Geometry with Professor Burger
A companion book for Thinkwell's Geometry online video course. Volume #1 covers Chapters 1 ‒ 5.
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Before we get too far into the fun, I wanted to personally introduce myself: I'm your virtual instructor, Professor Burger. Welcome to Thinkwell's wonderful world of Geometry!
Together, you and I will explore the ideas that are foundational to Geometry. I hope you ’ ll let me help you in MAKING MEANING of the math ideas we ’ ll see together — there ’ s no need to memorize if you focus on deep understanding. I invite (urge) you to take the time to truly think through the math we ’ ll explore together and to mindfully practice the skills behind the ideas every day. If you do, you ’ ll not only succeed in Geometry, but you ’ ll also be on solid ground for all the other math in your future! About This Book This is a companion book to Thinkwell's Geometry online video course. Use this book as a complement to the online materials. I always say that to learn math you must DO MATH! I encourage you to put pencil to paper and • take notes in this book • highlight key concepts and earmark ideas you want to remember • doodle, sketch, and visualize the math ideas presented in each topic. In a nutshell, make this book your own and keep it by your side as you study the concepts in this course.
This book is divided into chapters. Within each chapter are a series of Geometry topics. Every topic online contains my Video Lessons along with an electronic version of the Notes, Practice questions and Worksheet questions (although I wish we ’ d call them “ Funsheets ” , but that ’ s another story). How To Use This Book Use this book alongside the online course and TAKE NOTES here while watching my video lessons. Your own notes are a key to your own success — I promise. Since this book summarizes the concepts, vocabulary, and key examples presented in the Video Lessons, it is a great tool to help you navigate the videos — but this companion book is not intended as a shortcut to replace the Video Lessons. To get the most out of this learning experience, I urge you to watch (and think through) all of the online Video Lessons. Maybe even watch some videos twice! The online Geometry course offers lots of opportunities to practice the skills you ’ ll need for success in Geometry. Each topic ’ s Worksheet Practice and Interactive Practice is a collection of questions connected to the content presented in the Video Lessons. I've included those questions here in this book, so you can explore them offline and spend time really thinking through each question. I always say that the best way to learn math is to DO MATH. So, take advantage of all the opportunities to practice what you've learned! Finally, at the end of each section, I've included a Review Worksheet to give you even more opportunities to review and practice the concepts you learned in the Video Lessons. Put pencil to paper (or pen, if you dare) to answer each question. I recommend you complete these Reviews before taking the course Quizzes and Tests online.
LET'S GO! I look forward to our Geometry journey together! Remember to make meaning and focus on deep understanding … and also remember that YOU CAN DO IT! Have fun! If you have any questions, please reach out to my friends at Thinkwell. Email them at email@example.com. Also, I ’ m on Twitter @ebb663, if you want to say, “ hello ” .
I wish you all the best in your Geometry success!
Your virtual teacher,
— Prof. B.
