1.2 Informal Geometry

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1.2 Informal Geometry"

Transcription

1 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 of an angle Definitions: give meaning to new terms xioms or Postulates: statements accepted as true Theorems: If P, then Q need to be proved. Undefined terms: Point Line; collinear points Plane Space Terms that will be formally defined latter ngle Triangle Rectangle Line Segment Definition: line segment is the part of a line that consists of two end points, and all the points between them. Notation Measuring The Midpoint ongruent line segments Measuring angles ongruent angles isect an angle (trisect ) Straight angle Right angle ngles formed by intersection of two lines Perpendicular lines Parallel lines (informally) Empty set onstructions: onstruct a segment congruent to a given segment onstruct the midpoint of a segment

2 1.3 Early Definitions and Postulates Definition: gives meaning to new terms; described precisely a term. Postulate: a statement assumed to be true Postulate 1. Two points determine a line Through two distinct points, there is exactly one line. Postulate 2. Ruler Postulate The measure of any line segment is a unique positive number. Postulate 3 Segment -ddition Postulate If X is a point on and -X-, then X+X=. Postulate 4 If two lines intersect, they intersect at a point. Postulate 5 If two lines intersect, they intersect at a point. Postulate 6 If two distinct planes intersect, then their intersection is a line. Postulate 7 Given two distinct points in a plane, the line containing these points also lies in the plane. Definition: The distance between two points and is the line segment that joins the two points. Definition: ongruent line segments are two segments that have the same length. Definition: The midpoint of a line segment is the point that separates the line segment into two congruent parts. Definition: Ray denoted is the union of and all points X on such that is between and X. Definition: Parallel lines are lines that lie in the same plane but do not intersect. Definition: oplanar points are points that lie in the same plane. Theorem The midpoint of a line segment is unique

3 1.4 ngles and their relationships Definition: n angle is the union of two rays that share a common endpoint. Postulate 8 The measure of an angle is a unique positive number. Types of angles: cute angles Right angles Obtuse angles Straight angles Reflex angles Interior, Exterior, and the sides of an angle. Postulate 9 ngle-ddition Postulate If a point D lies in the interior of an angle, then m D+m D=m. Definition: Two angles are adjacent (adj. s) if they have a common vertex and a common side between them. Definition: Two angles are congruent ( s) if they have the same measure. Definition: The bisector of an angle is the ray that separate the given angle into two congruent angles. Definition: Two angles are complementary if the sum of their measures is 90 Definition: Two angles are supplementary if the sum of their measures is 180. Vertical ngles onstruct an angle congruent to a given angle. onstruct the bisector for a given angle.

4 1.5 Introduction to geometric proof To believe certain geometric principles, it is necessary to have proof Two column proof: 1. Statement: State the theorem to be proved 2. Drawing: Represents the hypothesis of the theorem 3. Given: Describes the drowning according to the information found in the hypothesis of the theorem 4. Prove: Describes the drowning according to the claim found in the conclusion of the theorem 5. Proof: Orders in a logical flow, a list of claims (Statements) and justifications (Reasons), beginning with the Given and ending with the Prove Selected properties from lgebra are used as reasons to justify statements: Properties of equality (a, b, and c are real numbers) ddition Property of Equality If =, then + = + Subtraction Property of Equality If =, then = Multiplication Property of Equality If =, then = Division Property of Equality If =, then = Reflexive Property = Symmetric Property If =, then = Distributive Property + = + Substitution Property If =, then replaces b in any equation Transitive Property If =, and = then = Properties of inequality (a, b, and c are real numbers) ddition Property of Inequality If >, then + > + (<) Subtraction Property of Inequality If >, then > (<)

5 lgebraic Proof Ex: Given: 2 1 = 3 Prove: = 2 Proof Statements Reasons = 3 1. Given = ddition Property of Equality 3. 2 = 4 3. Substitution 4. = 4. Division Property of Equality 5. = 2 5. Substitution Geometric Proof Given P on the segment, prove that = Drawing Given: Prove: = P Proof Statements Reasons Given 2. + = 2. Segment ddition Postulate 3 3. = 3. Subtraction property of addition

