Geometry Chapter 5 Relationships Within Triangles
|
|
- Valentine Sutton
- 7 years ago
- Views:
Transcription
1 Objectives: Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 To use properties of midsegments to solve problems. To use properties of perpendicular bisectors and angle bisectors. To identify properties of perpendicular bisectors. To identify properties of medians and altitudes of a triangle. To write the negation of a statement and the inverse and contrapositive of a conditional statement. To use indirect reasoning. To use inequalities involving angles of triangles. To use inequalities involving sides of triangles. Theorems and Postulates: Theorem 5-1: If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. Theorem 5-2: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5-3: If a point is equidistant from the endpoint of a segment, then it is on the perpendicular bisector of the segment. Theorem 5-4: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 5-5: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Theorem 5-8: The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. Theorem 5-9: The lines that contain the altitudes of a triangle are concurrent. Corollary to the Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. m1 m2 and m1 m 3 Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer sides. If XZ XY, then my m Z
2 Theorem 5-11: If two angles of a triangle are not congruent then the longer side lies opposite the larger side. If ma mb, then BC AC. Theorem 5-12: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. XY YZ XZ YZ ZX YX ZX XY ZY Vocabulary: Midsegment Coordinate Proof Circumcenter of a triangle Circumscribed Incenter of a triangle Median of a triangle Centroid Orthocenter of a triangle Altitude of a triangle Inverse Contrapositive Equivalent Statements Indirect Reasoning Indirect Proof Daily Work 5 pts. each Homework 5.1 p #1-19odd, 22-25, 26, 29, p #1-4, 6, 12-15, 19-25, p #7, 8-11, 17, 29, 42, p #1-2, 3-15odd, 19-22all, 27,29, 31 Checkpoint Quiz p. 263 #1-10 (Check answers in the back of your book.) p #1-15odd, 17-19, 23, 29, 31-35odd 5.5 p #1-27odd Triangle Inequality Worksheet Review 1 p. 694 #1-30 Review 2 p. 281 #1-10, 11-41odd Points Earned
3 Journal (5 points per Question) Vocabulary (15 points) Chapter Project (30 points) Answer each of the following using complete sentences. 1. Draw a right triangle and its midsegments. Compare the four triangles created by the midsegments with the original triangle. Make a conjecture about your observations. 2. Explain and give an example of the difference between a median and a bisector of a triangle. 3. Give an example of a time where the median and the bisector are the same. 4. Use a conditional statement to write the inverse and the contrapositive of the statement. 1. Make a PowerPoint of 5 vocabulary terms, include definitions and pictures. 2. Make a poster of 5 vocabulary terms, include definitions and pictures. 3. Make vocabulary note cards for 10 of the vocabulary words with the word on one side and a picture and definition on the reverse. Attached is the information for the Chapter 5 Project. We will work on it for the first few days of Chapter 5. Points Earned
4
5 Investigating Special Segments of a Triangle Name Hour Date Terms you will need: Perpendicular bisector of a triangle Angle bisector of a triangle Median Altitude Your group will need to work together to do the following: Cut four large acute scalene triangles out of paper (use a ruler and protractor). Label the vertices of each triangle A, B, and C and number the triangles 1-4. For triangle #1 fold the triangle to produce the perpendicular bisectors of all three sides. For triangle #2 fold the triangle to produce the angle bisectors of all three angles. For triangle #3 fold the triangle to produce all three medians. For triangle #4 fold the triangle to produce all three altitudes. You may discuss the following questions with your group, but everyone needs to write the answers individually. Please use complete sentences. You will be investigating the results of the four folded triangles. Use a ruler and protractor to measure things and feel free to draw on your triangle with a compass. When you are measuring or drawing on your triangle you may find it handy to tape or glue it to another sheet of paper. Look at triangle #1. What do you observe to be true about the perpendicular bisectors of the sides of your triangle? Look for three things.
6 Look at triangle #2. What do you observe to be true about the angle bisectors of your triangle? Look for three things. Look at triangle #3. What do you observe to be true about the medians of your triangle? Please list all the things you can find. Look at triangle #4. What do you observe to be true about the altitudes of your triangle? Please list all the things you can find.
