Download 1 An Approach to Geometry (stolen in part from Moise and Downs

An Approach to Geometry (stolen in part from Moise and Downs: Geometry)
Undefined terms: point, line, plane
The rules, axioms, theorems, etc. of elementary algebra are assumed as prior knowledge, and
apply to any measurements.
(P is a postulate, T is a theorem, Ex is an exercise)
P1 Distance Postulate To every pair of different points, there corresponds a unique positive
number (called the distance between the points).
Notation: The distance between points A and B is denoted AB
P2. The Ruler Postulate The points of a line can be placed in 1-1 correspondence with the real
numbers in such a way that the distance between any two points is the absolute value of the
difference of the corresponding numbers.
Defn. A correspondence that satisfies the conditions of the ruler postulate will be called a
coordinate system for the line, and the number corresponding to a point will be called the
coordinate of the point.
T0 Every line has infinitely many points.
T1 Ruler Placement Given two points P and Q on a line, there is a coordinate system for the line
such that the coordinate of P is 0, and the coordinate of Q is positive.
Defn. B is between A and C if
(1) A, B, and C are points of the same line and
(2) AB + BC = AC
Notation: A-B-C and C-B-A both denote that B is between A and C
T2. Let A, B, C be points of a line, with coordinates x, y, z. If x<y<z then A-B-C.
T3. If A, B, and C are points of the same line, then exactly one of them is between the other two.
P3. The Line Postulate For every two points there is exactly one line that contains both points.
T4. Two (different) lines intersect in at most one point.
Definitions and notation: The line that contains A and B is denoted AB
Definitions and notation: For any two points A and B, the segment AB is the union of A and B
and all of the points that are between A and B. The points A and B are called the endpoints of
AB . The distance AB is the length of the segment AB .
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Definitions and notation: For any two points A and B, the ray AB is the union of AB and all of
the points C such that A-B-C. The point A is called the endpoint of AB . If A is between B and D
then AB and AD are called opposite rays.
T5. Point Plotting Theorem Let AB be a ray, and let x be a positive number. Then there is
exactly one point P of AB such that AP =x.
Defn. A point B is called a midpoint of a segment AC if A-B-C and AB=BC. The midpoint of a
segment is said to bisect the segment. Also, any line, plane, ray or segment that contains the
midpoint (B) and does not contain segment ( AC ) is called a bisector of the segment ( AC ).
T6. Midpoint theorem Every segment has exactly one midpoint.
Defn. Space is the set of all points (note: if one is using the usual model for Euclidean geometry,
then space is 3-dimensional)
Defn. A set of points is collinear if there is a line which contains all of the points in the set; a set
of points is coplanar if there is a plane which contains all of the points in the set.
P4. The Plane-Space Postulate Every plane contains at least 3 (different) non-collinear points.
Space contains at least 4 (different) non-coplanar points.
P5. The Flat Plane Postulate If two points of a line lie in a plane, then the line lies in the same
plane
T7. If a line intersects a plane not containing it, then the intersection contains only one point.
P6. The Plane Postulate Any three points lie in at least one plane, and any three non-collinear
points lie in exactly one plane.
T8. Given a line and a point not on the line, there is exactly one plane containing both
T9. Given two intersecting lines, there is exactly one plane containing both lines
P7. Intersection of Planes Postulate If two different planes intersect, then their intersection is a
line.
Ex. 1 Prove that P7 is independent from P1-P6.
Ex. 2 Which postulate makes it impossible for this axiom system to be satisfied by a finite
model.
Defn. A set α is convex if for every two points P and Q in α, the entire segment PQ lies in α
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P8. The Plane Separation Postulate Given a line and a plane containing it, the points of the
plane that do not lie on the line form two sets such that
• each of the sets is convex
• if P lies on one set, and Q lies in the other, then PQ intersects the line.
Defn. Given a line l and a plane α containing it, the two sets described in the plane separation
postulate are called half-planes or sides of l, and l, is called the edge of each of them. If P lies in
one of the half planes, and Q lies in the other, then we say that P and Q lie on opposite sides of l.
Defn If two rays have the same endpoint, but do not lie on the same line, then their union is an
angle. The two rays are called its sides and their common endpoint is called its vertex. If the
rays are AB and AC , then the angle is denoted by ∠BAC of ∠CAB .