6 1.6 Perpendicular Lines Definition: Perpendicular lines are two lines that meet to form congruent adjacent angles. Theorem If two lines are perpendicular, then they meet to form right angles. Drawing Given: intersecting at E Prove: is a right angle E D Proof Statements Reasons Given Definition: 3. = 3.Two congruent angles have equal measurements 4. = Measure of a straight angle 5. + = 5. ngle addition postulate 6. + = Substitution 7. + = Substitution 8. 2 = Substitution 9. = Division Property 10. is a right angle 10. Definition of right angle

7 Theorem If two lines intersect, then the vertical angles formed are congruent. Given: intersecting at O Prove: 2 4 Drawing 2 3 O 1 4 D Proof Statements Reasons 1. intersects at O 1. Given 2. = Definition: measure of straight angles 3. = 3. Substitution = and = 4. ngle addition Postulate = Substitution 6. 4 = 2 7. Subtraction Property If two angles are equal in measure the angles are congruent Symmetric Property of ongruence of angles

8 onstruction 1. onstruct a perpendicular to a given line at a specified point. Theorem (NO Proof) In a plane, there is exactly one line perpendicular to a given line at any point on the line. onstruction 2. onstruct a perpendicular bisector to a line segment.

9 1.7 Geometric Proof of a Theorem Theorems: Statements that can be proved Hypothesis: given onclusion: what we need to establish onverse of a Theorem The converse of the statement If P, then Q is If Q, then P. Interchange the hypothesis with the conclusion. Exemple: Theorem If two lines are perpendicular, then they meet to form right angles. The converse: Theorem If two lines meet to form right angles, then they are perpendicular lines.

10 Proof of Theorem Theorem If two lines meet to form right angles, then they are perpendicular lines. Drawing Given: and intersecting at E so that is a right angle Prove: E D Proof Statements Reasons 1. and intersecting at E so that is a right angle 1. Given 2. = If an is a right angle, its measure is is a straight, so = If an is a straight angle, its measure is = 4. ngle addition postulate = Substitution (2), (3), (4) 6. = Subtraction Property of = (5) 7. = 7. Substitution (2), (^) If two have = measures, the s are If two lines form adjacent s the lines are

11 Proof of Theorem Theorem If two angles are complementary to the same angle ( or to congruent angles), then these angles are congruent. Given: 1 and 3are complementary 2 and 4 are complementary 1 = 2 Prove: 3 4 Plan: 1 and 3=90 2 and 4= = = 4 Drawing Proof Statements Reasons 1. 1 and 3 are complementary, 2 and 4 are complem. 1. Given 2. m 1 + 3=90 and 2 + 4= Defn of complementary angles 3. m 1 + 3= Substitution ( 2) 4. 1 = 2 4. Given 5. m 1 + 3= Substitution, (3), (4) 6. 3 = 4 6. Subtraction Property of = (5) Defn

12 Theorem If two angles are supplementary to the same angle ( or to congruent angles), then these angles are congruent. Theorem ny two right angles are congruent. Theorem If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. Picture Proof of Theorem Given: Prove: 1 and 2 are complementary Proof: With, we see that 1 and 2 are parts of a right angle. Then m 1 + m 2 = 90, so 1 and 2 are complementary. Drawing 1 2 D

13 Proof of Theorem Theorem If the exterior sides of two adjacent acute angles form a straight line, then these angles are supplementary. Given: 3 and 4 and Prove: 3 and 4 are supplementary Plan: Drawing H E 3 4 F G Proof Statements Reasons 1. 3 and 4 and 1. Given 2. m = 2. ngle - ddition Postulate 3. is a straight angle 3. Defn of a straight angle 4. = The measure of a straight angle 5. m 3 + 4= Substitution, (2), (4) 6. 3 and 4 are supplementary 6. Defn of supplementary angles Theorem If two line segments are congruent, then their midpoint separate these segments into four congruent segments. Theorem If two angles are congruent, then their bisector separate these angles into four congruent angles.