TABLE OF CONTENTS
Tips for Success������������������������������������������������������������������������������������������������������������������������� 1 Suggested Pacing Guide��������������������������������������������������������������������������������������������������������������3 Chapter 1: Fundamentals of Geometry ���������������������������������������������������������������������������������������5 1.1 Points, Lines, Planes, and Angles������������������������������������������������������������������������������ 7 1.1.1 Understanding Points, Lines, and Planes ��������������������������������������������������9 1.1.2 Measuring and Constructing Segments ����������������������������������������������������16 1.1.3 Measuring and Constructing Angles �������������������������������������������������������� 24 1.1.4 Pairs of Angles ����������������������������������������������������������������������������������������� 31 1.1 Review Worksheet ��������������������������������������������������������������������������������������38 1.2 Coordinate and Transformation Tools ��������������������������������������������������������������������� 43 1.2.1 Using Formulas in Geometry ��������������������������������������������������������������������45 1.2.2 Midpoint and Distance in the Coordinate Plane ����������������������������������������51 1.2.3 Transformations in the Coordinate Plane ������������������������������������������������� 58 1.2 Review Worksheet ��������������������������������������������������������������������������������������64 Chapter 2: Reasoning and Writing Geometric Proofs����������������������������������������������������������������67 2.1 Inductive and Deductive Reasoning�������������������������������������������������������������������������69 2.1.1 Using Inductive Reasoning to Make Conjectures ������������������������������������71 2.1.2 Conditional Statements ����������������������������������������������������������������������������77 2.1.3 Using Deductive Reasoning to Verify Conjectures ������������������������������������83 2.1.4 Biconditional Statements and Definitions �������������������������������������������������90 2.1 Review Worksheet ��������������������������������������������������������������������������������������95 2.2 Mathematical Proof �������������������������������������������������������������������������������������������������99 2.2.1 Algebraic Proof �������������������������������������������������������������������������������������� 101 2.2.2 Geometric Proof ��������������������������������������������������������������������������������������108 2.2.3 Flowchart and Paragraph Proofs ������������������������������������������������������������115 2.2 Review Worksheet ������������������������������������������������������������������������������������122 Chapter 3: Parallel and Perpendicular Lines���������������������������������������������������������������������������127 3.1 Lines with Transversals ������������������������������������������������������������������������������������������ 129 3.1.1 Planes, Lines, and Angles ���������������������������������������������������������������������� 131 3.1.2 Angles, Parallel Lines, and Transversals ����������������������������������������������� 136 3.1.3 Proving that Lines are Parallel ���������������������������������������������������������������142 3.1.4 Properties of Perpendicular Lines ����������������������������������������������������������149 3.1 Review Worksheet ������������������������������������������������������������������������������������154 3.2 Slope and the Equation of a Line ��������������������������������������������������������������������������159 3.2.1 Finding the Slope Given Two Points ����������������������������������������������������� 161 3.2.2 Slope-Intercept Form ����������������������������������������������������������������������������� 166 3.2.3 Point-Slope Form �����������������������������������������������������������������������������������173
TABLE OF CONTENTS
3.2.4 Slopes of Parallel and Perpendicular Lines �������������������������������������������������182 3.2 Review Worksheet ������������������������������������������������������������������������������������������188 Chapter 4: Triangle Congruence����������������������������������������������������������������������������������������������������191 4.1 Triangles and Congruence��������������������������������������������������������������������������������������������193 4.1.1 Classifying Triangles ������������������������������������������������������������������������������������195 4.1.2 Angle Relationships in Triangles ������������������������������������������������������������������200 4.1.3 Congruent Triangles �������������������������������������������������������������������������������������207 4.1 Review Worksheet ������������������������������������������������������������������������������������������213 4.2 Proving Triangle Congruence ��������������������������������������������������������������������������������������217 4.2.1 Triangle Congruence: SSS and SAS ������������������������������������������������������������219 4.2.2 Triangle Congruence: ASA, AAS, and HL ����������������������������������������������������225 4.2.3 Triangle Congruence: CPCTC ���������������������������������������������������������������������230 4.2.4 Introduction to Coordinate Proof ������������������������������������������������������������������236 4.2.5 Isosceles and Equilateral Triangles �������������������������������������������������������������241 4.2 Review Worksheet ������������������������������������������������������������������������������������������247 Chapter 5: Properties and Attributes of Triangles��������������������������������������������������������������������������253 5.1 Segments in Triangles��������������������������������������������������������������������������������������������������255 5.1.1 Perpendicular and Angle Bisector Theorems ������������������������������������������������257 5.1.2 Medians, Altitudes, and Midsegments in Triangles ��������������������������������������263 5.1 Review Worksheet ������������������������������������������������������������������������������������������269 5.2 Relationships in Triangles ��������������������������������������������������������������������������������������������271 5.2.1 Indirect Proof and Inequalities in One Triangle ��������������������������������������������273 5.2.2 Inequalities in Two Triangles ������������������������������������������������������������������������280 5.2.3 The Pythagorean Theorem ��������������������������������������������������������������������������285 5.2.4 Applying Special Right Triangles��������������������������������������������������������������������291 5.2 Review Worksheet ������������������������������������������������������������������������������������������296 Blank Graph Paper����������������������������������������������������������������������������������������������������������������������� 301 Formulas & Symbols����������������������������������������������������������������������������������������������������������������������307
Use this book to help you stay organized.