7 Build a Mobile Goal: Work with a partner to create a mobile with triangles that demonstrates how an object can be balanced at its centroid (point where the three medians of the triangle intersect). Your mobile should be designed to meet the following requirements: Consist of at least three levels. Contain at least two pairs of congruent triangles (make it clear which are congruent). Include a right triangle, equilateral triangle, isosceles triangle, and a scalene triangle. Balanced so it moves in a gentle breeze or when you touch it. Neat, well made with attention to details. It should be interesting to look at. Materials you will have available to you: String Sticks of some kind Thin cardboard Protractors Rulers Compasses How you will be graded: Levels within your mobile Congruent triangles Right, equilateral, isosceles, and scalene triangles Balance Craftsmanship Total Possible 3 points 2 points 1 point 0 points All 3 levels present 2 levels 1 level Nothing functional Both pairs of congruent triangles present and congruent. All for present and accurately made. All three levels balance and move when lightly tapped. Neat, well made and interesting to look at. Use of pattern and/or color. 15 points Either one pair of triangles perfectly congruent or 2 pairs with minor errors. Three of the four present and accurately made or all four made with a few minor errors. Two levels balance and move when lightly tapped. Two out of neat, well made, or interesting. Attempted to make one pair of congruent triangles, but they contain errors. At least two attempted with only minor errors or 3-4 with more errors. One level balances and moves when lightly tapped. One out of neat, well made, or interesting. Not attempted. Not attempted. Not Balanced. A mess.
Name Period 10/22 11/1 10/31 11/1. Chapter 4 Section 1 and 2: Classifying Triangles and Interior and Exterior Angle Theorem
Name Period 10/22 11/1 Vocabulary Terms: Acute Triangle Right Triangle Obtuse Triangle Scalene Isosceles Equilateral Equiangular Interior Angle Exterior Angle 10/22 Classify and Triangle Angle Theorems
More informationLesson 5-3: Concurrent Lines, Medians and Altitudes
Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special
More informationChapter 6 Notes: Circles
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
More informationChapters 6 and 7 Notes: Circles, Locus and Concurrence
Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of
More informationDuplicating Segments and Angles
CONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson, you Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using patty
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 information5.1 Midsegment Theorem and Coordinate Proof
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle - A midsegment of a triangle is a segment that connects
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationGeometry. Relationships in Triangles. Unit 5. Name:
Geometry Unit 5 Relationships in Triangles Name: 1 Geometry Chapter 5 Relationships in Triangles ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK.
More informationLesson 3.1 Duplicating Segments and Angles
Lesson 3.1 Duplicating Segments and ngles In Exercises 1 3, use the segments and angles below. Q R S 1. Using only a compass and straightedge, duplicate each segment and angle. There is an arc in each
More informationUnit 2 - Triangles. Equilateral Triangles
Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical
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 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 informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationCenters of Triangles Learning Task. Unit 3
Centers of Triangles Learning Task Unit 3 Course Mathematics I: Algebra, Geometry, Statistics Overview This task provides a guided discovery and investigation of the points of concurrency in triangles.
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 informationGEOMETRY. Constructions OBJECTIVE #: G.CO.12
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
More informationCAIU Geometry - Relationships with Triangles Cifarelli Jordan Shatto
CK-12 FOUNDATION CAIU Geometry - Relationships with Triangles Cifarelli Jordan Shatto CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12
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 informationVisualizing Triangle Centers Using Geogebra
Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will
More informationNAME DATE PERIOD. Study Guide and Intervention
opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-1 M IO tudy Guide and Intervention isectors, Medians, and ltitudes erpendicular isectors and ngle isectors perpendicular bisector
More informationThe Triangle and its Properties
THE TRINGLE ND ITS PROPERTIES 113 The Triangle and its Properties Chapter 6 6.1 INTRODUCTION triangle, you have seen, is a simple closed curve made of three line segments. It has three vertices, three
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 Module 4 Unit 2 Practice Exam
Name: Class: Date: ID: A Geometry Module 4 Unit 2 Practice Exam Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which diagram shows the most useful positioning
More informationMathematics Geometry Unit 1 (SAMPLE)
Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This
More informationSemester Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question.
Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Are O, N, and P collinear? If so, name the line on which they lie. O N M P a. No,
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 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 informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
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 non-collinear points are connected by segments. Each
More informationConjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)
Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name
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 informationThis is a tentative schedule, date may change. Please be sure to write down homework assignments daily.
