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.

Accepted as

axioms.

We will not

attempt to

prove them

Accepted as

axioms.

We will not

attempt to

prove them

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

FromWikimedia 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 CB

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?

CBA

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

Conditional Statements

• Viviani’s TheoremIF a point P is interior toan 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 trapezoidTHEN 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

IF two segments are diagonals of a trapezoidTHEN the diagonals do not bisect each other

P Q P Q TRUE st

atem

ent

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 situationsthe text asks foruse of straight edge & compassonly

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, non-collinear 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:

• Use Pasch’s theorem to prove

If AD is between AC and AB ,

then AD intersects segment BC.

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Chapter 2

Construction Proving