Check out the suggested pacing guide in this book or download the online Lesson Plan and create a study schedule for yourself. Your schedule will be your plan for Geometry success!
Be an active learner. Before you begin studying, collect the tools you'll need: a pencil, scratch paper, highlighters, or graph paper are great things to have on-hand.
As you watch the Video Lessons online, work out the examples along with Prof. Burger on the Lesson Notes here (or on your own paper). Highlight important points in the Lesson Notes, and earmark topics you want to go back to review before a Quiz or Test.
Practice as you go. After each Video example, complete the Worksheet questions for that example. Once you've watched all the video lessons and answered all the Worksheet questions, check your understanding by completing the Practice question set. Go online to check your answers and to see answer feedback with step- by-step explanations. Review to remember.
Before a Quiz or Test, complete the Review Worksheet and re-do any exercises you need extra practice to master.
Reach out if you need help! Have questions? Need help? Reach out to us at firstname.lastname@example.org. We're here to help!
Suggested Pacing guide
This pacing guide follows a 36-week plan to sequentially progress through Thinkwell's online Geometry course. Since the course is self-paced, feel free to go as quickly or as slowly through the material as you need to – this guide is just a suggestion. The list below corresponds with Thinkwell's Geometry online sequence.
口 WEEK 1: – Algebra Review Part I 口 WEEK 2: – Algebra Review Part II 口 WEEK 3: – 1.1 Points, Lines, Planes, and Angles 口 WEEK 4: – 1.1 Points, Lines, Planes, and Angles (Cont.) 口 WEEK 5: – 1.2 Coordinate and Transformation Tools 口 WEEK 6: – 1.1 - 1.2 Online Assessment 口 WEEK 7: – 2.1 Inductive and Deductive Reasoning 口 WEEK 8: – 2.2 Mathematical Proof 口 WEEK 9: – 2.1 - 2.2 Online Assessment 口 WEEK 10: – 3.1 Lines with Transversals 口 WEEK 11: – 3.2 Slope and the Equation of a Line 口 WEEK 12: – 3.2 Slope and the Equation of a Line (Cont.) 口 WEEK 13: – 3.1 - 3.2 Online Assessment 口 WEEK 14: – 4.1 Triangles and Congruence 口 WEEK 15: – 4.2 Proving Triangle Congruence 口 WEEK 16: – 4.2 Proving Triangle Congruence (Cont.) 口 WEEK 17: – 4.1 - 4.2 Online Assessment 口 WEEK 18: – Modules 1 - 17 Online Assessment
口 WEEK 19: – 5.1 Segments in Triangles 口 WEEK 20: – 5.2 Relationships in Triangles 口 WEEK 21: – 5.2 Relationships in Triangles (Cont.) 口 WEEK 22: – 5.1 - 5.2 Online Assessment 口 WEEK 23: – 6.1 Polygons and Parallelograms 口 WEEK 24: – 6.2 Other Special Quadrilaterals 口 WEEK 25: – 6.1 - 6.2 Online Assessment 口 WEEK 26: – 7.1 Similarity Relationships 口 WEEK 27: – 8.1 Trigonometric Ratios 口 WEEK 28: – 7.1 - 8.1 Online Assessment 口 WEEK 29: – 9.1 Developing Geometric Formulas 口 WEEK 30: – 10.1 Surface Area and Volume 口 WEEK 31: – 9.1 - 10.1 Online Assessment 口 WEEK 32: – 11.1 Lines and Arcs in Circles 口 WEEK 33: – 11.2 Angles and Segments in Circles 口 WEEK 34: – 12.1 Congruence Transformations 口 WEEK 35: – 11.1 - 11.2, 12.1 Online Assessment 口 WEEK 36: – Modules 19-35 Online Assessment
Chapter 1 F undamentals of Geometry
Points, Lines, Planes, and Angles
1.1.1 Understanding Points, Lines, and Planes
Key Objectives • Identify, name, and draw points, lines, segments, rays, and planes. • Apply basic facts about points, lines, and planes. Key Terms • A basic geometric figure that cannot be defined in terms of other figures is an undefined term . • Points that lie on the same line are collinear . • Points that lie in the same plane are coplanar . • A segment or line segment is the part of a line consisting of two points and all points between them. • An endpoint is a point at the end of a segment or the starting point of a ray. • A ray is a part of a line that starts at an endpoint and extends forever in one direction. • Opposite rays are two rays that have a common endpoint and form a line. • A postulate , or axiom, is a statement that is accepted as true without proof. • An intersection is the set of all points that two or more figures have in common. Theorems, Postulates, Corollaries, and Properties • Postulate Through any two points there is exactly one line. • Postulate Through any three noncollinear points there is exactly one plane. • Postulate If two points lie in a plane, then