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Chapter 2
Construction  Proving
Historical Background
• Euclid’s Elements
• Greek mathematicians used
 Straightedge
 Compass – draw circles, copy distances
 No measurement
Euclid’s Postulates
1. Given two distinct points P and Q, there
is a line ( that is, there is exactly one line)
that passes through P and Q.
2. Any line segment can be extended
indefinitely.
3. Given two distinct points P and Q, a circle
centered at P with radius PQ can be
drawn.
4. Any two right angles are congruent.
Euclid’s Postulates
5. If two lines are intersected by a
transversal in such a way that the sum of
the degree measures of the two interior
angles on one side of the transversal is
less than the sum of two right angles,
then the two lines meet on that side of the
transversal.
(Accepted as an axiom for now)
Playfair’s Postulate
• Given any line l and any point P not on l ,
there is exactly one line through P that is
parallel to l .
Euclid’s Postulates
From
Wikimedia
Commons
Congruence
• Ordinary meaning:
 Two things agree in nature or quality
• Mathematics:
 Exactly same size and shape
 Note: all circles have same shape, but not
same size
A
B
C
Congruence
• What does it take to guarantee two
triangles congruent?
 SSS?
 ASA?
 SAS?
 SSA?
 AAS?
 AAA?
Congruence Criteria for Triangles
• SAS: If two sides and the included angle
of one triangle are congruent to two sides
and the included angle of another triangle,
then the two triangles are congruent.
• We will accept this axiom without proof
Angle-Side-Angle Congruence
• State the Angle-Side-Angle criterion for
triangle congruence (don’t look in the
book)
• ASA: If two angles and the included side
of one triangle are congruent respectively
to two angles and the included angle of
another triangle, then the two triangles are
congruent
Angle-Side-Angle Congruence
• Proof
• Use negation
• Justify the steps in the proof on next slide
ASA
• Assume
AB DE
x  DE AB  DX
ABC  DXF
C  XFD
But given C  EFD
AB  DX  DE
ABC  DEF
Similarity
• Definition
 Exactly same shape, perhaps different size
 Note: What if A and B are same height or
same area?
 What does it take to guarantee similar
triangles? Any two polygons similar?
A
B
C
Similarity
• Similar triangles can be used to prove the
Pythagorean theorem
 Note which triangles are similar
 Note the resulting ratios
Constructions
• Be sure to use Geogebra to construct
robust figures
 If a triangle is
meant to be
equilateral,
moving a vertex
should keep it
equilateral
Constructions
• Classic construction challenges
 Doubling a cube
 Squaring a circle
 Trisecting an angle
Geometric Language Revisited
• Reminder
 Constructions limited to straight edge &
compass
• Straight edge for
 Line, line segment, ray
Geometric Language Revisited
• Typical constructions
 Finding midpoint
 Finding “center” (actually centers) of different
polygons
 Tangent to a circle (must be  to radius)
 Angle bisector
• Note Geogebra has tools to do some of
these without limits of compass,
straightedge … OK to use most of time
Conditional Statements
• Implication P implies Q
if P then Q
Possible to have either
If the hypothesis is a true or a false
false, an implication conclusion
tells us nothing.
Conditional Statements
• Viviani’s Theorem
IF a point P is interior to
an equilateral triangle THEN the sum of the
lengths of the perpendiculars from P to the sides
of the triangle is equal to the altitude.
Conditional Statements
• What would make the hypothesis false?
• With false hypothesis, it still might be
possible for the lengths to equal the
altitude
Conditional Statements
• Consider a false conditional statement
 IF two segments are diagonals of a trapezoid
THEN the diagonals bisect each other
• How can we rewrite this as a true
statement
Conditional Statements
• Where is this on the
truth table?
• We want the opposite
 P  Q   P  Q
 IF two segments are diagonals of a trapezoid
THEN the diagonals do not bisect each other
Robust Constructions & Proofs
• Robust construction in Geogebra
 Dynamic changes of vertices keep properties
that were constructed
• Shows specified relationship holds even
when some of points, lines moved
 Note: robust sketch is technically not a proof
• Robust sketch will help formulate proof
Angles & Measuring
• Classifications of angles
 Right
 Acute
 Obtuse
 Straight
• Measured with
 Degrees
 Radians
 Gradients
Constructing Perpendiculars,
Parallels
• Geogebra has tools for doing this
• In certain situations
the text asks for
use of straight
edge & compass
only
Properties of Triangles
• Classifications
 Equilateral
 Isosceles
 Scalene
 Right
 Obtuse
 Acute
 Similar
Properties of Triangles
• Consider relationships between interior
angles and exterior angles.
• State your observations, conjectures
Properties of Triangles
• Conjecture 1
 If an exterior angle is formed by extending
one side of a triangle, then this exterior angle
will be larger than the interior angles at each
of the other two vertices.
Properties of Triangles
• Conjecture 2
 If an exterior angle is formed by extending
one side of a triangle, then the measure of
this exterior angle will be the same as the
sum of the measures of the two remote
interior angles of the same triangle.
Properties of Triangles
• Corollary to Exterior Angle Theorem
 A perpendicular line from a point to a given
line is unique. In other words, from a specified
point, there is only one line that is
perpendicular to a given line.
• Proof by contradiction … assume two ’s
Euclid’s Fifth Postulate
• If a straight line falling on two straight lines
makes the sum of the interior angles on
the same side less than the sum of two
right angles, then the two straight lines, if
produced indefinitely, meet on that side on
which the angles are less than two right
angles.
Clavius’ Axiom
• The set of points equidistant from a given
line on one side of it forms a straight line
( Hartshorne, 2000, 299).
Playfair’s Postulate
• Given any line and any point P not on ,
there is exactly one line through P that is
parallel to .
Recall Euclid’s Postulates
1. Given two distinct points P and Q, there
is a line ( that is, there is exactly one line)
that passes through P and Q.
2. Any line segment can be extended
indefinitely.
3. Given two distinct points P and Q, a circle
centered at P with radius PQ can be
drawn.
4. Any two right angles are congruent.
Use of Postulates for
Constructions
• Use to prove possibility of construction
 Then use that result to establish next
• Example: Equilateral triangles can be
constructed with a straight edge and
compass
 Based on Proposition 1 in Elements
Use of Postulates for
Constructions
• A line segment can be copied from one
location to another with a straightedge and
a compass.
 Based on Propositions 2 and 3 in the
Elements
This figure specified
a “floppy” compass
for the construction
Ideas about “Betweenness”
• Euclid took this for granted
 The order of points on a line
• Given any three collinear points
 One will be between the other two
Ideas about “Betweenness”
• When a line enters a triangle crossing side
AB
 What are all the ways it can leave the
triangle?
Ideas about “Betweenness”
• Pasch’s theorem:
If A, B, and C are distinct, noncollinear points and L is a line that
intersects segment AB, then L also
intersects either segment AC or segment
BC.
• Note proof on pg 43
Ideas about “Betweenness”
• Crossbar Theorem:
If AD is between AC and AB ,
then AD intersects segment BC.
• Use Pasch’s theorem to prove
Chapter 2
Construction  Proving