Defn The interior of ∠BAC is the set of all point P in the plane of ∠BAC such that
1. P and B are on the same side of AC , and
2. P and C are on the same side of AB .
The exterior of ∠BAC is the set of all points of the plane of ∠BAC that lie neither on the angle
nor in its interior.
Defn If A, B, and C are any three noncollinear points, then the union of the segments AB , AC
and BC is called a triangle, and is denoted by ABC . The points A, B, and C are called its
vertices, and the segments AB , AC and BC are called its sides. Every triangle ABC
determines three angles, namely, ∠BAC , ∠ABC and ∠ACB . These are called the angles of
ABC . The perimeter of a triangle is the sum of the lengths of its sides.
Defn A point lies in the interior of a triangle if it lies in the interior of each of the angles of the
triangle. A point lies in the exterior of a triangle if it lies in the plane of the triangle if it lies in
the plane of the triangle, but it does not lie on the triangle or in the interior.
P9 Angle Measurement Postulate To every angle ∠BAC there corresponds a real number
between 0 and 180.
Defn The number given by the Angle Measurement Postulate is called the measure of ∠BAC ,
and is written m∠BAC .
P10 Angle Construction Postulate Let AB be a ray on the edge of the half-plane H. For every
number r between 0 and 180, there is exactly one ray AP , with P in H, such that ∠PAB =r.
P11 Angle Addition Postulate If D is in the interior of ∠BAC , then
m∠BAC = m∠BAD + m∠DAC
Defn If AB and AD are opposite rays and AC is any other ray, then ∠BAC and ∠CAD form a
linear pair.
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Defn If the sum of the measures of two angles is 180, then the angles are called supplementary,
and each is called a supplement of the other.
P12 If two angles form a linear pair, then they are supplementary.
Ex. 3. Why is P12 a postulate and not a theorem?
Defn A right angle is an angle having measure 90. An angle with measure less than 90 is called
acute. An angle with measure greater than 90 is called obtuse.
Defn If the sum of the measures of two angles is 90, then they are called complementery and
each of them is called a complement of the other.
Defn. Two angles with the same measure are called congruent.
Defn Two rays are perpendicular if they are the sides of a right angle. Two lines are
perpendicular if they contain a pair of perpendicular rays. Two sets are perpendicular if each of
them is a line, ray or segment, they intersect, and the lines containing them are perpendicular.
Defn A binary relation ~ is called an equivalence relation on a set A if it satisfies the properties
1. a ~ a for all a ∈ A (reflexive)
2. If a ~ b , then b ~ a (symmetric)
3. If a ~ b and b ~ c then a ~ c (transitive)
T10. Congruence between angles is an equivalence relation.
T11. If the angles in a linear pair are congruent, then each of them is a right angle.
T12. If two angles are complementary, then both are acute
T13 Any two right angles are congruent,
T14 If two angles are both congruent and supplementary, then each is a right angle.
Ex. 4. How are theorems 11 and 14 different?
T15 Supplements of congruent angles are congruent.
T16 Complements of congruent angles are congruent.
Defn. Two angles are vertical angles if their sides form two pairs of opposite rays.
T17 Vertical angles are congruent.
T18 If two lines are perpendicular, they form four right angles.
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Defn If D is in the interior of ∠BAC and ∠BAD ≅ ∠DAC , then AD bisects ∠BAC , and AD is
the bisector of ∠BAC .
T19 Given x, y,θ ∈R such that 0 < θ < 180 , and a ray AB on the edge of a half-plane H, then
there exist points C and D such that C lies on AB , AC = x , D lies in H, m∠CAD = θ and
AD = y .
T20 Given the conditions in T19, the points C and D are unique.
Defn Segments are congruent if they have the same measure.
Defn A correspondence ABC ↔ DEF between the vertices of two triangles ABC and DEF
assigns the following pairings, which can be thought of as a 1-1 onto map between the parts of
the triangle:
A↔D
B↔E
C↔F
AB ↔ DE
∠BAC ↔ ∠EDF
BC ↔ EF ∠ABC ↔ ∠DEF
AC ↔ DF ∠ACB ↔ ∠DFE
Note that being a correspondence does not imply congruence of any of these pairings
Defn Given a correspondence ABC ↔ DEF between the vertices of two triangles ABC and
DEF , if every pair of corresponding sides are congruent, and every pair of corresponding
angles are congruent, then the correspondence is called a congruence between the two triangles.
T21 Congruence is an equivalence relation for segments
T22 Congruence is an equivalence relation for triangles.
