Notes on Congruence 1


 George Joseph
 2 years ago
 Views:
Transcription
1 ongruence1 Notes on ongruence 1 xiom 1 (1). If and are distinct points and if is any point, then for each ray r emanating from there is a unique point on r such that =. xiom 2 (2). If = and = F, then = F. Moreover, every segment is congruent to itself. xiom 3 (3: Segment ddition). If,, =, and =, then =. xiom 4 (4). Given any, and given and ray emanating from a point, then there is a unique ray on a given side of line such that =. xiom 5 (5). If = and =, then =. Moreover, every angle is congruent to itself. xiom 6 (6: SS). 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. orollary 1 (orollary to SS). Given and segment =, there is a unique point F on a given side of such that = F. Proof. y 4 there is a unique ray G on the given side of such that = G. y 1, there is a unique point F on G such that = F. y SS, = F. Proposition 10. If in we have =, then =. Proof. onsider the correspondence of vertices,,. y hypothesis =, and by 5 =. Then by SS =. y the definition of congruent triangles =. Proposition 11 (Segment Subtraction). If, F, =, and = F, then = F. Proof. Suppose = F (R). y 1, there is a point G on F such that = G. y the R hypothesis F G. Since = by hypothesis, and = G we have = G by 3. y hypothesis, = F, so by 2, G = F. y the uniqueness in 1 G = F. This is a contradiction, so = F. Proposition 12. Given = F, then for any point between and, there is a unique point between and F such that =. Proof. y 1 there is a unique point on F such that =. Suppose is not between and F (R). y the definition of ray, either = F or F. Suppose = F. and are distinct points on by 1, but = and =. This contradicts 1, so F. Now suppose F. y 1 there is a unique point G on the ray opposite to such that G = F. y 3, G =. This contradicts the uniqueness part of 1, since =. Hence is between and F. efinition. < (or > ) means that there exists a point between and such that =. Proposition 13 (Segment Ordering). 1. xactly one of the following conditions holds (trichotomy): <, =, or >. 2. If < and = F, then < F. 3. If > and = F, then > F. 4. If < and < F, then < F (transitivity). Proof. 1 The statements of the propositions and many proofs are taken from the book uclidean and Nonuclidean Geometries by M. Greenberg.
2 ongruence2 1. Suppose and are not congruent. y 1, there exists a point on such that =. y definition of ray, either or. If, then < by definition of <. Suppose. y Proposition 3.12, there is a unique point F on between and such that = F. Then > by definition of <. 2. Since <, there is a point P between and such that = P. Since = F, by Proposition 3.12, there is a unique point Q between and F such that P = Q. y 2, = Q, so < F. 3. Since >, there is a point P between and such that P =. Suppose = F. Then by 2, P = F, so > F. 4. Since <, there is a point P between and such that = P. Since < F, there is a point Q between and F such that = Q. y Proposition 3, there is a point R between and Q such that P = R. y 2, = R. Since R Q and Q F, we know by Proposition 3.3 that R F. Hence, < F. Proposition 14. Supplements of congruent angles are congruent. Proof. Suppose = F. Let P be the ray opposite to and let Q be the ray opposite to. We want to show P = FQ. F P Since the points,, and P are given arbitrarily on the sides of and P, by 1 we can choose the points, F, and Q on F and FQ such that =, = F, and P = Q. Then = F by SS. y the definition of congruent triangles, = F, and =. y 3, P = Q. Then again by SS, P = FQ. y the definition of congruent triangles P = FQ and P = Q. nd again by SS, P = FQ, so P = FQ. Proposition Vertical angles are congruent to each other. 2. n angle congruent to a right angle is a right angle. Proof. 1. y definition two angles are vertical if they allow labeling and where and are opposite, and and are opposite. Q
3 ongruence3 isthesupplementto and isthesupplementto. y5 =, so by Proposition 5 =. 2. Suppose is a right angle and F =. Suppose P is a point on the ray opposite to and Q is a point on the ray opposite F. We need to show F = Q. Since = F, by Proposition 5, their supplements are congruent, i.e. P = Q. y the definition of right angle = P, so by 5 = Q. gain by 5 F = Q. Proposition 16. For every line l and every point P there exists a line through P perpendicular to l. Proof. ither P lies on l or it does not. ssume first that P does not lie on l, and let and be any two points on l by I2. y 4 there is a ray X such that X is on the opposite side of l as P and X = P. y 1 there is a point P on X such that P = P. Since P and P are on opposite sides of l, PP intersects l at a point Q between P and P. If Q =, P and P are supplementary. Since these angles are congruent, they are right