the line containing those points lies in the plane. • Postulate If two unique lines intersect, then they intersect at exactly one point. • Postulate If two unique planes intersect, then they intersect at exactly one line. • A point names a location and has no size. It is represented by a dot. • A line is a straight path that has no thickness and extends forever. • A plane is a flat surface that has no thickness and extends forever. Geometry is an area of mathematics concerned with the study of two-dimensional and three-dimensional figures. Before a figure can be studied in geometry, even the most basic parts of the figure must be identified and named. For example, consider a triangle. A triangle is a very simple figure with three sides where each pair of sides meets at a corner. Even though a triangle is a simple figure, its parts must be identified using geometric terms before a triangle can be explicitly defined. The most elementary geometric figures are discussed in the examples below.
Example 1 Naming Points, Lines, and Planes The three fundamental geometric figures are points, lines, and planes. These three figures cannot be defined in terms of other figures, so they are referred to as undefined terms. 1.1.1 Understanding Points, Lines, and Planes (continued)
A point names a location and is represented by a dot. The dot must have some thickness or size when drawn, but an idealized point has no size because it is only a location. A point is named with a capital letter, such as P . A line is a straight path of points that extends infinitely in two opposite directions. An idealized line contains an infinite number of points and has length, but no thickness. A line is named with a lowercase letter or by drawing a double-headed arrow over the names of two points on that line. For example, the line here, which passes through the points X and Y , can be named XY or . A plane is a flat surface that extends infinitely. Points and lines are contained on a plane. A plane is named with a capital script letter, such as R , or with any three points on that plane, and these points are not on a single line. For example, ABC names a plane that contains points A , B , and C if A , B , and C are not contained on a single line. The points used for naming a plane can be listed in any order.
Points that lie on the same line are collinear points, and points that lie in the same plane are coplanar points. There are also terms to describe points that do not lie on the same line or plane: noncollinear and noncoplanar.
The given figure shows two planes, F and N ; five points, A , B , C , D , and E ; and two lines. Points B , C , D , and E are contained on F . So, B , C , D , and E are four coplanar points. Furthermore, N contains points A , B , C , and D , so these four points are also coplanar.
1.1.1 Understanding Points, Lines, and Planes (continued)
Lines are drawn with arrows at each end. There are only two lines drawn in this figure; one line is a vertical line and the other is a horizontal line. (Note that lines do not have to be horizontal or vertical.) A line is named using any two points that the line passes through. Since the vertical line passes through A and C , it can be named AC . The horizontal line passes through B , C , and D . So, the horizontal line can be named BC CD , , or by using any pair of the points it passes through.
Example 2 Drawing Segments and Rays
A section of a line defined by two endpoints that does not extend infinitely in either direction is called a segment. A segment is named by drawing a line (with no arrows) over the names of the two endpoints. For example, AB names a segment where A and B are the endpoints. Either endpoint can be listed first in the name of a segment. A section of a line defined by one endpoint that extends infinitely in one direction is called a ray. A ray is named by drawing a single-headed arrow over the name of the endpoint and any other point on that ray. For example, AB names a ray where A is the endpoint and B is a point on the ray. When two rays have a common endpoint and form a line, they are called opposite rays. To draw a segment with endpoints A and B , draw a straight line with a point at the beginning and another point at the end. Identify these endpoints as A and B . The segment can be named AB since its endpoints are A and B .
