# NAME DATE PERIOD. Study Guide and Intervention

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 of a side of a triangle is a line, segment, or ray in the same plane as the triangle that is perpendicular to the side and passes through its midpoint. nother special segment, ray, or line is an angle bisector, which divides an angle into two congruent angles. wo properties of perpendicular bisectors are: (1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment, and (2) the three perpendicular bisectors of the sides of a triangle meet at a point, called the circumcenter of the triangle, that is equidistant from the three vertices of the triangle. wo properties of angle bisectors are: (1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides of the angle, and (2) the three angle bisectors of a triangle meet at a point, called the incenter of the triangle, that is equidistant from the three sides of the triangle. xample 1 is the perpendicular xample 2 M is the angle bisector bisector of. ind x. of M. ind x if m 1 5x 8 and m 2 8x 16. 5x 6 3x 8 is the perpendicular bisector of, so. 3x 8 5x x 7 x 1 2 M M is the angle bisector of M, so m 1 m 2. 5x 8 8x x 8 x ind the value of each variable y 3x 8x 6x 2 7x 9 6x 10y 4 (4x 30) is the perpendicular is equilateral. bisects. bisector of. 4. or what kinds of triangle(s) can the perpendicular bisector of a side also be an angle bisector of the angle opposite the side? 5. or what kind of triangle do the perpendicular bisectors intersect in a point outside the triangle? hapter 5 6 Glencoe Geometry

2 5-1 M IO tudy Guide and Intervention (continued) isectors, Medians, and ltitudes Medians and ltitudes median is a line segment that connects the vertex of a triangle to the midpoint of the opposite side. he three medians of a triangle intersect at the centroid of the triangle. entroid heorem he centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. centroid L xample oints,, and are the midpoints of, and, respectively. ind x, y, and z. U 2 3 U 2 3 U 2 3 6x 2 3 (6x 15) (24 3y 3) 6z 4 2 (6z 4 11) 3 9x 6x y 3 3 (6z 4) 6z x y 9z 6 6z 15 x y 3z 9 5 y z 3 L 2 3, L 2 3, L U 11 6z 4 3y 3 6x Lesson 5-1 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. ind the value of each variable x 7x x 3 9x 6 3y is a median. ;,, and are midpoints G 7x 4 H 5y H H HG M y z 8y 24 6z x V 6x 32 G V is the centroid of ; is the centroid of. 18; M 15; 24 9z 6 9x 2 J 3y 5 O z 2x K M L 7. or what kind of triangle are the medians and angle bisectors the same segments? 8. or what kind of triangle is the centroid outside the triangle? hapter 5 7 Glencoe Geometry

3 5-2 M IO tudy Guide and Intervention Inequalities and riangles ngle Inequalities roperties of inequalities, including the ransitive, ddition, ubtraction, Multiplication, and ivision roperties of Inequality, can be used with measures of angles and segments. here is also a omparison roperty of Inequality. or any real numbers a and b, either a b, a b, or a b. he xterior ngle heorem can be used to prove this inequality involving an exterior angle. xterior ngle Inequality heorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. 1 m 1 m, m 1 m opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. xample List all angles of G whose measures are less than m 1. he measure of an exterior angle is greater than the measure of either remote interior angle. o m 3 m 1 and m 4 m 1. List all angles that satisfy the stated condition. 1. all angles whose measures are less than m 1 2. all angles whose measures are greater than m 3 3. all angles whose measures are less than m 1 4. all angles whose measures are greater than m 1 5. all angles whose measures are less than m 7 6. all angles whose measures are greater than m 2 7. all angles whose measures are greater than m 5 8. all angles whose measures are less than m 4 9. all angles whose measures are less than m all angles whose measures are greater than m 4 G H L M J K 1 2 U X W V Q O 9 10 Lesson 5-2 hapter 5 13 Glencoe Geometry

4 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-2 M IO tudy Guide and Intervention (continued) Inequalities and riangles ngle-ide elationships When the sides of triangles are not congruent, there is a relationship between the sides and angles of the triangles. If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. If, then m m. If m m, then. xample 1 List the angles in order xample 2 List the sides in order from least to greatest measure. from shortest to longest. 6 cm 7 cm 35 9 cm,, 20,, 125 List the angles or sides in order from least to greatest measure cm cm cm etermine the relationship between the measures of the given angles. 4., U 5., U U V UV, etermine the relationship between the lengths of the given sides. 7., , , hapter 5 14 Glencoe Geometry