Mon Tue Wed Thu Fri Aug 26 Aug 27 Aug 28 Aug 29 Aug 30 Introductions, Expectations, Course Outline and Carnegie Review summer packet Topic: (1-1) Points, Lines, & Planes Topic: (1-2) Segment Measure Quiz
More informationGeometry Regents Review
Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest
More informationTIgeometry.com. Geometry. Angle Bisectors in a Triangle
Angle Bisectors in a Triangle ID: 8892 Time required 40 minutes Topic: Triangles and Their Centers Use inductive reasoning to postulate a relationship between an angle bisector and the arms of the angle.
More informationCeva s Theorem. Ceva s Theorem. Ceva s Theorem 9/20/2011. MA 341 Topics in Geometry Lecture 11
MA 341 Topics in Geometry Lecture 11 The three lines containing the vertices A, B, and C of ABC and intersecting opposite sides at points L, M, and N, respectively, are concurrent if and only if 2 3 1
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 mid-point and segment bisector M If a line intersects another line segment
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 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 informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXMINTION GEOMETRY Thursday, January 26, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name
More informationSection 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18
Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,
More informationThe Euler Line in Hyperbolic Geometry
The Euler Line in Hyperbolic Geometry Jeffrey R. Klus Abstract- In Euclidean geometry, the most commonly known system of geometry, a very interesting property has been proven to be common among all triangles.
More informationIncenter and Circumcenter Quiz
Name: lass: ate: I: Incenter and ircumcenter Quiz Multiple hoice Identify the choice that best completes the statement or answers the question.. The diagram below shows the construction of the center of
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationMathematics Georgia Performance Standards
Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your
More informationCircle Theorems. This circle shown is described an OT. As always, when we introduce a new topic we have to define the things we wish to talk about.
Circle s circle is a set of points in a plane that are a given distance from a given point, called the center. The center is often used to name the circle. T This circle shown is described an OT. s always,
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 informationAnalytical Geometry (4)
Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line
More information5-1 Perpendicular and Angle Bisectors
5-1 Perpendicular and Angle Bisectors Equidistant Distance and Perpendicular Bisectors Theorem Hypothesis Conclusion Perpendicular Bisector Theorem Converse of the Perp. Bisector Theorem Locus Applying
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 informationGeometry of 2D Shapes
Name: Geometry of 2D Shapes Answer these questions in your class workbook: 1. Give the definitions of each of the following shapes and draw an example of each one: a) equilateral triangle b) isosceles
More informationMathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3
Mathematics 3301-001 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 informationThe Geometry of Piles of Salt Thinking Deeply About Simple Things
The Geometry of Piles of Salt Thinking Deeply About Simple Things PCMI SSTP Tuesday, July 15 th, 2008 By Troy Jones Willowcreek Middle School Important Terms (the word line may be replaced by the word
More information11.3 Curves, Polygons and Symmetry
11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon
More informationSpecial Segments in Triangles
HPTER 10 Special Segments in Triangles c GOL Identify the altitudes, medians, and angle bisectors in a triangle. You will need a protractor a ruler Learn about the Math Every triangle has three bases and
More informationGeometry - Semester 2. Mrs. Day-Blattner 1/20/2016
Geometry - Semester 2 Mrs. Day-Blattner 1/20/2016 Agenda 1/20/2016 1) 20 Question Quiz - 20 minutes 2) Jan 15 homework - self-corrections 3) Spot check sheet Thales Theorem - add to your response 4) Finding
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationGPS GEOMETRY Study Guide
GPS GEOMETRY Study Guide Georgia End-Of-Course Tests TABLE OF CONTENTS INTRODUCTION...5 HOW TO USE THE STUDY GUIDE...6 OVERVIEW OF THE EOCT...8 PREPARING FOR THE EOCT...9 Study Skills...9 Time Management...10
More information15. Appendix 1: List of Definitions
page 321 15. Appendix 1: List of Definitions Definition 1: Interpretation of an axiom system (page 12) Suppose that an axiom system consists of the following four things an undefined object of one type,
More informationWeek 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test
Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan
More informationPUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION GEOMETRY HONORS. Middle School and High School
PUBLIC SCHOOLS OF EDISON TOWNSHIP OFFICE OF CURRICULUM AND INSTRUCTION GEOMETRY HONORS Length of Course: Elective/Required: Schools: Term Required Middle School and High School Eligibility: Grades 8-12
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 informationHeron s Formula. Key Words: Triangle, area, Heron s formula, angle bisectors, incenter
Heron s Formula Lesson Summary: Students will investigate the Heron s formula for finding the area of a triangle. The lab has students find the area using three different methods: Heron s, the basic formula,
More informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationIMO Training 2008 Circles Yufei Zhao. Circles. Yufei Zhao.
ircles Yufei Zhao yufeiz@mit.edu 1 Warm up problems 1. Let and be two segments, and let lines and meet at X. Let the circumcircles of X and X meet again at O. Prove that triangles O and O are similar.
