Triangles and Transversals Triangles

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Triangles and Transversals Triangles A three-sided polygon. Symbol → ▵ You name it ▵ ABC. Total Angle Sum of a Triangle The interior angles of a triangle add up to 180 degrees. Symbol = Acute Triangle A triangle that contains only angles that are less than 90 degrees. Obtuse Triangle A triangle with one angle greater than 90 degrees (an obtuse angle). Right Triangle A triangle with one right angle (90 degrees). Vertex The common endpoint of two or more rays or line segments. Complementary Angles Two angles whose measures have a sum of 90 degrees. Supplementary Angles Two angles whose measures have a sum of 180 degrees. Perpendicular Lines Lines that intersect to form a right angle (90 degrees). Symbol = Parallel Lines Lines that never intersect. Arrows are used to indicate lines are parallel. Symbol = || Transversal Lines A line that cuts across two or more (usually parallel) lines. Intersect The point where two lines meet or cross. Vertical Angles Angles opposite one another at the intersection of two lines. Vertical angles have the same angle measurements. Interior Angles An angle inside a shape. Exterior Angles Angles outside of a shape. Alternate Interior Angles The pairs of angles on opposite sides of the transversal but inside the two lines. Alternate Exterior Angles Each pair of these angles are outside the lines, and on opposite sides of the transversal. Corresponding Angles The angles in matching corners. Reflexive Angles An angle whose measure is greater than 180 degrees and less that 360 degrees. Straight Angles An angle that measures exactly 180 degrees. Adjacent Angles Angles with common side and common vertex without overlapping. Angles that are next to one another..

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Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: Date
Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: ________________________________________ Date: ______________ Block: ________ Identifying Pairs of Lines and Angles Lines that intersect are coplanar Lines that do not intersect o are parallel (coplanar) OR o are skew (not coplanar) In the figure, name: o intersecting lines:___________ o parallel lines:_______________ o skew lines:________________ o parallel planes:_____________ Example: Think of each segment in the figure as part of a line. Find: Line(s) parallel to CD and containing point A _________ Line(s) skew to and containing point A _________ Line(s) perpendicular to and containing point A _________ Plane(s) parallel to plane EFG and containing point A _________ Parallel and Perpendicular Lines If two lines are in the same plane, then they are either parallel or intersect a point. There exist how many lines through a point not on a line? __________ Only __________ of them is parallel to the line. Parallel Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Perpendicular Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Angles and Transversals Interior angles are on the INSIDE of the two lines Exterior angles are on the OUTSIDE of the two lines Alternate angles are on EITHER SIDE of the transversal Consecutive angles are on the SAME SIDE of the transversal Corresponding angles are in the same position on each of the two lines Alternate interior angles lie on either side of the transversal inside the two lines Alternate exterior angles lie on either side of the transversal outside the two lines Consecutive interior angles lie on the same side of the transversal inside the two lines (same side interior) Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs.
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Proofs with Perpendicular Lines
3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units.
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Parallel Lines and Transversals
5.5 Parallel Lines and Transversals How can you use STATES properties of parallel lines to solve real-life problems? STANDARDS MA.8.G.2.2 1 ACTIVITY: A Property of Parallel Lines Work with a partner. ● Talk about what it means for two lines 12 1 cm 56789 to be parallel. Decide on a strategy for 1234 drawing two parallel lines. 01 91 8 9 9 9 9 10 11 12 13 10 10 10 10 ● Use your strategy to carefully draw 7 67 two lines that are parallel. 14 15 5 16 17 18 19 20 3421 22 23 2 24 25 26 1 ● Now, draw a third line 27 28 in. parallel that intersects the two 29 30 lines parallel lines. This line is called a transversal. 2 1 3 4 6 ● The two parallel lines and 5 7 8 the transversal form eight angles. Which of these angles have equal measures? transversal Explain your reasoning. 2 ACTIVITY: Creating Parallel Lines Work with a partner. a. If you were building the house in the photograph, how could you make sure that the studs are parallel to each other? b. Identify sets of parallel lines and transversals in the Studs photograph. 212 Chapter 5 Angles and Similarity 3 ACTIVITY: Indirect Measurement Work with a partner. F a. Use the fact that two rays from the Sun are parallel to explain why △ABC and △DEF are similar. b. Explain how to use similar triangles to ﬁ nd the height of the ﬂ agpole. x ft Sun’s ray C Sun’s ray 5 ft AB3 ft DE36 ft 4.