5 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-3 M IO tudy Guide and Intervention Indirect roof Indirect roof with lgebra One way to prove that a statement is true is to assume that its conclusion is false and then show that this assumption leads to a contradiction of the hypothesis, a definition, postulate, theorem, or other statement that is accepted as true. hat contradiction means that the conclusion cannot be false, so the conclusion must be true. his is known as indirect proof. teps for Writing an Indirect roof 1. ssume that the conclusion is false. 2. how that this assumption leads to a contradiction. 3. oint out that the assumption must be false, and therefore, the conclusion must be true. xample Given: 3x 5 8 rove: x 1 tep 1 ssume that x is not greater than 1. hat is, x 1 or x 1. tep 2 Make a table for several possibilities for x 1 or x 1. he contradiction is that when x 1 or x 1, then 3x 5 is not greater than 8. tep 3 his contradicts the given information that 3x 5 8. he assumption that x is not greater than 1 must be false, which means that the statement x 1 must be true. x 3x Write the assumption you would make to start an indirect proof of each statement. 1. If 2x 14, then x or all real numbers, if a b c, then a c b. omplete the proof. Given: n is an integer and n 2 is even. rove: n is even. 3. ssume that 4. hen n can be expressed as 2a 1 by 5. n 2 ubstitution 6. Multiply. 7. implify. 8. 2(2a 2 2a) (2a 2 2a) 1 is an odd number. his contradicts the given that n 2 is even, so the assumption must be 10. herefore, hapter 5 22 Glencoe Geometry

6 5-3 M IO tudy Guide and Intervention (continued) Indirect roof Indirect roof with Geometry o write an indirect proof in geometry, you assume that the conclusion is false. hen you show that the assumption leads to a contradiction. he contradiction shows that the conclusion cannot be false, so it must be true. xample Given: m 100 rove: is not a right angle. tep 1 ssume that is a right angle. tep 2 how that this leads to a contradiction. If is a right angle, then m 90 and m m hus the sum of the measures of the angles of is greater than 180. tep 3 he conclusion that the sum of the measures of the angles of is greater than 180 is a contradiction of a known property. he assumption that is a right angle must be false, which means that the statement is not a right angle must be true. Write the assumption you would make to start an indirect proof of each statement. 1. If m 90, then m 45. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 2. If V is not congruent to V, then V is not isosceles. omplete the proof. Given: 1 2 and G is not congruent to G. rove: is not congruent to. 3. ssume that ssume the conclusion is false. 4. G G 5. G G his contradicts the given information, so the assumption must 1 2 G Lesson 5-3 be 8. herefore, hapter 5 23 Glencoe Geometry

7 5-4 M IO tudy Guide and Intervention he riangle Inequality he riangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle with them, you will find that it is not possible. his illustrates the riangle Inequality heorem. riangle Inequality heorem he sum of the lengths of any two sides of a triangle is greater than the length of the third side. b c a xample he measures of two sides of a triangle are 5 and 8. ind a range for the length of the third side. y the riangle Inequality, all three of the following inequalities must be true. 5 x 8 8 x x x 3 x 3 13 x herefore x must be between 3 and 13. etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. 1. 3, 4, , 9, 15 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 3. 8, 8, , 4, , 8, , 2.5, 3 ind the range for the measure of the third side given the measures of two sides and and and and uppose you have three different positive numbers arranged in order from least to greatest. What single comparison will let you see if the numbers can be the lengths of the sides of a triangle? Lesson 5-4 hapter 5 29 Glencoe Geometry

8 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-4 M IO tudy Guide and Intervention (continued) he riangle Inequality istance etween a oint and a Line he perpendicular segment from a point to a line is the shortest segment from the point to the line. is the shortest segment from to. he perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Q Q is the shortest segment from Q to plane. xample Given: oint is equidistant from the sides of an angle. rove: roof: 1. raw and to 1. ist. is measured the sides of. along a. 2. and are right angles. 2. ef. of lines 3. and are right triangles. 3. ef. of rt t. angles are. 5. is equidistant from the sides of. 5. Given ef. of equidistant eflexive roperty HL omplete the proof. Given: ; U rove: U roof: 1. ; U and are a linear pair; 4. ef. of and U are a linear pair. 5. and are supplementary; 5. and U are supplementary ngles suppl. to angles are. 7. U U hapter 5 30 Glencoe Geometry

9 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5-5 M IO tudy Guide and Intervention Inequalities Involving wo riangles Inequality he following theorem involves the relationship between the sides of two triangles and an angle in each triangle. Inequality/Hinge heorem If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle If,, and m m, then. xample Write an inequality relating the lengths of and. wo sides of are congruent to two sides of and m m. y the Inequality/Hinge heorem, Write an inequality relating the given pair of segment measures. 1. M M,, J G H K M G, HK M, Write an inequality to describe the possible values of x (4x 10) cm cm cm 24 cm 1.8 cm cm 1.8 cm (3x 2.1) cm hapter 5 36 Glencoe Geometry

