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Transcript
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