14 2.1: The Parallel Postulate 2.2: Indirect Proof 2.3: Proving Lines Parallel 2.4: The ngles of a Triangle hapter 2 2.1: The Parallel Postulate 1. Perpendicular Lines 2. Parallel Lines 3. Euclidian Geometry 4. Parallel Lines and ongruent ngles 5. Parallel Lines and ongruent ngles 1. Perpendicular Lines Recall:. Definition: Two lines are perpendicular if they meet to form congruent adjacent angles. Theorem 1.6.1: Perpendicular lines meet to form right angles. onstruction 5: onstruct a perpendicular line to a given line at a given point D. onstruct a bisector to a given line onstruction 6: onstruct a perpendicular line to a given line from a point not on the given line Theorem From a point not on a given line, there is exactly one line perpendicular to the given line. 2. Parallel Lines Parallel Lines and planes Definition: Parallel lines are lines in the same plane that do not intersect 3. Euclidian Geometry the plane is flat, two dimensional surface in which the line segment joining any two points of the plane lies entirely within the plane Postulate 10 Through a point not on a line, exactly one line is parallel to the given line.

15 Special ngles Transversal: a line that intersect two (or more) other lines at distinct points Interior angles: ngles formed between line m and n: 3, 4, 5, 6 Exterior angles: ngles formed outside line m and n: 1, 2, 7, l m orresponding angles: ngles that lie in the same relative position: left, right, above, below 1 and 5 above, left 3 and 7 below, left 2 and 6 above, right 4 and 8 below, right n lternate Interior ngles: interior angles that have different vertices and lie on opposite sides of the transversal: lternate Exterior ngles: exterior angles that have different vertices and lie on opposite sides of the transversal: 3 and 6 3 and 7 1 and 8 2 and 7

16 Postulate 11 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. orresponding angles: l 1 and 5, 3 and 7, 2 and 6, 4 and m Ex: If 1 = 110 and, find 2, 4, 5, and n Theorem If two parallel lines are cut by a transversal line, then the pairs of alternate interior angles are congruent. Given:, transversal l Prove: l m n Proof Statements Reasons 1., transversal l 1. Given Postulate Vertical ngles, Theorem Substitution

17 Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. Given:, transversal T W Prove: 1 and 3 are supplementary R U X S V Y Proof Statements Reasons 1., transversal 1. Given lternate interior angles 3. m 1 = 2 3. Defn of congruent angles 4. m = Straight angle 5. m = 5. ngle ddition Postulate 6. m = Substitution, (4) and (5) 7. m = Substitution, (3) and (6) 8. 1 and 3 are supplementary 8. Definition Suppl angles

18 Theorem If two parallel line are cut by a transversal, then the pairs of exterior angles on the same side of the transversal are supplementary.

19 2.2 Indirect Proof Let represents the conditional statement If P then Q and ~ =not P. The following statements are related to the given statement onditional If P then Q onverse If Q then P Inverse ~ ~ If not P then not Q ontrapositive ~ ~ If not Q then not P Example: If Juan lives in L, then he lives in onditional: : If Juan lives in L, then he lives in onverse: If Juan lives in, then he lives in L Inverse : If Juan doesn t live in L, then he doesn t lives in ontrapositive : If Juan doesn t lives in, then he doesn t live in L Given : P Prove : Q Indirect Proof : 1. Suppose that ~Q is true. 2. Reason from the supposition until you reach a contradiction 3. Note that the supposition claiming that ~ is true must be false and that Q must therefore be true