angles, so PP l. If Q, then Q = Q by 1, so PQ = P Q by SS. y the definition of congruent triangles PQ = P Q. Hence PP l. Now suppose P lies on l. y Proposition 2.3 there is a point not on l. y the above argument we can construct a line perpendicular to l through this point, thereby obtaining a right angle. y 4, there is a unique ray on a particular side of l emanating from P such that P with one side contained in l is congruent to a right angle. y Proposition 3.15, P is a right angle. The side of this angle not contained in l is contained in a line perpendicular to l through P. Proposition 17 (S). Given and F with =, = F, and = F. Then = F. Proof. y 1, there is a unique point X on such that = X. y hypothesis = and = F, so = XF by SS. y the definition of congruent triangles = FX. y hypothesis = F, so by the uniqueness of 4 F = FX. y Proposition 2.3, = X. Hence = F. Proposition 18. If in we have =, then = and is isosceles. Proof. onsider the correspondence of vertices,, and. y 1 =. y hypothesis =, so by Proposition 3.17, =. y the definition of congruent triangles =, so by the definition of isosceles triangle is isosceles. Proposition 19 (ngle ddition). Given G between and, H between and F, G = FH, and G = H. Then = F. Proof. y the crossbar theorem we may choose G so that G. y 1 we may choose, F, and H so that =, = F, and G = H. y hypothesis G = H, so by SS G = H. Similarly, by hypothesis G = FH, so by SS G = FH. y the definition of congruent triangles G = H and G = HF. We next need to show, H, and F are collinear. y the definition of congruent triangles G = H and G = FH. Since, G, and are collinear G and G are supplementary. y Proposition 5 G is congruent to the supplement of H. enote this supplement by HX. y 4 HX is unique, so HX = HF. Then H is supplementary to FH, so F, H, and are collinear. Since H is between and F, H is in the interior of F, so H F by Proposition 3.7. Since G = H and G = HF, by 3 = F. y SS = F, so by the definition of congruent triangles = F. Proposition 20 (ngle Subtraction). Given G between and, H between and F, G = FH, and = F. Then G = H. Proof. We proceed as in the proof of Proposition Suppose G = H (R). y 4 there is a unique ray X on the same side of H such that G = HX. y the R hypothesis X. y hypothesis FH = G, and by R hypothesis XH = G, so by Proposition 10 = XF. y hypothesis = F, so by 5 F = XF. y the uniqueness part of 4 X =, but this is a contradiction, so G = H.
4 ongruence4 Lemma 1. Given = F, then for any ray G between and, there is a unique ray H between and F such that G = FH. G H F Proof. y the rossbar Theorem we can choose G so that G. y 1 we can choose points and F such that = and = F. y SS = F, and by the definition of congruent triangles = F and = F. Then by Proposition 3.12 there is a unique point H on F such that G = FH. gain by SS, G = FH, so by the definition of congruent triangles G = FH. We only need to show that H is between and F. Since H is on F, H is between and F, so by Proposition 3.7 H is between and F. efinition. < F means there exists a ray G between and F such that = GF. Proposition 21 (Ordering of ngles). 1. xactly one of the following conditions holds (trichotomy): P < Q, P = Q, or P > Q. 2. If P < Q and Q = R, then P < R. 3. If P > Q and Q = R, then P > R. 4. If P < Q and Q < R, then P < R. (transitivity). Proof. This proofis verysimilarto the proofofproposition3.13. Forlabeling purposeswe say P =, Q = F, and R = GHI. 1. Suppose = F. y 4 there exists a unique ray X on the same side of F as such that = XF. X either is between F and or X is not between F and. If X is between F and, then < F. Suppose X is not between F and. Since X is on the same side of F as and X is not between F and, we know that X and F are on opposite sides of. y Lemma 2 and orollary 1 every point except on F is on the opposite side of as every point of X, so segment F does not intersect X. y a similar argument with the ray opposite X and F, we can show that segment F does not meet X. Hence is on the same side of X as F, so is interior to XF. Then is between X and F. y Lemma 4 there exists a unique ray Y between and such that F = Y. Hence > F. 2. Since < F, there exists a ray X between and F such that = XF. y Lemma 4 there is a unique ray HY between HG and HI such that YHI = XF. y 5 = YHI, so < GHI. 3. Since > F, there exists a ray X between and such that X = F. y 5 X = GHI, so > GHI. 4. Suppose < F and F < GHI. Since < F, there is a unique ray X between and F such that = XF. Since F < GHI, there is a unique ray HY between HG and HI such that F = YHI. y Lemma 4, there is a unique ray HZ between HY and HI such that XF = ZHI. y 5 = ZHI. Since HI HZ HY and HI HY HZ, by Lemma 3 HI HZ HZ. Then by definition < GHI.