1.1.1 Understanding Points, Lines, and Planes (continued)
To draw opposite rays with a common endpoint W , begin by drawing a point and labeling it W , the endpoint for both rays. Opposite rays extend in opposite directions from a common endpoint to form a line. So, if the first ray extends from W to the left, then in order for the two rays to form a line, the second ray must extend from W to the right. (Note that opposite rays do not have to form a horizontal line; any line can be formed.)
Example 3 Identifying Points and Lines in a Plane Many facts regarding geometric figures will be presented as a postulate. A postulate, or axiom, is a statement that is accepted as true without proof. Three postulates regarding the relationship between points, lines, and planes are listed below. • Through any two points there is exactly one line. • Through any three noncollinear points there is exactly one plane. • If two points lie in a plane, then the line containing those points lies in the plane.
The given figure shows one plane, F ; three points, C , D , and E ; and one line. The green line passes through two points, C and D . This line can be named or it can be named with a reference to the two points it passes through, C and D .
Example 4 Representing Intersections When two lines or planes cross each other, they are said to intersect. Consider two intersecting lines. The point at which the two lines intersect is the point that the two lines have in common. This common point is called the intersection of the lines. Generally, the intersection of two figures is the set of all points that two figures have in common. An intersection may contain one point, a limited number of points, or an infinite number of points, depending on the figures. The following postulates describe the intersections of lines and planes.
• If two unique lines intersect, then they intersect at exactly one point. • If two unique planes intersect, then they intersect at exactly one line.
1.1.1 Understanding Points, Lines, and Planes (continued)
In order for two lines to intersect, they must not have the same slant. So, to draw two intersecting lines, draw two lines (each with arrows at the ends) such that the degree of slant in the first line is different from that of the second line. It is possible to draw two lines that do not intersect. Non-intersecting lines will have the same degree of slant. For example, two horizontal lines will never intersect. Here, draw two planes that intersect in one line. So, draw two unique planes that cross each other. The line formed by the intersection of the two planes is contained on both planes.
To draw a line that intersects the two planes, but does not lie in either plane, draw a line that intersects with each plane at a single point. The vertical line drawn here intersects with both of the planes, but it does not lie in either plane.
1.1.1 Understanding Points, Lines, and Planes − Worksheet
Example 1: Use the figure to name each of the following. 1. five points 2. two lines
3. two planes
4. point on
Example 2: Draw and label each of the following. 5. a segment with endpoints M and N
6. a ray with endpoint F that passes through G
Example 3: Use the figure to name each of the following. 7. a line that contains A and C
B D C
8. a plane that contains A, D , and C
Example 4: Sketch a figure that shows each of the following. 9. three coplanar lines that intersect in a common point