10 5-5 M IO tudy Guide and Intervention (continued) Inequalities Involving wo riangles Inequality he converse of the Hinge heorem is also useful when two triangles have two pairs of congruent sides. Inequality If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. M If M, M, and, then m M m. 38 xample Write an inequality relating the measures of and. wo sides of are congruent to two sides of, and. y the Inequality, m m Write an inequality relating the given pair of angle measures. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 1. M m M, m m, m 3. X 4. X 50 Y Z 30 Y W Z m, m Z m XYW, m WYZ Write an inequality to describe the possible values of x ( 1 2 x 6) cm 60 cm 33 (3x 3) 30 cm 60 cm Lesson 5-5 hapter 5 37 Glencoe Geometry

### Picture. Right Triangle. Acute Triangle. Obtuse Triangle

Name Perpendicular Bisector of each side of a triangle. Construct the perpendicular bisector of each side of each triangle. Point of Concurrency Circumcenter Picture The circumcenter is equidistant from

### Picture. Right Triangle. Acute Triangle. Obtuse Triangle

Name Perpendicular Bisector of each side of a triangle. Construct the perpendicular bisector of each side of each triangle. Point of Concurrency Circumcenter Picture The circumcenter is equidistant from

### 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

### A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Perpendicular Bisector Theorem

Perpendicular Bisector Theorem A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. Converse of the Perpendicular Bisector Theorem If a

### #2. Isosceles Triangle Theorem says that If a triangle is isosceles, then its BASE ANGLES are congruent.

1 Geometry Proofs Reference Sheet Here are some of the properties that we might use in our proofs today: #1. Definition of Isosceles Triangle says that If a triangle is isosceles then TWO or more sides

### To use properties of perpendicular bisectors and angle bisectors

5-2 erpendicular and ngle isectors ommon ore tate tandards G-O..9 rove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment

### 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

### 1.2 Informal Geometry

1.2 Informal Geometry Mathematical System: (xiomatic System) Undefined terms, concepts: Point, line, plane, space Straightness of a line, flatness of a plane point lies in the interior or the exterior

### 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

### Geometry. 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.

### 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

### Duplicating Segments and Angles

ONDENSED LESSON 3.1 Duplicating Segments and ngles In this lesson you will Learn what it means to create a geometric construction Duplicate a segment by using a straightedge and a compass and by using

### acute angle adjacent angles angle bisector between axiom Vocabulary Flash Cards Chapter 1 (p. 39) Chapter 1 (p. 48) Chapter 1 (p.38) Chapter 1 (p.

Vocabulary Flash ards acute angle adjacent angles hapter 1 (p. 39) hapter 1 (p. 48) angle angle bisector hapter 1 (p.38) hapter 1 (p. 42) axiom between hapter 1 (p. 12) hapter 1 (p. 14) collinear points

### 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

### Geometry Chapter 5 Review Relationships Within Triangles. 1. A midsegment of a triangle is a segment that connects the of two sides.

Geometry Chapter 5 Review Relationships Within Triangles Name: SECTION 5.1: Midsegments of Triangles 1. A midsegment of a triangle is a segment that connects the of two sides. A midsegment is to the third

### 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

### 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

### 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

### 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

### 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

### 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

### 5.6 Angle Bisectors and

age 1 of 8 5.6 ngle isectors and erpendicular isectors oal Use angle bisectors and perpendicular bisectors. ey Words distance from a point to a line equidistant angle bisector p. 61 perpendicular bisector

### 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

### PARALLEL LINES CHAPTER

HPTR 9 HPTR TL OF ONTNTS 9-1 Proving Lines Parallel 9-2 Properties of Parallel Lines 9-3 Parallel Lines in the oordinate Plane 9-4 The Sum of the Measures of the ngles of a Triangle 9-5 Proving Triangles

### EXPECTED BACKGROUND KNOWLEDGE

MOUL - 3 oncurrent Lines 12 ONURRNT LINS You have already learnt about concurrent lines, in the lesson on lines and angles. You have also studied about triangles and some special lines, i.e., medians,

### 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.

### Lesson 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

### 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

### Int. Geometry Unit 2 Quiz Review (Lessons 1-4) 1

Int. Geometry Unit Quiz Review (Lessons -4) Match the examples on the left with each property, definition, postulate, and theorem on the left PROPRTIS:. ddition Property of = a. GH = GH. Subtraction Property