5 ongruence5 Proposition 22 (SSS). Given and F. If =, = F, and = F, then = F. Proof. y orollary 4, since = F we can pick a point G uniquely on the opposite side of as such that F = G. y the definition of congruent triangles = G and F = G. Then by 2 = G and = G. We will show that = G. Since and G are on opposite sides of, segment G intersects at X. y 3, X, = X, or X. The circumstances X = and X are equivalent to X = and X, respectively. Suppose X. onsider G. Since = G, G, by Proposition 3.10 G = G. Now consider G. Since = G, by Proposition 3.10 G = G. Then by Proposition 3.20 (angle subtraction) = G. y SS = G. Suppose = X. Since G = in G, by Proposition 10 = G. Then by SS = G. Suppose X. onsider G. Since = G, G, by Proposition 3.10 G = G. Now consider G. Since = G, by Proposition 3.10 G = G. Then by Proposition 3.19 (angle addition) = G. y SS = G. In all three cases = G. y the definition of congruent triangles = G. Since F = G, = G. y 5 =, so by SS = F. Proposition 23. ll right angles are congruent to each other. Proof. Suppose = and HF = HG are two pairs of right angles. ssume = HF (R). y Proposition 3.21 (a), either > HF or < HF. Without loss of generality suppose > HF. Then there is X between and such that X = HF. y Proposition 3.14 X = HG. Since = by hypothesis, and > HF by by R hypothesis, we have > HF by Proposition 3.21 (c). Since HF = HG by hypothesis, we have again by Proposition 3.21 (c) > HG. From above since X = HG we have > X by Proposition 3.21 (c). y Proposition 3.8 (c), since X is between and, we know that is between X and, since is the ray opposite to. Then < X, but this contradicts Proposition 3.21 (a). Hence = HF.
1.2 Informal Geometry
1.2 Informal Geometry Mathematical System: (xiomatic System) Undefined terms, concepts: Point, line, plane, space Straightness of a line, flatness of a plane point lies in the interior or the exterior
More informationNeutral Geometry. April 18, 2013
Neutral Geometry pril 18, 2013 1 Geometry without parallel axiom Let l, m be two distinct lines cut by a third line t at point on l and point Q on m. Let be a point on l and a point on m such that, are
More informationBetweenness of Points
Math 444/445 Geometry for Teachers Summer 2008 Supplement : Rays, ngles, and etweenness This handout is meant to be read in place of Sections 5.6 5.7 in Venema s text [V]. You should read these pages after
More informationEuclidean Geometry  The Elements
hapter 2 Euclidean Geometry  The Elements Goal: In this chapter we will briefly discuss the first thirty two propositions of Euclid s: The Elements. You can find the Elements on the Web at: http://aleph0.clarku.edu/
More informationChapter 4 Circles, TangentChord Theorem, Intersecting Chord Theorem and Tangentsecant Theorem
Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangenthord Theorem, Intersecting hord Theorem and Tangentsecant Theorem utline asic definitions and facts on circles The
More informationThe Protractor Postulate and the SAS Axiom. Chapter The Axioms of Plane Geometry
The Protractor Postulate and the SAS Axiom Chapter 3.43.7 The Axioms of Plane Geometry The Protractor Postulate and Angle Measure The Protractor Postulate (p51) defines the measure of an angle (denoted
More informationLast revised: November 9, 2006
MT 200 OURSE NOTES ON GEOMETRY STONY ROOK MTHEMTIS DEPRTMENT Last revised: November 9, 2006 ontents 1. Introduction 3 1.1. Euclidean geometry as an axiomatic theory 3 1.2. asic objects 3 2. Incidence xioms
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More informationGiven: ABCD is a rhombus. Prove: ABCD is a parallelogram.
