3.S1). If P ∈ Int ∠ABC ∩ ∠BCA, then P ∈ Int 4ABC
Theorem (3.3.S2). If P ∈ Int 4ABC, then either P A 6= CA or P B 6= CB.
Theorem (3.3.3: SSS Congruence Theorem). Suppose that 4ABC and 4DEF are triangles so that
AB ∼
= DE, BC ∼
= EF and AC ∼
= DF . Then 4ABC ∼
= 4DEF .
Definition (3.5.1). A line ` is called the perpendicular bisector of a segment s, if it bisects s and s ⊥ `.
Theorem (3.5.1: Perpendicular Bisector Theorem). Given two points A and B so that 0 < AB < α,
the set of points equidistant from A and B is a perpendicular bisector of AB.
Theorem (3.5.2). Given a line ` and a point P not on `, there exists a line m so that P is on ` and
` ⊥ m. If there is a point on ` which is not distance α/2 from P , then the line m is the only such line
with these properties.
Theorem (3.5.A). If α = ∞, there is a unique perpendicular to a line which passes through a point not
on the line.
←→
Theorem (3.5.B). If P Q = α/2 for some Q on line `, and P Qis perpendicular to `, then for any X
←→
on `, P X = α/2 and P X ⊥ `.
Theorem (3.5.C). If the legs of a right triangle have length α/2, the base angles are right angles.
Conversely, if the base angles are right angles then the legs have length α/2.
Theorem (3.5.D). If a right triangle has legs of length α/2, then it is an equilateral triangle with three
right angles.
Definition (3.5.2). If ` is a line and A a point not on `, then F is called a foot of A on ` if F is on `,
←→
AF ≤ α/2 and AF ⊥ `. The distance of a point A to a line ` is 0 if A is on `, and the distance of A
to its foot on ` if A is not on `.
Inequalities
Definition (3.6.1). In 4ABC, if D is any point such that (BCD) is true, then ∠ACD is called an
exterior angle of the triangle, and ∠BAC, ∠ABC are called the opposite interior angles of that exterior
angle.
Theorem (3.6.S1). If A, B, C and D are points so that A, B and C are not collinear AB < α/2,
AC ≤ α/2 and (BDC) is true, then AD < α/2.
Theorem (3.6.1: Exterior Angle Inequality). In any triangle whose sides are all less than α/2 any
exterior angle has a greater measure than its opposite interior angles.
Theorem (3.3.A). If the sides of a triangle are of length less than α/2, then the sum of the measures
of any two of its angles is strictly less than β.
Theorem (3.3.B). Any triangle having sides of length α/2 has at most one right or obtuse angle.
8
Theorem (3.3.C). In an isosceles triangle whose sides have length less than α/2, the base angles are
acute.
Theorem (3.3.D). If a right triangle has sides of length less than α/2, it has only one right angle and
no obtuse angles.
−−→
Theorem (3.6.2). If ∠ABC is acute and AB < α/2, the foot D of A is on the ray BC
Theorem (3.3.E). In 4ABC having sides of length less than α/2 and with acute angles angles ∠ABC
and ∠BCA, the foot of A falls on the segment BC.
Theorem (3.7.S1: Triangle Inequality Part One). Suppose that A, B, C are three distinct, noncollinear
points so that AB, BC and AC are all less than α/2. Then AC < AB + BC.
Theorem (3.7.S2: Full Triangle Inequality). IF A, B and C are any three points, then AC ≤ AB +BC.
Furthermore, if A, B and C are distinct, then AC = AB + BC if and only if (ABC) is true.
Theorem (3.7.S3: Scalene Inequality). In a triangle 4ABC, if AB > BC then m∠BCA > m∠CAB
and if m∠BCA > m∠CAB then AB > BC.
Theorem (3.7.S4: SAS Inequality/Hinge Theorem). Suppose that in 4ABC and 4DEF , AB ∼
= DE,
BC ∼
EF
.
Then
m∠ABC
>
m∠DEF
iff
AC
>
DF
.