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 informationTImath.com. Geometry. Points on a Perpendicular Bisector
Points on a Perpendicular Bisector ID: 8868 Time required 40 minutes Activity Overview In this activity, students will explore the relationship between a line segment and its perpendicular bisector. Once
More informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationName: Chapter 4 Guided Notes: Congruent Triangles. Chapter Start Date: Chapter End Date: Test Day/Date: Geometry Fall Semester
Name: Chapter 4 Guided Notes: Congruent Triangles Chapter Start Date: Chapter End Date: Test Day/Date: Geometry Fall Semester CH. 4 Guided Notes, page 2 4.1 Apply Triangle Sum Properties triangle polygon
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 informationIntermediate Math Circles October 10, 2012 Geometry I: Angles
Intermediate Math Circles October 10, 2012 Geometry I: Angles Over the next four weeks, we will look at several geometry topics. Some of the topics may be familiar to you while others, for most of you,
More informationMath 531, Exam 1 Information.
Math 531, Exam 1 Information. 9/21/11, LC 310, 9:05-9:55. Exam 1 will be based on: Sections 1A - 1F. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/531fa11/531.html)
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More information39 Symmetry of Plane Figures
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
More informationGeometry EOC Practice Test #2
Class: Date: Geometry EOC Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Rebecca is loading medical supply boxes into a crate. Each supply
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of
More informationShow all work for credit. Attach paper as needed to keep work neat & organized.
Geometry Semester 1 Review Part 2 Name Show all work for credit. Attach paper as needed to keep work neat & organized. Determine the reflectional (# of lines and draw them in) and rotational symmetry (order
More informationReasoning and Proof Review Questions
www.ck12.org 1 Reasoning and Proof Review Questions Inductive Reasoning from Patterns 1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...? (a) 81 (b) 64 (c) 121 (d) 56 2. What is the next
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationObjectives. Cabri Jr. Tools
^Åíáîáíó=NO Objectives To learn how to construct all types of triangles using the Cabri Jr. application To reinforce the difference between a construction and a drawing Cabri Jr. Tools fåíêççìåíáçå `çåëíêìåíáåö
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationPerpendicular and Angle Bisectors
Perpendicular and Angle Bisectors Mathematics Objectives Students will investigate and define perpendicular bisector and angle bisector. Students will discover and describe the property that any point
More informationLine Segments, Rays, and Lines
HOME LINK Line Segments, Rays, and Lines Family Note Help your child match each name below with the correct drawing of a line, ray, or line segment. Then observe as your child uses a straightedge to draw
More informationGEOMETRY COMMON CORE STANDARDS
1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
More informationGeometry Progress Ladder
Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes
More informationDiscovering Math: Exploring Geometry Teacher s Guide
Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional
More informationCevians, Symmedians, and Excircles. MA 341 Topics in Geometry Lecture 16
Cevians, Symmedians, and Excircles MA 341 Topics in Geometry Lecture 16 Cevian A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). B cevian
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 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 information1.1 Identify Points, Lines, and Planes
1.1 Identify Points, Lines, and Planes Objective: Name and sketch geometric figures. Key Vocabulary Undefined terms - These words do not have formal definitions, but there is agreement aboutwhat they mean.
More informationCumulative Test. 161 Holt Geometry. Name Date Class
Choose the best answer. 1. P, W, and K are collinear, and W is between P and K. PW 10x, WK 2x 7, and PW WK 6x 11. What is PK? A 2 C 90 B 6 D 11 2. RM bisects VRQ. If mmrq 2, what is mvrm? F 41 H 9 G 2
More informationGeoGebra Workshop for the Initial Teacher Training in Primary Education
GeoGebra Workshop for the Initial Teacher Training in Primary Education N. Ruiz natalia.ruiz@uam.es Facultad de Formación de Profesorado y Educación (Faculty of Education) Universidad Autónoma de Madrid
More information