Given: is a rhombus. Prove: is a parallelogram. 1. &. 1. Property of a rhombus. 2.. 2. Reflexive axiom. 3.. 3. SSS. + o ( + ) =180 4.. 4. Interior angle sum for a triangle. 5.. 5. PT + o ( + ) =180 6..
More informationMath 330A Class Drills All content copyright October 2010 by Mark Barsamian
Math 330A Class Drills All content copyright October 2010 by Mark Barsamian When viewing the PDF version of this document, click on a title to go to the Class Drill. Drill for Section 1.3.1: Theorems about
More informationPARALLEL LINES CHAPTER
HPTR 9 HPTR TL OF ONTNTS 91 Proving Lines Parallel 92 Properties of Parallel Lines 93 Parallel Lines in the oordinate Plane 94 The Sum of the Measures of the ngles of a Triangle 95 Proving Triangles
More informationThis supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.
Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.
More information1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationName: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Name: lass: _ ate: _ I: SSS Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Given the lengths marked on the figure and that bisects E, use SSS to explain
More informationA quick review of elementary Euclidean geometry
C H P T E R 0 quick review of elementary Euclidean geometry 0.1 MESUREMENT ND CONGRUENCE 0.2 PSCH S XIOM ND THE CROSSR THEOREM 0.3 LINER PIRS ND VERTICL PIRS 0.4 TRINGLE CONGRUENCE CONDITIONS 0.5 THE EXTERIOR
More informationPicture. Right Triangle. Acute Triangle. Obtuse Triangle
Name Perpendicular Bisector of each side of a triangle. Construct the perpendicular bisector of each side of each triangle. Point of Concurrency Circumcenter Picture The circumcenter is equidistant from
More informationPicture. Right Triangle. Acute Triangle. Obtuse Triangle
Name Perpendicular Bisector of each side of a triangle. Construct the perpendicular bisector of each side of each triangle. Point of Concurrency Circumcenter Picture The circumcenter is equidistant from
More information#2. Isosceles Triangle Theorem says that If a triangle is isosceles, then its BASE ANGLES are congruent.
1 Geometry Proofs Reference Sheet Here are some of the properties that we might use in our proofs today: #1. Definition of Isosceles Triangle says that If a triangle is isosceles then TWO or more sides
More information14 add 3 to preceding number 35 add 2, then 4, then 6,...
Geometry Definitions, Postulates, and Theorems hapter 2: Reasoning and Proof Section 2.1: Use Inductive Reasoning Standards: 1.0 Students demonstrate understanding by identifying and giving examples of
More informationA summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:
summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs: efinitions: efinition of midpoint and segment bisector M If a line intersects another line segment
More informationPOTENTIAL REASONS: Definition of Congruence:
Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides
More informationElementary triangle geometry
Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors
More informationMathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3
Mathematics 3301001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs
More informationCoordinate Plane Project
Coordinate Plane Project C. Sormani, MTTI, Lehman College, CUNY MAT631, Fall 2009, Project XI BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent and Similar Triangle Theorems,
More informationInt. Geometry Unit 2 Quiz Review (Lessons 14) 1
Int. Geometry Unit Quiz Review (Lessons 4) Match the examples on the left with each property, definition, postulate, and theorem on the left PROPRTIS:. ddition Property of = a. GH = GH. Subtraction Property
More informationStudent Name: Teacher: Date: District: MiamiDade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1
Student Name: Teacher: Date: District: MiamiDade County Public Schools Assessment: 9_12 Mathematics Geometry Exam 1 Description: GEO Topic 1 Test: Tools of Geometry Form: 201 1. A student followed the
More information1. Determine all real numbers a, b, c, d that satisfy the following system of equations.
altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c
More informationLesson 2: Circles, Chords, Diameters, and Their Relationships
Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct
More informationacute angle adjacent angles angle bisector between axiom Vocabulary Flash Cards Chapter 1 (p. 39) Chapter 1 (p. 48) Chapter 1 (p.38) Chapter 1 (p.
Vocabulary Flash ards acute angle adjacent angles hapter 1 (p. 39) hapter 1 (p. 48) angle angle bisector hapter 1 (p.38) hapter 1 (p. 42) axiom between hapter 1 (p. 12) hapter 1 (p. 14) collinear points
More informationChapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.
Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.
More informationChapter 4. Outline of chapter. 1. More standard geometry (interior and exterior angles, etc.) 3. Statements equivalent to the parallel postulate
Chapter 4 Outline of chapter 1. More standard geometry (interior and exterior angles, etc.) 2. Measurement (degrees and centimeters) 3. Statements equivalent to the parallel postulate 4. Saccheri and Lambert
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationGeometry Unit 1. Basics of Geometry
Geometry Unit 1 Basics of Geometry Using inductive reasoning  Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture an unproven statement that is based
More informationAdvanced Euclidean Geometry
dvanced Euclidean Geometry What is the center of a triangle? ut what if the triangle is not equilateral?? Circumcenter Equally far from the vertices? P P Points are on the perpendicular bisector of a line
More informationNAME DATE PERIOD. Study Guide and Intervention
opyright Glencoe/McGrawHill, a division of he McGrawHill ompanies, Inc. 51 M IO tudy Guide and Intervention isectors, Medians, and ltitudes erpendicular isectors and ngle isectors perpendicular bisector
More informationLogic Rule 0 No unstated assumptions may be used in a proof. Logic Rule 1 Allowable justifications.
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and NonEuclidean Geometries, 4th Ed by Marvin Jay Greenberg (Revised: 18 Feb 2011) Logic Rule 0 No unstated assumptions may be
More informationSOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses
CHAPTER SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY For the review sessions, I will try to post some of the solved homework since I find that at this age both taking notes and proofs are still a burgeoning
More informationGeometry. Unit 6. Quadrilaterals. Unit 6
Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections
More information**The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle.
Geometry Week 7 Sec 4.2 to 4.5 section 4.2 **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle. Protractor Postulate:
More informationA polygon with five sides is a pentagon. A polygon with six sides is a hexagon.
Triangles: polygon is a closed figure on a plane bounded by (straight) line segments as its sides. Where the two sides of a polygon intersect is called a vertex of the polygon. polygon with three sides
More information11 th Annual HarvardMIT Mathematics Tournament
11 th nnual HarvardMIT Mathematics Tournament Saturday February 008 Individual Round: Geometry Test 1. [] How many different values can take, where,, are distinct vertices of a cube? nswer: 5. In a unit
More informationTriangle Congruence and Similarity: A CommonCoreCompatible Approach
Triangle Congruence and Similarity: A CommonCoreCompatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry curriculum in grades 8 to 10:
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More information2. Sketch and label two different isosceles triangles with perimeter 4a + b. 3. Sketch an isosceles acute triangle with base AC and vertex angle B.
Section 1.5 Triangles Notes Goal of the lesson: Explore the properties of triangles using Geometer s Sketchpad Define and classify triangles and their related parts Practice writing more definitions Learn
More informationTheorem Prove Given. Dates, assignments, and quizzes subject to change without advance notice.
Name Period GP GOTRI PROOFS 1) I can define, identify and illustrate the following terms onjecture Inductive eductive onclusion Proof Postulate Theorem Prove Given ates, assignments, and quizzes subject
More informationThree Lemmas in Geometry
Winter amp 2010 Three Lemmas in Geometry Yufei Zhao Three Lemmas in Geometry Yufei Zhao Massachusetts Institute of Technology yufei.zhao@gmail.com 1 iameter of incircle T Lemma 1. Let the incircle of triangle
More information4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.
14 Perpendicularity and Angle Congruence Definition (acute angle, right angle, obtuse angle, supplementary angles, complementary angles) An acute angle is an angle whose measure is less than 90. A right
More information1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always 3. 1 4. 3 5. 1 6. 1 7a. True 7b. True 7c. True 7d. True 7e. True 8.
1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always 3. 1 4. 3 5. 1 6. 1 7a. True 7b. True 7c. True 7d. True 7e. True 8. 3 and 13 9. a 4, c 26 10. 8 11. 20 12. 130 13 12 14. 10 15.