Defn A side of a triangle is said to be included by the angles whose vertices are the endpoints of
the segment.
Defn An angle of a triangle is said to be included by the sides of the triangle which lie in the
sides of the angle.
P13 SAS Given a correspondence between two triangles such that two sides and the included
angle of one are congruent to the corresponding sides and angle of the other, then the
correspondence is a congruence.
T23 Isosceles triangles (sides implies angles) If two sides of a triangle are congruent, then the
angles opposite those sides are congruent.
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Lemma to T24 Given A-B-C and D that is not on AC , then A,B lie in the same half-plane of
CD .
T24 Given ray AB on the edge of the half-plane H and distinct rays AC and AD such that C
and D are in H, then either D lies in the interior of ∠BAC or C lies in the interior of ∠BAD .
T25 Given ray AB on the edge of the half-plane H, with distinct rays AC and AD such that C
and D are in H, and angles ∠BAC and ∠BAD such that m∠BAC < m∠BAD , then C lies in the
interior of ∠BAD .
T26 Every angle has a unique angle bisector.
Defn A triangle with two congruent sides is isosceles. The remaining side is the base. The two
angles that include the base are base angles. The angle opposite the base is the vertex angle.
Defn A triangle whose three sides are congruent is equilateral and a triangle whose three angles
are congruent is equiangular.
Defn A triangle not two of whose sides are congruent is called scalene.
T27 Every equilateral triangle is equiangular
T28 ASA Given a correspondence between two triangles such that two angles and the included
side of one are congruent to the corresponding angles and side of the other, then the
correspondence is a congruence.
T29 Isosceles triangles (angles implies sides) If two angles of a triangle are congruent, then the
sides opposite those angles are congruent.
T30 Every equiangular triangle is equilateral.
T31 Isosceles triangle thm. Given
ABC , ∠ABC ≅ ∠ACB if and only if AB = AC
Lemma to SSS: Given A-B-C and a point D not on line AB , then
m∠ADB + m∠BDC = m∠ADC
T32 SSS Given a correspondence between two triangles such that three sides of one are
congruent to the corresponding sides of the other, then the correspondence is a congruence.
T33 In a given plane, through a given point of a given line, there is one and only one line
perpendicular to the given line.
Defn A median of a triangle is a segment whose endpoints are a vertex of the triangle, and the
midpoint of the opposite side.
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Defn A segment is an angle bisector of a triangle if it
1. lies in the ray which bisects an angle of the triangle and
2. its end points are the vertex of this angle and a point of the opposite side.
Defn An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line
containing the opposite side.
Ex 5 Explain the difference between an angle bisector of a triangle, and an angle bisector of an
angle.
Ex 6 Show that SSA cannot be a congruence theorem.
T34 In an isosceles triangle, if a segment is a median to the base, then it is perpendicular to the
base.
NEW T35 If two distinct lines l and m are both intersected by a transversal line t, and the
alternate interior angles formed between l, m, and t are congruent, then l and m do not intersect.
NEW T36 (=converse of T34, old T35.5) In an isosceles triangle, an altitude to the base is also
the median to the base.
T37 (old T36) In an isosceles triangle, the angle bisector of the vertex angle is also the median to
the base.
T38 (old T37) If an altitude of a triangle is also a median, then the triangle is isosceles.
T39 (old T38) If an altitude of a triangle is also an angle bisector, then the triangle is isosceles.
T40 In an isosceles triangle, the medians to the congruent sides are congruent
T41 In an isosceles triangle, the angle bisectors of the congruent angles are congruent
T42 If two triangles are congruent, then medians to corresponding sides are congruent.
Defn In a given plane, the perpendicular bisector of a segment is the line which is perpendicular
to the segment at its midpoint
T44 If given distinct point C and D and angles ∠ABC and ∠ABD such that
m∠ABC = m∠ABD = 90 , then B,C,D are collinear.
T45 Every segment has a unique perpendicular bisector
T46 If a point P is on the perpendicular bisector of AB then AP=BP.
T47 If a point P is equidistant from points A and B, then P lies on the perpendicular bisector of
AB
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T48 HL If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and
corresponding leg of another right triangle, then the triangles are congruent.
T52 (Nichole’s conjecture: AAS) Given two triangles and a correspondence between them, such
that two pairs of corresponding angles are congruent, and a pair of corresponding sides that are
not included in the angles are congruent, then the triangles are congruent.