10. two lines that do not intersect
1.1.1 Understanding Points, Lines, and Planes − Practice
2. Identify the plane containing D , F , and G .
1. Identify a point on
4. Identify the intersection of plane A and plane B .
3. Identify two opposite rays.
6. Draw a picture that represents a ray with endpoint B that passes through A .
5. Draw a picture that represents a line intersecting a plane at one point.
1.1.2 Measuring and Constructing Segments
Key Objectives • Use length and midpoint of a segment. • Construct midpoints and congruent segments. Key Terms • A coordinate is a number used to identify the location of a point. On a number line, a point corresponds to one number and this number is called a coordinate. • The distance between any two points is the absolute value of the difference of the coordinates. • The distance between two points A and B is also called the length of AB , or AB . • Congruent segments are segments that have the same length. • A construction is a way of creating a figure that is more precise than a sketch. • In order for you to say that a point B is between two points A and C , all three of the points must lie on the same line, and AB + BC = AC . • The midpoint M of a segment AB is the point that bisects, or divides, the segment into two congruent segments. If M is the midpoint of segment AB , then AM = MB . • To bisect is to divide into two congruent parts. • A segment bisector is any ray, segment, or line that intersects a segment at its midpoint. Formulas • Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. Theorems, Postulates, Corollaries, and Properties • Segment Addition Postulate If B is between A and C , then AB + BC = AC . Example 1 Finding the Length of a Segment When a segment is drawn on a number line, each of the segment’s endpoints corresponds with exactly one real number on a number line, called a coordinate. The relationship between points and real numbers is described in the Ruler Postulate. The points on a line can be put into a one-to-one correspondence with the real numbers. In other words, when a segment is on a number line, there is exactly one number on that number line that corresponds with each endpoint. A segment is a part of a line that has a defined length. The length of a segment on a number line is the distance between its two endpoints, and the distance between any two points on a number line is the absolute value of the difference of the coordinates. So, to find the length of a segment on a number line, find the difference between the coordinates (the numbers at the endpoints), and then take the absolute value of that difference.
1.1.2 Measuring and Constructing Segments (continued)
In this example a number line is given with the points X , Z , Y , and W , named on the number line. The coordinate of each point is the number that corresponds with that point. So, the coordinate of W is 3. To find XW , first identify coordinates of its endpoints: − 7 and 3. Then subtract these coordinates (in either order) and take the absolute value of the difference. Here, the length of the segment was found by subtracting − 7 from 3. Note that subtracting in the reverse order results in the same length. | − 7 − 3| = | − 10| = 10 Notice that the segment from X to W is represented here as XW and as XW . The notation XW refers to the length of the segment, so XW is equal to a number (in this case XW = 10). The notation XW refers to the segment itself. When two segments have the same length, they are congruent. The symbol for congruent is similar to the symbol = , but the congruent symbol has a squiggle over the = . The difference between congruent and equal is that segments themselves can be congruent, while the lengths of segments can be equal. Small dashes, called tick marks, are drawn in figures to indicate that segments are congruent.
Example 2 Copying a Segment
A sketch of a figure is an approximate version of a figure. A construction is a way of creating a figure that is more precise than a sketch. One way to make a geometric construction is to use a compass and straightedge.
1.1.2 Measuring and Constructing Segments (continued)
Use a straightedge and a compass to construct a segment that is congruent to the given segment, PQ . Step 1 Using a straightedge, draw a segment that is longer than the given segment. Then, near one end of the drawn segment, draw and label a point. This is one of the constructed segment’s endpoints. Here, the drawn segment is longer than PQ and the endpoint is labeled A . Step 2 Place the compass so that its ends are at the given segment’s endpoints. Then, confirm that the compass is placed correctly by drawing a small arc that passes through the given segment’s endpoint. The compass is open to Q and P in this example. The drawn arc passes through P , so the compass is placed correctly. Step 3 Without moving the arms of the compass , place it on the drawn segment so that its tip is on the labeled point. Then use the compass to sketch an arc that intersects the drawn segment. Here, the tip of the compass is placed at A and the arc intersects the drawn segment at the opposite end. Step 4 Label the point where the arc intersects the drawn segment. This is the constructed segment’s second endpoint. In this example, the point where the arc intersects the drawn segment is labeled B . So, the constructed segment is AB . Since PQ and AB are congruent segments, they have the same length. So, AB = PQ .
1.1.2 Measuring and Constructing Segments (continued) Example 3 Using the Segment Addition Postulate
In order for you to say that a point B is between two points A and C , all three of the points must lie on the same line, and AB + BC = AC .