More informationBC AB = AB. The first proportion is derived from similarity of the triangles BDA and ADC. These triangles are similar because
150 hapter 3. SIMILRITY 397. onstruct a triangle, given the ratio of its altitude to the base, the angle at the vertex, and the median drawn to one of its lateral sides 398. Into a given disk segment,
More informationPoincaré s Disk Model for Hyperbolic Geometry
Chapter 9 oincaré s Disk Model for Hyperbolic Geometry 9.1 Saccheri s Work Recall that Saccheri introduced a certain family of quadrilaterals. Look again at Section 7.3 to remind yourself of the properties
More informationTerminology: When one line intersects each of two given lines, we call that line a transversal.
Feb 23 Notes: Definition: Two lines l and m are parallel if they lie in the same plane and do not intersect. Terminology: When one line intersects each of two given lines, we call that line a transversal.
More information12. Parallels. Then there exists a line through P parallel to l.
12. Parallels Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails
More informationGeometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
More informationMath 3372College Geometry
Math 3372College Geometry Yi Wang, Ph.D., Assistant Professor Department of Mathematics Fairmont State University Fairmont, West Virginia Fall, 2004 Fairmont, West Virginia Copyright 2004, Yi Wang Contents
More informationMost popular response to
Class #33 Most popular response to What did the students want to prove? The angle bisectors of a square meet at a point. A square is a convex quadrilateral in which all sides are congruent and all angles
More informationA (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. beginning or end.
Points, Lines, and Planes Point is a position in space. point has no length or width or thickness. point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. (straight)
More informationSeattle Public Schools KEY to Review Questions for the Washington State Geometry End of Course Exam
Seattle Public Schools KEY to Review Questions for the Washington State Geometry End of ourse Exam 1) Which term best defines the type of reasoning used below? bdul broke out in hives the last four times
More informationTheorem 3.1. If two circles meet at P and Q, then the magnitude of the angles between the circles is the same at P and Q.
3 rthogonal circles Theorem 3.1. If two circles meet at and, then the magnitude of the angles between the circles is the same at and. roof. Referring to the figure on the right, we have A B AB (by SSS),
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More information/27 Intro to Geometry Review
/27 Intro to Geometry Review 1. An acute has a measure of. 2. A right has a measure of. 3. An obtuse has a measure of. 13. Two supplementary angles are in ratio 11:7. Find the measure of each. 14. In the
More informationGEOMETRY FINAL EXAM REVIEW
GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.
More informationGeometry Final Assessment 1112, 1st semester
Geometry Final ssessment 1112, 1st semester Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name three collinear points. a. P, G, and N c. R, P, and G
More informationCopyright 2014 Edmentum  All rights reserved. 04/01/2014 Cheryl Shelton 10 th Grade Geometry Theorems Given: Prove: Proof: Statements Reasons
Study Island Copyright 2014 Edmentum  All rights reserved. Generation Date: 04/01/2014 Generated By: Cheryl Shelton Title: 10 th Grade Geometry Theorems 1. Given: g h Prove: 1 and 2 are supplementary
More information73 Parallel and Perpendicular Lines
Learn to identify parallel, perpendicular, and skew lines, and angles formed by a transversal. 73 Parallel Insert Lesson and Perpendicular Title Here Lines Vocabulary perpendicular lines parallel lines
More informationHow Do You Measure a Triangle? Examples
How Do You Measure a Triangle? Examples 1. A triangle is a threesided polygon. A polygon is a closed figure in a plane that is made up of segments called sides that intersect only at their endpoints,
More informationFormal Geometry S1 (#2215)
Instructional Materials for WCSD Math Common Finals The Instructional Materials are for student and teacher use and are aligned to the Course Guides for the following course: Formal Geometry S1 (#2215)
More informationA segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Perpendicular Bisector Theorem
Perpendicular Bisector Theorem A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Converse of the Perpendicular Bisector Theorem If a
More informationChapter 4.1 Parallel Lines and Planes
Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about
More informationPOINT OF INTERSECTION OF TWO STRAIGHT LINES
POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationThe Inscribed Angle Alternate A Tangent Angle
Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is
More informationGeometry Unit 10 Notes Circles. Syllabus Objective: 10.1  The student will differentiate among the terms relating to a circle.