=
Restricted Congruence Results
Theorem (3.8.S1: AAS Congruence Criterion). Let 4ABC and 4DEF be two triangles which have
all sides of length less than α/2, so that ∠BAC ∼
= ∠EDF , ∠ABC ∼
= ∠DEF and AC ∼
= DF . Then
∼
4ABC = 4DEF .
Theorem (3.8.S2: SSA Theorem). Suppose that in 4ABC, 4DEF that AB ∼
= DE, BC ∼
= EF and
∼
∼
∠BCA = ∠EF D. Then either ∠BAC and ∠EDF are supplementary, or 4ABC = 4DEF .
Theorem (3.8.S3: HL Congruence in Right Triangles). Suppose that 4ABC and 4DEF are right
triangles with right angle ∠ABC and ∠DEF so that AB ∼
= DE, AC ∼
= DF and so that every side has
∼
a length of less than α/2. Then 4ABC = 4DEF .
Polygons
Definition (4.1.1). Let P0 , P1 , . . . , Pn be n distinct points then the set:
[P0 P1 . . . Pn ] = P0 P1 ∪ P1 P2 ∪ · · · ∪ Pn−1 Pn
is called a polygonal path of order n. The n + 1 points are called the vertices of the path, the segments
are called sides, and two sides are called consecutive or adjacent if they share a vertex. A polygonal path
is called closed if P0 = Pn and is called simple if its sides do not intersect, except for the shared vertices
of adjacent sides.
Definition (4.1.2). An n-gon is a simple closed polygonal path of order n so that no two adjacent sides
are collinear. A 4-gon is called a quadrilateral, and if A, B, C, D are four distinct points so that no
three of them are collinear, then ♦ABCD will indicate the quadrilateral defined by the polygonal path
[ABCDA]. The sides and vertices of ♦ABCD are the sides and vertices of the polygonal path, and the
angels of ♦ABCD are the angles ∠ABC, ∠BCD, ∠CDA and ∠DAB. The sides AB and CD are
called opposite, as are the sides AC and BD. The angles ∠ABC and ∠CDA are called opposite, as are
∠DAB and ∠BCD. The segments AC and BD are called the diagonals of the quadrilateral.
9
Definition (4.1.3). Let P be a polygon, having the following property: if s is a side of the polygon,
` the line containing that side and H1 , H2 the two half-planes determined by `, then P ⊂ ` ∪ H1 or
P ⊂ ` ∪ H2 . Then P is called a convex polygon.
Theorem (4.1.L). Let ♦ABCD be a quadrilateral (not necessarily a convex quadrilateral) and suppose
←→
←→
that H1 is a side of AB which contains both C and D. Then ♦ABCD ⊂ AB ∪ H1 .
Theorem (4.1.1). Let ♦ABCD be a convex quadrilateral. Then the following betweenness relations of
−→−→−−→ −→−−→−−→ −−→−→−−→
−−→−−→−−→
rays hold: (AB AC AD), (BABDBC), (CB CACD), and (DC DB DA).
Theorem (4.1.C). The diagonals of a convex quadrilateral intersect each other.
Theorem (4.1.2). Suppose that ♦ABCD is a quadrilateral with the property that C and D lie on the
←→
←→
same side of the line AB, that D and A lie on the same side of the line BC and that A and B lie on
←→
←→
the same side of the line DA. Then B and C lie on the same side of the line DA, and ♦ABCD is a
convex quadrilateral.
Definition (4.2.S1). The statement ♦ABCD ∼
= ♦EF GH to indicate that the following eight statements
are true:
AB ∼
= EF , BC ∼
= F G, CD ∼
= GH, DA ∼
= HE
∠ABC ∼
= ∠EF G, ∠BCD ∼
= ∠F GH, ∠CDA ∼
= ∠GHE, ∠DAB ∼
= ∠HEF
Theorem (4.2.S1: SASAS Congruence). Suppose that in the convex quadrilaterals ♦ABCD, ♦EF GH,
we have AB ∼
= EF , ∠ABC ∼
= ∠EF G, BC ∼
= F G, ∠BCD ∼
= ∠F GH and CD ∼
= GH. Then ♦ABCD ∼
=
♦EF GH.