### 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,

### 6.1. Perpendicular and Angle Bisectors

6.1 T TI KOW KI.2..5..6. TI TOO To be proficient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data. erpendicular and ngle isectors

### 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

### 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

### A polygon with five sides is a pentagon. A polygon with six sides is a hexagon.

Triangles: polygon is a closed figure on a plane bounded by (straight) line segments as its sides. Where the two sides of a polygon intersect is called a vertex of the polygon. polygon with three sides

### 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

### 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

### Lesson 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

### 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

### 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

### 8.2 Angle Bisectors of Triangles

Name lass Date 8.2 ngle isectors of Triangles Essential uestion: How can you use angle bisectors to find the point that is equidistant from all the sides of a triangle? Explore Investigating Distance from

### 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

### 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.

### The Four Centers of a Triangle. Points of Concurrency. Concurrency of the Medians. Let's Take a Look at the Diagram... October 25, 2010.

Points of Concurrency Concurrent lines are three or more lines that intersect at the same point. The mutual point of intersection is called the point of concurrency. Example: x M w y M is the point of

### Isosceles triangles. Key Words: Isosceles triangle, midpoint, median, angle bisectors, perpendicular bisectors

Isosceles triangles Lesson Summary: Students will investigate the properties of isosceles triangles. Angle bisectors, perpendicular bisectors, midpoints, and medians are also examined in this lesson. A

### 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

### State the assumption you would make to start an indirect proof of each statement.

1. State the assumption you would make to start an indirect proof of each statement. Identify the conclusion you wish to prove. The assumption is that this conclusion is false. 2. is a scalene triangle.

### 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

### 5.2 Use Perpendicular Bisectors

5.2 Use Perpendicular isectors Goal p Use perpendicular bisectors to solve problems. Your Notes VOULRY Perpendicular bisector Equidistant oncurrent Point of concurrency ircumcenter THEOREM 5.2: PERPENIULR

### Geometry 8-1 Angles of Polygons

. Sum of Measures of Interior ngles Geometry 8-1 ngles of Polygons 1. Interior angles - The sum of the measures of the angles of each polygon can be found by adding the measures of the angles of a triangle.

### NCERT. In examples 1 and 2, write the correct answer from the given four options.

MTHEMTIS UNIT 2 GEOMETRY () Main oncepts and Results line segment corresponds to the shortest distance between two points. The line segment joining points and is denoted as or as. ray with initial point

### A (straight) line has length but no width or thickness. A line is understood to extend indefinitely to both sides. beginning or end.

Points, Lines, and Planes Point is a position in space. point has no length or width or thickness. point in geometry is represented by a dot. To name a point, we usually use a (capital) letter. (straight)

### 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

### CK-12 Geometry: Perpendicular Bisectors in Triangles

CK-12 Geometry: Perpendicular Bisectors in Triangles Learning Objectives Understand points of concurrency. Apply the Perpendicular Bisector Theorem and its converse to triangles. Understand concurrency

### 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

### 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

### 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.

### Notes on Perp. Bisectors & Circumcenters - Page 1

Notes on Perp. isectors & ircumcenters - Page 1 Name perpendicular bisector of a triangle is a line, ray, or segment that intersects a side of a triangle at a 90 angle and at its midpoint. onsider to the

### Conjectures 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:

### Geometry Final Assessment 11-12, 1st semester

Geometry Final ssessment 11-12, 1st semester Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name three collinear points. a. P, G, and N c. R, P, and G

### Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: Activity 24

Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 24 esources: Springoard- Geometry Unit Overview In this unit, students will study formal definitions of basic figures,

### Neutral Geometry. Chapter Neutral Geometry

Neutral Geometry Chapter 4.1-4.4 Neutral Geometry Geometry without the Parallel Postulate Undefined terms point, line, distance, half-plane, angle measure Axioms Existence Postulate (points) Incidence

### Geometry. Unit 6. Quadrilaterals. Unit 6

Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections

### Ch 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and Angles [and Triangles]

h 3 Worksheets S15 KEY LEVEL 2 Name 3.1 Duplicating Segments and ngles [and Triangles] Warm up: Directions: Draw the following as accurately as possible. Pay attention to any problems you may be having.