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Understand the Principles and Properties of Axiomatic (Synthetic
Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length
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Parallel Lines Cut by a Transversal
Parallel Lines Cut by a Transversal I. UNIT OVERVIEW & PURPOSE: The goal of this unit is for students to understand the angle theorems related to parallel lines. This is important not only for the mathematics course, but also in connection to the real world as parallel lines are used in designing buildings, airport runways, roads, railroad tracks, bridges, and so much more. Students will work cooperatively in groups to apply the angle theorems to prove lines parallel, to practice geometric proof and discover the connections to other topics including relationships with triangles and geometric constructions. II. UNIT AUTHOR: Darlene Walstrum Patrick Henry High School Roanoke City Public Schools III. COURSE: Mathematical Modeling: Capstone Course IV. CONTENT STRAND: Geometry V. OBJECTIVES: 1. Using prior knowledge of the properties of parallel lines, students will identify and use angles formed by two parallel lines and a transversal. These will include alternate interior angles, alternate exterior angles, vertical angles, corresponding angles, same side interior angles, same side exterior angles, and linear pairs. 2. Using the properties of these angles, students will determine whether two lines are parallel. 3. Students will verify parallelism using both algebraic and coordinate methods. 4. Students will practice geometric proof. 5. Students will use constructions to model knowledge of parallel lines cut by a transversal. These will include the following constructions: parallel lines, perpendicular bisector, and equilateral triangle. 6. Students will work cooperatively in groups of 2 or 3. VI. MATHEMATICS PERFORMANCE EXPECTATION(s): MPE.32 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two lines are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal.
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Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used
Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the ﬁrst 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Deﬁnitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent.
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Neutral Geometry
Neutral geometry And then we add Euclid’s parallel postulate Saccheri’s dilemma Options are: Summit angles are right Wants Summit angles are obtuse Was able to rule out, and we’ll see how Summit angles are acute The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of the straight line! Rule out obtuse angles: If we knew that a quadrilateral can’t have the sum of interior angles bigger than 360, we’d be fine. We’d know that if we knew that a triangle can’t have the sum of interior angles bigger than 180. Hold on! Isn’t the sum of the interior angles of a triangle EXACTLY180? Theorem: Angle sum of any triangle is less than or equal to 180º Suppose there is a triangle with angle sum greater than 180º, say anglfle sum of ABC i s 180º + p, w here p> 0. Goal: Construct a triangle that has the same angle sum, but one of its angles is smaller than p. Why is that enough? We would have that the remaining two angles add up to more than 180º: can that happen? Show that any two angles in a triangle add up to less than 180º. What do we know if we don’t have Para lle l Pos tul at e??? Alternate Interior Angle Theorem: If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parallel. Converse of AIA Converse of AIA theorem: If two lines are parallel then the aliilblternate interior angles cut by a transversa l are congruent.
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Equivalents to the Euclidean Parallel Postulate in This Section We Work
Equivalents to the Euclidean Parallel Postulate In this section we work within neutral geometry to prove that a number of different statements are equivalent to the Euclidean Parallel Postulate (EPP). This has historical importance. We noted before that Euclid’s Fifth Postulate was quite different in tone and complexity from his first four postulates, and that Euclid himself put off using the fifth postulate for as long as possible in his development of geometry. The history of mathematics is rife with efforts to either prove the fifth postulate from the first four, or to find a simpler postulate from which the fifth postulate could be proved. These efforts gave rise to a number of candidates that, in the end, proved to be no simpler than the fifth postulate and in fact turned out to be logically equivalent, at least in the context of neutral geometry. Euclid’s fifth postulate indeed captured something fundamental about geometry. We begin with the converse of the Alternate Interior Angles Theorem, as follows: If two parallel lines n and m are cut by a transversal l , then alternate interior angles are congruent. We noted earlier that this statement was equivalent to the Euclidean Parallel Postulate. We now prove that. Theorem: The converse of the Alternate Interior Angles Theorem is equivalent to the Euclidean Parallel Postulate. ~ We first assume the converse of the Alternate Interior Angles Theorem and prove that through a given line l and a point P not on the line, there is exactly one line through P parallel to l. We have already proved that one such line must exist: Drop a perpendicular from P to l; call the foot of that perpendicular Q.
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NATHAN SIDOLI and TAKANORI KUSUBA, Nasir Al-Din Al-Tusi's
Naṣīr al-Dīn al-Ṭūsī’s revision of Theodosius’s Spherics Nathan Sidoli and Takanori Kusuba Keywords: Spherical geometry, spherical astronomy, Spherics, Theodosius, Naṣīr al-Dīn al-Ṭūsī, Greek to Arabic translation, medieval manuscript trans- mission, editorial practices. Abstract We examine the Arabic edition of Theodosius’s Spherics composed by Naṣīr al-Dīn al-Ṭūsī. Through a comparison of this text with earlier Arabic and Greek versions and a study of his editorial remarks, we develop a better un- derstanding of al-Ṭūsī editorial project. We show that al-Ṭūsī’s goal was to revitalize the text of Theodosius’s Spherics by considering it ﬁrstly as a prod- uct of the mathematical sciences and secondarily as a historically contingent work. His editorial practices involved adding a number of additional hy- potheses and auxiliary lemmas to demonstrate theorems used in the Spherics, reworking some propositions to clarify the underlying mathematical argu- ment and reorganizing the proof structure in a few propositions. For al-Ṭūsı, the detailed preservation of the words and drawings was less important than a mathematically coherent presentation of the arguments and diagrams. Introduction If the number of the manuscripts may be taken as any indication, during the medieval period, some of the most popular versions of ancient Greek mathe- matical texts were those made in the middle of the 13th century by Abū JaÒfar Suhayl 8 (2008) pp. 9–46 10 Nathan Sidoli and Takanori Kusuba Muḥammad ibn Muḥammad ibn al-Ḥasan Naṣīr al-Dīn al-Ṭūsī. During the course of a long scholarly career, al-Ṭūsī made new Arabic editions of the texts that were then the classics of the mathematical sciences.