Definition (4.3.S1). A Saccheri quadrilateral is a convex quadrilateral ♦ABCD so that ∠A and ∠B
are right and so that AD = BC. The angles ∠A and ∠B are called the base angles, while the remaining
two angles are called the summit angles. The side AB is called the base and the side CD is called the
summit.
←→ ←→ ←→ ←→
Definition (4.3.1). Let A, B, C and D be distinct points so that DA ⊥ AB, BC ⊥ AB and AD = BC.
Then ♦ABCD is a Saccheri quadrilateral.
Theorem (4.3.2). The summit angles of any Saccheri quadrilateral are congruent. The summit is
congruent to the base iff the summit angles are right.
Theorem (Saccheri’s Theorem). Let 4ABC be right with right angle at C and sides of length α/2.
Then it is possible to find a Saccheri quadrilateral so that the sum of angles in 4ABC is equal to the
sum of the measures of the summit angles of that quadrilateral (or equivalently, to twice the measure of
one summit angle).
Circles
Definition (4.5.1). A circle is the set of all points which lie at some distance 0 < r < α/2 from some
point O. The point O is called the center of the circle and the number r is called the radius of the circle.
Theorem (4.5.1). Suppose that ` is a line, that O is a point off the line, and C is the foot of O on `
←→
(i.e. a point such that OC ≤ α/2 and OC ⊥ `). Furthermore, suppose that A and B be points on ` such
that (ACB) is true and OA, OB and OC are each less than α/2. Then OA > OC and OB > OC.
Furthermore if X is any other point on the line then:
10
• If (XAC), then OX > OA > OC
• If (AXC), then OC < OX < OA
• If (CXB), then OC < OX < OB
• If (CBX), then OC < OB < OX
Definition (4.5.S1). Let c be a circle with center O. The interior of a circle is the set of points whose
distance from the center is strictly less than the radius. A line is called a tangent line of c if it intersects
c at exactly one point. The point of intersection is called the point of contact of the tangent line. A
line is called a secant line of c if it intersects c at exactly two points. A segment is called a chord of c
if its endpoints are on the circle. A chord is called a diameter if it contains the center of the circle. A
segment is called a radial segment of the circle if one endpoint is the center of c and the other point is
on the circle. An angle is called a central angle of the circle if it has the center as an endpoint, and an
inscribed angle if it contains an endpoint on the circle. An arc of a circle is the intersection of a circle,
with the union of a line and one side of the line. The interior of the arc is the part which lies on one
side of the line, and the endpoints are teh points which lie on the line. An arc is called an intercepted
arc of an angle if the interior of an arc lies in the interior of that angle and the endpoints lie on the
angle. A chord is said to be subtended by an angle if its endpoints are on the angle and its interior is
in the interior of the angle.
Theorem (4.5.S1). Let c be a chord of a circle and ` the perpendicular bisector of c. Then ` contains
the center of the circle.
Theorem (4.5.S2). Suppose that ` is a line which passes through the center of a circle, and c is a chord
of the circle which ` bisects. Then ` is perpendicular to c.
Theorem (4.5.S3). Let c be a chord on a circle, and ` the line containing c. Let O be the center of the
←→
circle, and F the foot of O on `, (i.e. the unique point of ` such that OF ⊥ `). Then F is the midpoint
of c.
Theorem (4.5.S4). Suppose that c and d are chords in the same circle. Then c ∼
= d iff c and d are
equidistant from the center.
Theorem (4.5.S5). Let AB and CD be chords of a circle with center O, so that ∠AOB and ∠COD
are central angles. Then AB ∼
= CD iff ∠AOD ∼
= ∠COD.
Theorem (4.5.S6). A line cannot intersect a circle at more than two points.
Definition (4.5.S7). The interior of a circle (the points of distance less than the radius from the center)
is a convex set.
←→
Theorem (4.5.A: Corollary A of 4.5.1). If AB is a chord of a circle of radius r and center O X ∈ AB,
then OX ≤ r iff X ∈ AB and OX > r otherwise.