20 Ex1: Given is not perpendicular to prove that 1 and 2 are not complementary angles Paragraph Proof: Writing like an essay; still each statement has to be justified. Suppose that 1 and 2 are complementary angles. Then m = 90 because the sum of the measures of two compl. angles is 90. We also know that m = by the ngle ddition Postulate. In turn, =90 by substitution. Then is a right angle. Thus,. ut this contradicts the given hypothesis; therefore, the supposition must be false, nd it follows that 1 and 2 are not complementary angles. 1 2 D Statements Reasons 1. Suppose that 1 and 2 are complementary angles 1. ontradict the hypothesis 2. m = The sum of the measures of two compl. angles is m = 3. ngle ddition Postulate 4. =90 4. substitution 5. is a right angle 5. Definition of right angle Theorem of perpendicular lines 7. ontradiction 7. Given is not perpendicular to 8. 1 and 2 are not complementary angles 8. ecause of contradiction

21 Ex2: omplete a formal proof of the following theorem: If two lines are cut by a transversal so that the corresponding angles are not congruent, then the two lines are not parallel lines Given: m and l are cut by the transversal t and 1 5 Prove: m ssume that. When these lines are cut by transversal t, any two corresponding angles (including 1 and 5 are congruent. ut 1 5 by hypothesis. Thus, the assumed statement, which claims that must be false. It follows that m t m l Ex3: omplete a formal proof of the following theorem: The bisector of an angle is unique Given: bisect Prove: is the only bisector for D ssume that : is also a bisector of and that m =. Given that bisect, it follows that m =. y the ngle ddition Postulate, m = + m. y substitution, = + ; but then = 0, by subtraction. n angle with measure 0 contradicts the Protractor Postulate, therefore, the assumed statement must be false, and it follows that the bisector of an angle is unique. E D

22 Recall from 2.1 the relevant Postulate and Theorems: 2.3 Proving line parallel Postulate 11 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Theorem If two parallel lines are cut by a transversal line, then the pairs of alternate interior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Theorem If two parallel lines are cut by a transversal, then the pairs of interior angles on the same side of the transversal are supplementary. Theorem If two parallel line are cut by a transversal, then the pairs of exterior angles on the same side of the transversal are supplementary. Note: The Postulate and each Theorem has 1. The same hypothesis: If two parallel line are cut by a transversal, and 2. conclusion involving angle relationship. If we wish to prove that two lines are parallel instead of a relation between angles we can use onverse. 1. hypothesis: an angle relationship, and 2. conclusion:, then the two lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of corresponding angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of the alternate interior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the pairs of the alternate exterior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the alternate interior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the alternate exterior angles on the same side of the transversal are supplementary, then these lines are parallel lines.

23 Theorem If two lines are cut by a transversal such that the pairs of corresponding angles are congruent, then these lines are parallel lines. Given: and Prove: cut by transversal ; 1 2 r t l Proof: 2 m Suppose that m. Then a line r can be drown through point P that is parallel to m; this follows from the Parallel Postulate. If, then 3 2 because these angles correspond. ut 1 2 by hypothesis. Now 3 1 by the Transitive Property of ongruence; therefore, m 3 = 1. ut m = 1 (see fig.). Substituting 1 for 3 leads to m = 1, and by subtraction, 4 = 0. This contradicts the Protractor Postulate, therefore r and l must coincide, and it follows that.

24 Theorem If two lines are cut by a transversal so that the pairs of the alternate interior angles are congruent, then these lines are parallel. Given: and cut by transversal ; 2 3 Prove: Paragraph Proof: t 1 3 l 2 m Proof Statements Reasons 1. and and transversal Given If two lines intersect, vertical angles are congruent Transitive property Theorem 2.3.1

25 Theorem If two lines are cut by a transversal such that the pairs of the alternate exterior angles are congruent, then these lines are parallel lines. Theorem If two lines are cut by a transversal such that the interior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Given: and Prove: cut by transversal ; 1 is supplementary to 2 t 3 l 1 2 m Proof Statements Reasons 1. and and transversal ; 1 is supplementary to 2 1. Given 2. 1 is supplementary to 3 2. Straight line If two angles are supplementary to the same angle they are Theorem 2.3.1

26 Theorem If two lines are cut by a transversal such that the exterior angles on the same side of the transversal are supplementary, then these lines are parallel lines. Theorem If two lines are each parallel to a third line, then these lines are parallel to each other. Theorem If two coplanar lines are each perpendicular to a third line, then these lines are parallel to each other. onstruction 7 onstruct the line parallel to a given line from a point not on that line.