In this example, the fact that B is between A and C is given. So, by the Segment Addition Postulate, it must be true that AB + BC = AC . Use the given information to sketch a segment. Since B is between A and C , the segment’s endpoints must be A and C . Now label the segment with the given lengths AC = 25 and BC = 7.2. From the sketched segment it is easy to see that AB + BC = AC , or AB + 7.2 = 25. Solve this equation by subtracting 7.2 from both sides to find AB. The length of AB is 25 − 7.2, or 17.8. The figure is given in this example. Begin by identifying the given information from the figure. The bar to the left of the segment extending from P to Q indicates that the corresponding expression, 9 x , represents PQ (read “ PQ ” as “the length of line segment PQ ”), or PQ = 9 x . To the right of the segment is 19 and 6 x − 1. The placement of 19 between P and R indicates that PR = 19, and the placement of 6 x − 1 between R and Q indicates that RQ = 6 x − 1. Furthermore, since P and Q are the endpoints of a segment containing R , R must be between P and Q . Therefore, by the Segment Addition Postulate, PR + RQ = PQ . Substitute the given lengths into this equation and solve for x . Note that x is not equal to PQ , but that PQ = 9 x . So, once the value of x is found, substitute that value into the expression 9 x to find PQ .
1.1.2 Measuring and Constructing Segments (continued) Example 4 Travel Application
The given information is organized on a series of segments, where the outermost endpoints represent Prof. Burger’s workplace, W , and his home, H . It is given that the distance from W to H is 3 kilometers and the distance from H to the party, M , is 900 meters. Notice that units of these two distances are different: kilometers (km) and meters (m). Only one unit can be used in calculations, so convert either 3 km to meters by multiplying, or 900 m to kilometers by dividing. Because multiplication is usually easier than division, convert the kilometers to meters using the fact that there are 1000 meters in 1 kilometer. WH = 3 km = 3(1000 m) = 3000 m Since M is between W and H in the segment, by the Segment Addition Postulate WM + MH = WH . Substitute the given lengths MH = 900 and WH = 3000 into the equation and solve to find WM . Now use the fact that G is the midpoint of WM to find GM . GM = WM /2 = 2100/2 = 1050 m The distance from the gas station, G to H , is GH = GM + MH = 1050 + 900 = 1950 m.
1.1.2 Measuring and Constructing Segments (continued) Example 5 Using Midpoints to Find Lengths
It is given that B is the midpoint of the segment from A to C . Therefore, the segment is divided into two equal parts, where AB = BC . Begin by using the fact that AB = BC to write an equation using the expressions for AB and BC given in the figure. Solve this equation for x and then substitute the value of x into the given expressions to find AB and BC . Once AB and BC are known, the Segment Addition Postulate can be used to find the length of AC .
1.1.2 Measuring and Constructing Segments − Worksheet
Example 1: Find each length. 1. AB
A C D –2–1012345 –2.5 B 3.5
Example 2: 3. Sketch, draw, and construct a segment congruent to RS .
Example 3: 4. B is between A and C , A C = 15.8, and AB = 9.9. Find BC.
5. Find MP .
5 y + 9
Example 4: 6. If a picnic area is located at the midpoint between Lubbock and Amarillo, find the distance to the picnic area from the sign.
Example 5: 7. K is the midpoint of JL, JL = 4 x − 2, and JK = 7. Find x , KL, and JL .
8. E bisects DF , DE = 2 y , and EF = 8 y − 3. Find DE , EF , and DF .
1.1.2 Measuring and Constructing Segments − Practice
1. Y is between X and Z , XY = 5.8, and YZ = 12.4. Find XZ .
2. PQ bisects ST at R . SR = 3 x + 3 and ST = 30. Find x .
3. GH bisects LM at K . LK = 5 x + 2 and LM = 64. Find x .
4. E is the midpoint of DF, DE = 6 x + 1 and EF = 7 x − 4. Find DE, EF, and DF .
5. Tell whether this statement is sometimes, always, or never true. Draw a sketch to support your answer. If I is not the midpoint of HJ , then H , I , and J are collinear.
6. Tell whether this statement is sometimes, always, or never true. Draw a sketch to support your answer. If B is a point between A and C , then AB + BC = AC .
1.1.3 Measuring and Constructing Angles Key Objectives • Name and classify angles. • Measure and construct angles and angle bisectors. Key Terms • An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex . • The set of all points between the sides of the angle is the interior of an angle . • The exterior of an angle is the set of all points outside the angle. • The measure of an angle is usually given in degrees.