Theorem (4.5.B: Corollary B of 4.5.1). If ` is tangent to a circle, then no point of ` is in the interior
of that circle.
Theorem (4.5.3: Secant Theorem). Let ` be a line formed by two opposite rays at the point A which is
interior to the circle C. Then ` intersects the circle twice, once on each opposite rays.
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Theorem (4.5.D: Corollary D to the Secant Theorem). If A is an exterior point of a circle, with B an
exterior point of the same circle and AB < α, then the segment AB intersects the circle at exactly one
point.
Theorem (4.5.4: Tangent Theorem). A line is tangent to a circle if and only if it intersects the circle
and is perpendicular to the radial segment which contains that point of intersection.
←→
←→
Theorem (4.5.5: Two Tangent Theorem). If P A and P B are two tangent lines to a circle which
intersect at P so that A and B are points on that circle, then P A = P B.
General Parallel Lines and Angle Sums
Definition (5.1.1). Two lines are said to be parallel if they have no point in common. If ` and m are
parallel, we will write ` k m and if they are not parallel we will write ` ∦ m.
Theorem (5.1.1). Let ` and m be lines and suppose that α < ∞. Then ` and m intersect.
Theorem (5.1.S1). If ` is a line, P is a point not on a line and α = ∞, then there is a line which
passes through P and is parallel to `.
Definition (5.1.1). Suppose that ` and m are lines which both intersect the transversal line t such that
A is a point of intersection between ` and t, B is a point of intersection between m and t, C and D are
points on ` and E and F are points on ` so that C and E are on the same side of t and D and F are
on the same side of t. Finally let H be a point on t so that (BAH) is true and G be a point on t so
that (ABG) is true.
Then ∠CAB and ∠ABF are called alternating interior angles and ∠EBF and ∠BAD are also called
alternating interior angles. The pairs ∠HAD and ∠ABF , ∠HAC and ∠ABE, ∠CAB and ∠EBG and
∠DAB and ∠F BG are called corresponding angles. The pairs ∠DAB and ∠F BA as well as ∠CAB
and ∠EBA are called interior angles on the same side of the transversal.
Theorem (5.1.2: Z Criterion). If α = ∞, and `, m and t are three lines so that a pair of alternate
interior angles are congruent (with t as the transversal line) then ` k m.
Theorem (5.1.A: F Criterion). If α = ∞ and `, m and t are three lines with t as transversal so that a
pair of corresponding angles are congruent, ` k m.
Theorem (5.1.B). If ` ⊥ m and m ⊥ n, then ` k n.
Theorem (5.1.C: C Criterion). If α = ∞ and `, m and t are three lines with t as transversal so that a
pair of interior angles on the same side of t are supplementary, then ` k m.
Theorem (4.5.3). If α < ∞, the angle-sum of any triangle is greater than β.
Theorem (4.5.4: Legendre’s First Theorem). If α = ∞, the angle-sum of any triangle is less than or
equal to β.
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General Parallel Lines and Angle-Sums
Definition (5.1.1). Two lines are said to be parallel if they have no point in common. If ` and m are
parallel, we will write ` k m and if they are not parallel we will write ` ∦ m.
Theorem (5.1.1). Let ` and m be lines and suppose that α < ∞. Then ` and m intersect.
Theorem (5.1.S1). If ` is a line, P is a point not on a line and α = ∞, then there is a line which
passes through P and is parallel to `.
Definition (5.1.2). Suppose that ` and m are lines which both intersect the transversal line t such that
A is a point of intersection between ` and t, B is a point of intersection between m and t, C and D are
points on ` and E and F are points on ` so that C and E are on the same side of t and D and F are
on the same side of t. Finally let H be a point on t so that (BAH) is true and G be a point on t so
that (ABG) is true.
Then ∠CAB and ∠ABF are called alternating interior angles and ∠EBF and ∠BAD are also called
alternating interior angles. The pairs ∠HAD and ∠ABF , ∠HAC and ∠ABE, ∠CAB and ∠EBG and
∠DAB and ∠F BG are called corresponding angles. The pairs ∠DAB and ∠F BA as well as ∠CAB
and ∠EBA are called interior angles on the same side of the transversal.