27 2.4 The ngles of a Triangle Definition: triangle (symbol ) is the union of three line segments that are determined by three noncollinear points. or or or or,, are vertices are sides of the triangle Points: D a point inside the triangle E point on the triangle F a point outside the triangle D E... F Triangles classified by ongruent sides Scalene Isosceles Equilateral Triangles classified by ongruent angles cute Obtuse Equilateral Right

28 Theorem In a triangle the sum of the measures of the interior angles is 180 Given: fig (a) Prove: + + = 180 (a) Picture Proof Through in fig (a), draw. We see that = 180. ut 1 = and 3 = (alternate interior angles) (b) D E Then + + = 180 in fig (a) orollary Each angle of an equiangular triangle measures 60 Proof: y Theorem m = 180 ; Since = =m by hypothesis, it follows that 3m = 180. Therefore m = 60, and = =m = 60 ; orollary The acute angles of a right triangle are complementary. orollary If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent orollary The measure of an exterior angle of a triangle equals the sum of the measures of the two nonadjacent interior angles

1. 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. 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 information

A summary of definitions, postulates, algebra rules, and theorems that are often used in geometry proofs:

A 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 information

Final Review Geometry A Fall Semester

Final 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 information

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1

Student Name: Teacher: Date: District: Miami-Dade County Public Schools. Assessment: 9_12 Mathematics Geometry Exam 1 Student Name: Teacher: Date: District: Miami-Dade 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 information

Euclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:

Euclidean 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 information

Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 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 information

Geometry Review Flash Cards

Geometry Review Flash Cards point is like a star in the night sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space. point may be represented by a dot on

More information

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

More information

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: 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 information

55 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 220 points.

55 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 information

Chapter 1: Essentials of Geometry

Chapter 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

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

Geometry Course Summary Department: Math. Semester 1

Geometry 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 information

Neutral Geometry. April 18, 2013

Neutral 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 information

Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: 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 information

Grade 4 - Module 4: Angle Measure and Plane Figures

Grade 4 - Module 4: Angle Measure and Plane Figures Grade 4 - Module 4: Angle Measure and Plane Figures Acute angle (angle with a measure of less than 90 degrees) Angle (union of two different rays sharing a common vertex) Complementary angles (two angles

More information

Definitions, Postulates and Theorems

Definitions, 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 information

POTENTIAL REASONS: Definition of Congruence:

POTENTIAL 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 information

Duplicating Segments and Angles

Duplicating 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 information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY. 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 information

NAME DATE PERIOD. Study Guide and Intervention

NAME 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 information

Incenter Circumcenter

Incenter 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 information

Foundations of Geometry 1: Points, Lines, Segments, Angles

Foundations 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 information

circle the set of all points that are given distance from a given point in a given plane

circle the set of all points that are given distance from a given point in a given plane Geometry Week 19 Sec 9.1 to 9.3 Definitions: section 9.1 circle the set of all points that are given distance from a given point in a given plane E D Notation: F center the given point in the plane radius

More information

Intermediate Math Circles October 10, 2012 Geometry I: Angles

Intermediate 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 information

GEOMETRY FINAL EXAM REVIEW

GEOMETRY 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 information

Algebra Geometry Glossary. 90 angle

Algebra 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 information

Seattle 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 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 information

GEOMETRY CONCEPT MAP. Suggested Sequence:

GEOMETRY 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 information

This is a tentative schedule, date may change. Please be sure to write down homework assignments daily.