T53 (old T35) In an isosceles triangle, an altitude to the base is also the angle bisector of the
vertex angle
T54 (old 49) In an isosceles triangle, the altitudes to the congruent sides are congruent.
T55 Given a line l and any point A not on l, there exists at least one line parallel to l through
point A.
F
A
Lemma 56 In the figure to the right, given BE ≡ EF, AE ≡ EC , then
m∠DCE > m∠BAE
Defn If A-C-D, then ∠BCD is an exterior angle of ABC
E
B
D
C
T57 Given A-C-D and B-C-E then the exterior angles ∠BCD and ∠ACE at C are congruent.
Defn The angles ∠A = ∠BAC and ∠B = ∠ABC are the remote interior angles of the exterior
angles ∠BCD and ∠ACE .
T58 An exterior angle of a triangle is greater than each of its remote interior angles.
T59 If a triangle has a right angle, then the other two angles are acute.
A
Lemma 60 If A-C-D and AB ≅ AD , then ∠ABC < ∠ADB and
∠ACB > ∠ADB
C
D
B
T61 If two sides of a triangle are not congruent, then the angles opposite
them are not congruent, and the larger angle is opposite the longer side.
T62 If two angles of a triangle are not congruent, then the sides opposite them are not congruent,
and the longer side is opposite the larger angle
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Defn Two lines are parallel if they do not intersect. Any combination of segments rays and lines
are parallel if the associated lines are parallel.
T63 Given a line l with point A on l and point B not on l, such that AB is not perpendicular to l,
then there exists and isosceles triangle with side AB such that the angle bisector of the vertex
angle lies on l.
T64 Given line l and point B not on l, there exists a unique line through B that is perpendicular to
l.
T65 The shortest segment joining a point to a line is the perpendicular segment.
P14 If two parallel lines l, and m, are intersected by a transversal line t, creating alternate interior
angles ∠1 and ∠2 , then ∠1 ≅ ∠2 .
Defn A quadrilateral is the union of 4 points A,B,C,D, and the segments AB , BC , CD and DA
such that no three of four points are collinear, and the segments intersect with each other only at
the endpoints. Non-intersecting segments of a quadrilateral are called opposite sides. And the
segments AC and BD are called the diagonals of the quadrilateral.
Defn A parallelogram is a quadrilateral such that the opposite sides are parallel.
Defn A rectangle is a quadrilateral such that all 4 angles are right angles.
Defn A rhombus is a quadrilateral with all 4 sides congruent.
Defn A square is a quadrilateral with all 4 sides congruent, and all 4 angles being right angles.
T66 The sum of the measures of the angles in a triangle is 180.
Defn If ABCD is a convex quadrilateral, then ∠ABC , ∠BCD , ∠CDA , ∠DAB are its interior
angles. Note: if ABCD is not convex, then defining the interior and interior angles is more
difficult because some of the “interior angles” will not be angles by our definition of angle.
T67 the sum of the measures of the interior angles of a convex quadrilateral is 360.
Note: The following have not been verified to be true yet. The “C” designation will change to
a “T” designation after they have been proved.
C68 If it is a parallelogram, then adjacent angles are supplementary
C69 If it’s a parallelogram, then opposite angles are congruent
C70 If it’s a parallelogram, then opposite sides are congruent.
C71 If it’s a parallelogram, then the diagonals are angle bisectors
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C72 If the diagonals of a quadrilateral bisect each other then it’s a parallelogram
C73 Every rhombus is a parallelogram
C74 If a quadrilateral is a rhombus, then its diagonals bisect each other
C75 If it’s a rhombus, then the diagonals bisect the angles
C76 It’s a rhombus if and only if diagonals are perpendicular bisectors.
C77 Every rectangle is a parallelogram
C78 The opposite sides of a rectangle are congruent
C79 The diagonals of a rectangle are congruent
C80 The diagonals of a rectangle bisect each other
C81 If it’s a quadrilateral with congruent diagonals and parallel opposites sides, then it’s a
rectangle.
C82 Every square is both a rhombus and a rectangle
C83 Every square is a parallelogram
C84 If it is a square then diagonals cross at right angles
C85 If it’s a square then diagonals are congruent
C86 A quadrilateral is a square if and only if its diagonals are congruent perpendicular bisectors
C87 If it’s a square then the diagonals are also angle bisectors
C88. If a quadrilateral is both a rectangle and a rhombus, then it is a square.
C89. If it is a rhombus, and all 4 angles are congruent, then it’s a square
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