Theorem (5.1.2: Z Criterion). If α = ∞, and `, m and t are three lines so that a pair of alternate
interior angles are congruent (with t as the transversal line) then ` k m.
Theorem (5.1.A: F Criterion). If α = ∞ and `, m and t are three lines with t as transversal so that a
pair of corresponding angles are congruent, ` k m.
Theorem (5.1.B). If α = ∞ and ` ⊥ m and m ⊥ n, then ` k n.
Theorem (5.1.C: C Criterion). If α = ∞ and `, m and t are three lines with t as transversal so that a
pair of interior angles on the same side of t are supplementary, then ` k m.
Theorem (5.1.3). If α < ∞, the angle-sum of any triangle is greater than β.
Theorem (5.1.4: Legendre’s First Theorem). If α = ∞, the angle-sum of any triangle is less than or
equal to β.
Euclidean Geometry
All results whose number begins with an “E” are only valid in Euclidean geometry.
Axiom (E: Euclidean Axiom). α = ∞. Furthermore, if ` is a line and P is a point not on `, then there
is exactly one line m which contains P and which is parallel to `.
Theorem (E.5.9.1). Suppose that ` and m are parallel lines which both intersect a transversal t. Then
the alternate interior angles formed by this transversal are congruent.
Theorem (E.5.9.S2: The Exterior Angle Theorem). The measure of an exterior angle in a triangle is
equal to the sums of the measures of the alternate interior angles.
Theorem (E.5.9.S1: Euclidean Angle Sum Theorem). In any triangle, the sum of the measures of all
angles in the triangle is β.
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Definition (5.10.S1). A trapezoid is a convex quadrilateral a set of opposite parallel sides. The parallel
sides are called the bases and the other sides are called the legs. The median of a trapezoid is the segment
connecting the midpoints of the legs. An isosceles trapezoid is a trapezoid with congruent legs.
A parallelogram is a convex quadrilateral so that both sets of opposite sides are parallel to each other. A
rectangle is a parallelogram with four right angles.
A rhombus is a convex quadrilateral with all sides congruent. A square is a rhombus which is also a
rectangle.
Theorem (E.5.10.S1). The opposite sides of a parallelogram are congruent to each other. Similarly,
opposite angles are congruent to each other.
Theorem (E.5.10.1). Suppose that 4ABC is a triangle with M the midpoint of AB. Then ∠ACB is
a right angle if and only if M C = AM = M B.
Theorem (E.5.10.C). Suppose that 4ABC is a right triangle with right angle ∠BCA, M is the midpoint
←−→ ←→
of the hypotenuse AB and N is the midpoint of the leg AC. Then M N k BC and M N = (1/2)BC.
Theorem (E.5.10.2: Midpoint-Connector Theorem for Triangles). Let 4ABC be any triangle, M the
←−→ ←→
midpoint of AB and N the midpoint of AC. Then M N k BC and M N = (1/2)AB.
Theorem (E.5.11.1: The Side Splitting Theorem). In 4ABC let D be a point on a side AB, and let
←→ ←→
E be the point on AC so that DE k BC. Then all of the following two equivalence between ratios hold:
AE AD
AE
AD
=
,
=
DB
EC AB
AC
Theorem (E.5.11.C). If 4ABC is a triangle, with D an interior point of AB and E the interior point
←→
of AC so that DE k BC, then:
AD
AE
DE
=
=
AB
AC
BC
Definition (5.11.S1). Suppose that 4ABC and 4DEF are two triangles so that ∠ABC ∼
= ∠DEF ,
∠BCA ∼
= ∠EF D, ∠CAB ∼
= ∠F DE and:
AB
BC
AC
=
=
DE
EF
DF
Then the triangles 4ABC and 4DEF are said to be similar, and we write 4ABC ∼ 4DEF . The
shared ratio (AB/DE) = (BC/EF ) = (AC/DF ) is called the constant of proportionality of the similarity.
Theorem (E.5.11.S1). Similarity is a congruence relation.
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