### of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

2901 Clint Moore Road #319, Boca Raton, FL 33496 Office: (561) 459-2058 Mobile: (949) 510-8153 Email: HappyFunMathTutor@gmail.com www.happyfunmathtutor.com GEOMETRY THEORUMS AND POSTULATES GEOMETRY POSTULATES:

### **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle.

Geometry Week 7 Sec 4.2 to 4.5 section 4.2 **The Ruler Postulate guarantees that you can measure any segment. **The Protractor Postulate guarantees that you can measure any angle. Protractor Postulate:

### CONGRUENT TRIANGLES 6.1.1 6.1.4

ONGUN INGL 6.1.1 6.1.4 wo triangles are congruent if there is a sequence of rigid transformations that carry one onto the other. wo triangles are also congruent if they are similar figures with a ratio

### 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,

### The Protractor Postulate and the SAS Axiom. Chapter The Axioms of Plane Geometry

The Protractor Postulate and the SAS Axiom Chapter 3.4-3.7 The Axioms of Plane Geometry The Protractor Postulate and Angle Measure The Protractor Postulate (p51) defines the measure of an angle (denoted

### 4.7 Triangle Inequalities

age 1 of 7 4.7 riangle Inequalities Goal Use triangle measurements to decide which side is longest and which angle is largest. he diagrams below show a relationship between the longest and shortest sides

### CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

### 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

### Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles

Chapters 4 and 5 Notes: Quadrilaterals and Similar Triangles IMPORTANT TERMS AND DEFINITIONS parallelogram rectangle square rhombus A quadrilateral is a polygon that has four sides. A parallelogram is

### CK-12 Geometry: Midpoints and Bisectors

CK-12 Geometry: Midpoints and Bisectors Learning Objectives Identify the midpoint of line segments. Identify the bisector of a line segment. Understand and the Angle Bisector Postulate. Review Queue Answer

### 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

### 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

### Name Period 11/2 11/13

Name Period 11/2 11/13 Vocabulary erms: ongruent orresponding Parts ongruency statement Included angle Included side GOMY UNI 6 ONGUN INGL HL Non-included side Hypotenuse Leg 11/5 and 11/12 eview 11/6,,

### 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

### Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition.

Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. 1. measures less than By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than

### 7.4 Showing Triangles are

age 1 of 7 7. howing riangles are imilar: and oal how that two triangles are similar using the and imilarity heorems. ey ords similar polygons p. he triangles in the avajo rug look similar. o show that

### 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

### Name Geometry Exam Review #1: Constructions and Vocab

Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make

### 5-1 Reteaching ( ) Midsegments of Triangles

5-1 Reteaching Connecting the midpoints of two sides of a triangle creates a segment called a midsegment of the triangle. Point X is the midpoint of AB. Point Y is the midpoint of BC. Midsegments of Triangles

### 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

### Geometry - Chapter 5 Review

Class: Date: Geometry - Chapter 5 Review 1. Points B, D, and F are midpoints of the sides of ACE. EC = 30 and DF = 17. Find AC. The diagram is not to scale. 3. Find the value of x. The diagram is not to

### 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

### (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 =

### Math 311 Test III, Spring 2013 (with solutions)

Math 311 Test III, Spring 2013 (with solutions) Dr Holmes April 25, 2013 It is extremely likely that there are mistakes in the solutions given! Please call them to my attention if you find them. This exam

### Geometry Unit 1. Basics of Geometry

Geometry Unit 1 Basics of Geometry Using inductive reasoning - Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture- an unproven statement that is based

### Name: Date: Hour: Target 5a (Day 1) Identify bisectors of angles and segments and use them to find segment measures.

Geometry Name: Date: Hour: Target 5a (Day 1) Identify bisectors of angles and segments and use them to find segment measures. Perpendicular Bisectors Theorem 5.1 Any point on the perpendicular of a segment

### Relevant Vocabulary. The MIDPOINT of a segment is a point that divides a segment into 2 = or parts.

im 9: How do we construct a perpendicular bisector? Do Now: 1. omplete: n angle bisector is a ray (line/segment) that divides an into two or parts. 48 Geometry 10R 2. onstruct and label D, the bi sector

### 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

### 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.

### Geometry: 1-1 Day 1 Points, Lines and Planes

Geometry: 1-1 Day 1 Points, Lines and Planes What are the Undefined Terms? The Undefined Terms are: What is a Point? How is a point named? Example: What is a Line? A line is named two ways. What are the

### GEOMETRY OF THE CIRCLE

HTR GMTRY F TH IRL arly geometers in many parts of the world knew that, for all circles, the ratio of the circumference of a circle to its diameter was a constant. Today, we write d 5p, but early geometers

### 5-1 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

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

### Chapter 1. Foundations of Geometry: Points, Lines, and Planes

Chapter 1 Foundations of Geometry: Points, Lines, and Planes Objectives(Goals) Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in

### Geometry: Euclidean. Through a given external point there is at most one line parallel to a

Geometry: Euclidean MATH 3120, Spring 2016 The proofs of theorems below can be proven using the SMSG postulates and the neutral geometry theorems provided in the previous section. In the SMSG axiom list,