This 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 information

(n = # of sides) One interior angle:

(n = # of sides) One interior angle: 6.1 What is a Polygon? Regular Polygon- Polygon Formulas: (n = # of sides) One interior angle: 180(n 2) n Sum of the interior angles of a polygon = 180 (n - 2) Sum of the exterior angles of a polygon =

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles 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 information

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected 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 information

37 Basic Geometric Shapes and Figures

37 Basic Geometric Shapes and Figures 37 Basic Geometric Shapes and Figures In this section we discuss basic geometric shapes and figures such as points, lines, line segments, planes, angles, triangles, and quadrilaterals. The three pillars

More information

ABC is the triangle with vertices at points A, B and C

ABC is the triangle with vertices at points A, B and C Euclidean Geometry Review This is a brief review of Plane Euclidean Geometry - symbols, definitions, and theorems. Part I: The following are symbols commonly used in geometry: AB is the segment from the

More information

Notes on Congruence 1

Notes on Congruence 1 ongruence-1 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

More information

Unit 3: Triangle Bisectors and Quadrilaterals

Unit 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 information

Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 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 information

Geometry Chapter 5 Relationships Within Triangles

Geometry Chapter 5 Relationships Within Triangles 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

More information

GEOMETRIC FIGURES, AREAS, AND VOLUMES

GEOMETRIC FIGURES, AREAS, AND VOLUMES HPTER GEOMETRI FIGURES, RES, N VOLUMES carpenter is building a deck on the back of a house. s he works, he follows a plan that he made in the form of a drawing or blueprint. His blueprint is a model of

More information

4. Prove the above theorem. 5. Prove the above theorem. 9. Prove the above corollary. 10. Prove the above theorem.

4. 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 information

CONGRUENCE BASED ON TRIANGLES

CONGRUENCE BASED ON TRIANGLES HTR 174 5 HTR TL O ONTNTS 5-1 Line Segments ssociated with Triangles 5-2 Using ongruent Triangles to rove Line Segments ongruent and ngles ongruent 5-3 Isosceles and quilateral Triangles 5-4 Using Two

More information

1.7 Find Perimeter, Circumference,

1.7 Find Perimeter, Circumference, .7 Find Perimeter, Circumference, and rea Goal p Find dimensions of polygons. Your Notes FORMULS FOR PERIMETER P, RE, ND CIRCUMFERENCE C Square Rectangle side length s length l and width w P 5 P 5 s 5

More information

Name: 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 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 information

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum 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 information

Informal Geometry and Measurement

Informal Geometry and Measurement HP LIN N NGL LIONHIP In xercises 56 and 57, P is a true statement, while Q and are false statements. lassify each of the following statements as true or false. 56. a) (P and Q) or b) (P or Q) and 57. a)

More information

A geometric construction is a drawing of geometric shapes using a compass and a straightedge.

A geometric construction is a drawing of geometric shapes using a compass and a straightedge. Geometric Construction Notes A geometric construction is a drawing of geometric shapes using a compass and a straightedge. When performing a geometric construction, only a compass (with a pencil) and a

More information

Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Conjunction 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 information

Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3

Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Geometry CP Lesson 5-1: Bisectors, Medians and Altitudes Page 1 of 3 Main ideas: Identify and use perpendicular bisectors and angle bisectors in triangles. Standard: 12.0 A perpendicular bisector of a

More information

Mathematics Geometry Unit 1 (SAMPLE)

Mathematics 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 information

1.1 Identify Points, Lines, and Planes

1.1 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 information

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1.

Chapter 1 Exam. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Name: lass: ate: I: hapter 1 Exam Multiple hoice Identify the choice that best completes the statement or answers the question. 1. bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = c. m = 40 b. m =

More information

half-line the set of all points on a line on a given side of a given point of the line

half-line the set of all points on a line on a given side of a given point of the line Geometry Week 3 Sec 2.1 to 2.4 Definition: section 2.1 half-line the set of all points on a line on a given side of a given point of the line notation: is the half-line that contains all points on the

More information

Find the measure of each numbered angle, and name the theorems that justify your work.