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Homework Section
Chapter Three Parallel Lines and Planes Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates and theorems in this chapter. D. Identify the special pairs of angles created by lines and transversals. E. Classify polygons according to sides and angles. F. Find the measures of interior and exterior angles. G. Compare/contrast inductive & deductive reasoning. H. Use inductive reasoning to reach a conclusion. HOMEWORK NOTE! In homework problems, all proofs MUST be done as two-column proofs. Although various homework problems will ask you to write a ‘paragraph proof’, you are to write all proofs as two-column proofs. Section 3-1 Definitions Homework Pages 76-77: 1-39 Excluding 18, 20, 22 Objectives A. Use the terms parallel lines, skew lines, and parallel planes correctly. B. Distinguish between parallel and skew lines. C. Use the term transversal correctly. D. Use the terms alternate interior angles, same-side interior angles, and corresponding angles correctly. E. Distinguish between alternate interior, same-side interior, and corresponding angles. F. Apply the terms in theorems and proofs. Concepts of a ‘parallel universe’ … • Parallel lines coplanar lines that do not intersect. – The symbol for parallel lines is || • Skew lines non-coplanar lines that do not intersect. • Parallel planes planes that do not intersect. • A line is parallel to a plane if they do not intersect. X Y m s n t m||n s is skew to t ‘Transversing’ the road of geometry … • Transversal A line that intersects two or more coplanar lines at different points.
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Islamic Mathematics
Islamic mathematics Bibliography of Mathematics in Medieval Islamic Civilization Version 13 January 1999. This bibliography is a revised, enlarged and updated version of the bibliography on Islamic mathematics by Richard Lorch on pp. 65-86 of Joseph W. Dauben's The History of Mathematics from Antiquity to the Present: A Selective Bibliography, New York and London: Garland, 1985. This bibliography of Islamic mathematics will appear as a chapter in the updated (1999?) version of Dauben's book which will be made available as a CD-Rom. Reactions and suggestions are very welcome, and can be sent to [email protected]. In this preliminary form, no attention has been paid to diacritical marks in Arabic names. The items in the bibliography have been numbered *1, *2, ... *122, *122a, *122b, *123 etc. and many cross-references have been provided. General Introduction Introductory Works Bibliographies and Handbooks Illustrated Works Texts and Commentaries (Specific Authors in Chronological Order) Studies on Specific Subjects Transmission of Mathematics Mathematics in Specific Areas in the Islamic World Arithmetic Irrational Magnitudes Algebra Number Theory, Indeterminate Equations and Magic Squares Geometry Trigonometry Timekeeping Interpolation, Tables, Analysis of Tables Cultural Context: Islamic Aspects Mathematical Astronomy and Astrology Instruments Mathematics, Art and Architecture Optics Geography Reprinted Works and Collections of Articles General Introduction file:///P|/Igitur%20archief_repository/PR&beleid%20Ig...bsites/HOGENDIJK/hogendijk_00_islamic_mathematics.htm (1 van 33)12-2-2007 14:36:27 Islamic mathematics Islamic mathematics and Arabic mathematics are modern historical terms for the mathematical sciences in Islamic civilization from the beginning of Islam (A.D. 622) until the 17th century.
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Euclidean Geometry Key Concepts • Classifying Angles • Parallel Lines and Transversal Lines • Classifying Triangles • Pr
Euclidean Geometry Key Concepts Classifying Angles Parallel lines and transversal lines Classifying Triangles Properties of Triangles . Relationship between angles . Congruency . Similarity . Pythagoras . Mid-point Theorem Properties of Quadrilaterals Terminology Acute angle: Greater than 00 but less than 900 Right angle: Angle equal to 900 Obtuse angle: Angle greater than 900 Straight angle: Angle equal to 1800 Reflex angle: Angle greater than 1800 but less than 3600 Revolution: Sum of the angles around a point, equal to 3600 Adjacent angles: Angles that share a vertex and a common side. Vertically opposite angles: Angles opposite each other when two lines intersect. They share a vertex and are equal. Supplementary angles: Two angles that add up to 1800. Complementary angles: Two angles that add up to 900. Parallel lines Lines that are always the same distance apart A transversal line A line that intersects two or more parallel lines. Interior angles Angles that lie in between the parallel lines. Exterior angles Angles that lie outside the parallel lines. Corresponding angles Angles on the same side of the lines and the same side of the transversal. Co-interior angles Angles that lie in between the lines and on the same side of the transversal. Alternate interior angles Interior angles that lie inside the line and on opposite sides of the transversal. X-planation Properties of the angles formed by a transversal line intersecting two parallel lines If the lines are parallel the corresponding angles will be equal the co-interior
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