Geometry Unit 0 Notes ircles Syllabus Objective: 0.  The student will differentiate among the terms relating to a circle. ircle the set of all points in a plane that are equidistant from a given point,
More informationNeutral Geometry. Chapter Neutral Geometry
Neutral Geometry Chapter 4.14.4 Neutral Geometry Geometry without the Parallel Postulate Undefined terms point, line, distance, halfplane, angle measure Axioms Existence Postulate (points) Incidence
More informationUnit 3: Triangle Bisectors and Quadrilaterals
Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties
More informationReflex Vertices A 2. A 1 Figure 2a
Reflex Vertices For any integer n 3, the polygon determined by the n points 1,..., n (1) in the plane consists of the n segments 1 2, 2 3,..., n1 n, n 1, (2) provided that we never pass through a point
More information3.1 Triangles, Congruence Relations, SAS Hypothesis
Chapter 3 Foundations of Geometry 2 3.1 Triangles, Congruence Relations, SAS Hypothesis Definition 3.1 A triangle is the union of three segments ( called its side), whose end points (called its vertices)
More informationGeometry 81 Angles of Polygons
. Sum of Measures of Interior ngles Geometry 81 ngles of Polygons 1. Interior angles  The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.
More informationProblems and Solutions, INMO2011
Problems and Solutions, INMO011 1. Let,, be points on the sides,, respectively of a triangle such that and. Prove that is equilateral. Solution 1: c ka kc b kb a Let ;. Note that +, and hence. Similarly,
More information55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.
Geometry Core Semester 1 Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which topics you need to review most carefully. The unit
More informationLesson 1: Introducing Circles
IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed
More informationLesson 18: Looking More Carefully at Parallel Lines
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
More informationNotes on Perp. Bisectors & Circumcenters  Page 1
Notes on Perp. isectors & ircumcenters  Page 1 Name perpendicular bisector of a triangle is a line, ray, or segment that intersects a side of a triangle at a 90 angle and at its midpoint. onsider to the
More informationCh 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and Angles [and Triangles]
h 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and ngles [and Triangles] Warm up: Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having.
More informationUnknown Angle Problems with Inscribed Angles in Circles
: Unknown Angle Problems with Inscribed Angles in Circles Student Outcomes Use the inscribed angle theorem to find the measures of unknown angles. Prove relationships between inscribed angles and central
More informationChapters 4 and 5 Notes: Quadrilaterals and Similar Triangles
Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles IMPORTANT TERMS AND DEFINITIONS parallelogram rectangle square rhombus A quadrilateral is a polygon that has four sides. A parallelogram is
More informationMath 311 Test III, Spring 2013 (with solutions)
Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam
More information1 Solution of Homework
Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,
More informationLet s Talk About Symmedians!
Let s Talk bout Symmedians! Sammy Luo and osmin Pohoata bstract We will introduce symmedians from scratch and prove an entire collection of interconnected results that characterize them. Symmedians represent
More informationFinal Review Geometry A Fall Semester
Final Review Geometry Fall Semester Multiple Response Identify one or more choices that best complete the statement or answer the question. 1. Which graph shows a triangle and its reflection image over
More informationLecture 24: Saccheri Quadrilaterals
Lecture 24: Saccheri Quadrilaterals 24.1 Saccheri Quadrilaterals Definition In a protractor geometry, we call a quadrilateral ABCD a Saccheri quadrilateral, denoted S ABCD, if A and D are right angles
More informationPlane transformations and isometries
Plane transformations and isometries We have observed that Euclid developed the notion of congruence in the plane by moving one figure on top of the other so that corresponding parts coincided. This notion
More informationConic Construction of a Triangle from the Feet of Its Angle Bisectors
onic onstruction of a Triangle from the Feet of Its ngle isectors Paul Yiu bstract. We study an extension of the problem of construction of a triangle from the feet of its internal angle bisectors. Given
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationHomework 9 Solutions and Test 4 Review
Homework 9 Solutions and Test 4 Review Dr. Holmes May 6, 2012 1 Homework 9 Solutions This is the homework solution set followed by some test review remarks (none of which should be surprising). My proofs
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationMI314 History of Mathematics: Episodes in NonEuclidean Geometry
MI314 History of Mathematics: Episodes in NonEuclidean Geometry Giovanni Saccheri, Euclides ab omni naevo vindicatus In 1733, Saccheri published Euclides ab omni naevo vindicatus (Euclid vindicated om
More informationPerformance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will
Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles
More informationChapter 5.1 and 5.2 Triangles
Chapter 5.1 and 5.2 Triangles Students will classify triangles. Students will define and use the Angle Sum Theorem. A triangle is formed when three noncollinear points are connected by segments. Each
More information