Find the measure of each numbered angle, and name the theorems that justify your work. Find the measure of each numbered angle, and name the theorems that justify your work. 1. The angles 2 and 3 are complementary, or adjacent angles that form a right angle. So, m 2 + m 3 = 90. Substitute.

More information

Semester 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. 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 information

Lesson 18: Looking More Carefully at Parallel Lines

Lesson 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 information

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem

Chapter 4 Circles, Tangent-Chord Theorem, Intersecting Chord Theorem and Tangent-secant Theorem Tampines Junior ollege H3 Mathematics (9810) Plane Geometry hapter 4 ircles, Tangent-hord Theorem, Intersecting hord Theorem and Tangent-secant Theorem utline asic definitions and facts on circles The

More information

Chapter 5: Relationships within Triangles

Chapter 5: Relationships within Triangles Name: Chapter 5: Relationships within Triangles Guided Notes Geometry Fall Semester CH. 5 Guided Notes, page 2 5.1 Midsegment Theorem and Coordinate Proof Term Definition Example midsegment of a triangle

More information

How Do You Measure a Triangle? Examples

How Do You Measure a Triangle? Examples How Do You Measure a Triangle? Examples 1. A triangle is a three-sided 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 information

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon.

A convex polygon is a polygon such that no line containing a side of the polygon will contain a point in the interior of the polygon. hapter 7 Polygons A polygon can be described by two conditions: 1. No two segments with a common endpoint are collinear. 2. Each segment intersects exactly two other segments, but only on the endpoints.

More information

Goal Find angle measures in triangles. Key Words corollary. Student Help. Triangle Sum Theorem THEOREM 4.1. Words The sum of the measures of EXAMPLE

Goal Find angle measures in triangles. Key Words corollary. Student Help. Triangle Sum Theorem THEOREM 4.1. Words The sum of the measures of EXAMPLE Page of 6 4. ngle Measures of Triangles Goal Find angle measures in triangles. The diagram below shows that when you tear off the corners of any triangle, you can place the angles together to form a straight

More information

Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

More information

The Triangle and its Properties

The 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

Unit 6 Grade 7 Geometry

Unit 6 Grade 7 Geometry Unit 6 Grade 7 Geometry Lesson Outline BIG PICTURE Students will: investigate geometric properties of triangles, quadrilaterals, and prisms; develop an understanding of similarity and congruence. Day Lesson

More information

Terminology: When one line intersects each of two given lines, we call that line a transversal.

Terminology: 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 information

Chapter 6 Notes: Circles

Chapter 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 information

/27 Intro to Geometry Review

/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 information

Bisections and Reflections: A Geometric Investigation

Bisections and Reflections: A Geometric Investigation Bisections and Reflections: A Geometric Investigation Carrie Carden & Jessie Penley Berry College Mount Berry, GA 30149 Email: ccarden@berry.edu, jpenley@berry.edu Abstract In this paper we explore a geometric

More information

22.1 Interior and Exterior Angles

22.1 Interior and Exterior Angles Name Class Date 22.1 Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other polygons? Resource Locker Explore 1 Exploring Interior

More information

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters,

More information

Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 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 information

Geometry Chapter 1 Review

Geometry Chapter 1 Review Name: lass: ate: I: Geometry hapter 1 Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name two lines in the figure. a. and T c. W and R b. WR and

More information

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.

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. 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 information

Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 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 information

Theorem Prove Given. Dates, assignments, and quizzes subject to change without advance notice.

Theorem 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 information

Given: ABCD is a rhombus. Prove: ABCD is a parallelogram.

Given: 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 information

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture. CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion

More information

2.1. Inductive Reasoning EXAMPLE A

2.1. Inductive Reasoning EXAMPLE A CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

More information

Lesson 2: Circles, Chords, Diameters, and Their Relationships

Lesson 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 information

5.1 Midsegment Theorem and Coordinate Proof

5.1 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 information

Chapter 4.1 Parallel Lines and Planes

Chapter 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 information

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS. 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 information

Geometry Chapter 5 - Properties and Attributes of Triangles Segments in Triangles

Geometry Chapter 5 - Properties and Attributes of Triangles Segments in Triangles Geometry hapter 5 - roperties and ttributes of Triangles Segments in Triangles Lesson 1: erpendicular and ngle isectors equidistant Triangle congruence theorems can be used to prove theorems about equidistant

More information

PROVING STATEMENTS IN GEOMETRY

PROVING STATEMENTS IN GEOMETRY CHAPTER PROVING STATEMENTS IN GEOMETRY After proposing 23 definitions, Euclid listed five postulates and five common notions. These definitions, postulates, and common notions provided the foundation for

More information

Geometry, Final Review Packet

Geometry, Final Review Packet Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a

More information

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd: GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

More information

alternate interior angles

alternate interior angles alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

More information

For the circle above, EOB is a central angle. So is DOE. arc. The (degree) measure of ù DE is the measure of DOE.

For the circle above, EOB is a central angle. So is DOE. arc. The (degree) measure of ù DE is the measure of DOE. efinition: circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. We use the symbol to represent a circle. The a line segment from the center

More information

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT!

GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! GEOMETRY 101* EVERYTHING YOU NEED TO KNOW ABOUT GEOMETRY TO PASS THE GHSGT! FINDING THE DISTANCE BETWEEN TWO POINTS DISTANCE FORMULA- (x₂-x₁)²+(y₂-y₁)² Find the distance between the points ( -3,2) and

More information

SOLVED PROBLEMS REVIEW COORDINATE GEOMETRY. 2.1 Use the slopes, distances, line equations to verify your guesses

SOLVED 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 information

Warm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example:

Warm Up #23: Review of Circles 1.) A central angle of a circle is an angle with its vertex at the of the circle. Example: Geometr hapter 12 Notes - 1 - Warm Up #23: Review of ircles 1.) central angle of a circle is an angle with its verte at the of the circle. Eample: X 80 2.) n arc is a section of a circle. Eamples:, 3.)

More information

Lines, Segments, Rays, and Angles

Lines, Segments, Rays, and Angles Line and Angle Review Thursday, July 11, 2013 10:22 PM Lines, Segments, Rays, and Angles Slide Notes Title Lines, Segment, Ray A line goes on forever, so we use an arrow on each side to indicate that.

More information

Chapter Two. Deductive Reasoning

Chapter Two. Deductive Reasoning Chapter Two Deductive Reasoning 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

More information

Elementary triangle geometry

Elementary 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 information

Geo - CH6 Practice Test

Geo - CH6 Practice Test Geo - H6 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the measure of each exterior angle of a regular decagon. a. 45 c. 18 b. 22.5

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. 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 information

100 Math Facts 6 th Grade

100 Math Facts 6 th Grade 100 Math Facts 6 th Grade Name 1. SUM: What is the answer to an addition problem called? (N. 2.1) 2. DIFFERENCE: What is the answer to a subtraction problem called? (N. 2.1) 3. PRODUCT: What is the answer

More information

Sum of the interior angles of a n-sided Polygon = (n-2) 180

Sum of the interior angles of a n-sided Polygon = (n-2) 180 5.1 Interior angles of a polygon Sides 3 4 5 6 n Number of Triangles 1 Sum of interiorangles 180 Sum of the interior angles of a n-sided Polygon = (n-2) 180 What you need to know: How to use the formula

More information

3.1. Angle Pairs. What s Your Angle? Angle Pairs. ACTIVITY 3.1 Investigative. Activity Focus Measuring angles Angle pairs

3.1. Angle Pairs. What s Your Angle? Angle Pairs. ACTIVITY 3.1 Investigative. Activity Focus Measuring angles Angle pairs SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Use Manipulatives Two rays with a common endpoint form an angle. The common endpoint is called the vertex. You can use a protractor to draw and measure

More information