• Since there are 360° in a circle, one degree is 1/360 of a circle. • An acute angle measures greater than 0° and less than 90°. • A right angle measures 90°. • An obtuse angle measures greater than 90° and less than 180°. • A straight angle is formed by two opposite rays and measures 180°. • Congruent angles are angles that have the same measure. • An angle bisector is a ray that divides an angle into two congruent angles. Theorems, Postulates, Corollaries, and Properties • Protractor Postulate Given AB and a point O on
AB , all rays that can be drawn from O can be put into
a one-to-one correspondence with the real numbers from 0 to 180. • Angle Addition Postulate If S is in the interior of ∠ PQR , then m ∠ PQS + m ∠ SQR = m ∠ PQR . Example 1 Naming Angles An angle is a figure formed when two rays meet at a common endpoint. The rays are called the angle’s sides, and the common endpoint is called the angle’s vertex. There are three ways to name an angle. If an angle’s vertex is not a point on any other angle, then the angle can be named using only its vertex. For example, if A is the vertex of an angle and A is not a point on any other angle (i.e., A is not a vertex of another angle and A is not on the side of another angle), then the angle can be named ∠ A , where ∠ is the symbol for angle. The second method for naming an angle is with a number. For example, if 1 is used to label an angle in a figure, then the angle can be named ∠ 1. The third method for naming an angle is explained in the example below.
1.1.3 Measuring and Constructing Angles (continued)
The third method for naming an angle is with its vertex and a point on each side, where the vertex is listed between the two points. This method is used when the angle’s vertex is either shared with another angle or it is a point on the side of another angle. D is the vertex of three angles in the given figure (one large angle and two smaller angles). Since D is the vertex of more than one angle, none of these three angles can be named simply ∠ D . And since none of these angles is labeled with a number, the third method for naming angles must be used. So, each of the three angles must be named with a point from each side and the vertex, D . To name the largest angle, first identify the sides of this angle. The sides are formed by rays DA and DC . So, A and C are each points on one of the angle’s sides. And since the vertex is D , the largest angle can be named either ∠ ADC or ∠ CDA .
Example 2 Measuring and Classifying Angles An angle is measured by relating the angle to a circle that is divided into 360 parts, shaped like pie pieces. Each one of these 360 pieces is called a degree. Imagine that an angle’s vertex is the center of a circle and that the circle itself intersects each side of the angle. The number of pie pieces, or degrees, that would be in the interior of the angle (i.e., between the two sides) is equal to the measure of the angle.
By the Protractor Postulate, a protractor can be used to find the measure of an angle that is between 0° and 180°.
1.1.3 Measuring and Constructing Angles (continued)
Angles can be classified by their measures. There are four types of angles with measures between 0° and 180°: straight, obtuse, right, and acute angles. A straight angle is also a line; its measure is 180°. An obtuse angle measures between 90° and 180°. The measure of a right angle is exactly 90°. A small box placed in an angle’s interior, near its vertex, is used to indicate that the angle is a right angle. All angles with a measure that is less than that of a right angle (so, less than 90°) are called acute angles. In this example, a protractor is placed over a given figure and then used to find m ∠ BOD (read as “the measure of angle BOD ”). The vertex of ∠ BOD is O . So, the center of the protractor is placed on O . To use the protractor to find m ∠ BOD , place it so that 0° is on one of the angle’s sides, ray OB . Then identify the degree measure on the protractor that corresponds with placement of the angle’s other side, ray OD . Since OD passes through the protractor at 65°, m ∠ BOD = 65°. The measure of ∠ BOD is less than 90°. Thus, ∠ BOD is an acute angle. The Segment Addition Postulate states that when a segment is divided into two parts, the sum of the lengths of those two parts is equal to the length of the whole segment. The Angle Addition Postulate relates the parts of an angle to each other in the same way that the Segment Addition Postulate relates the parts of a segment to each other. By the Angle Addition Postulate, when an angle is divided into two parts, the sum of the measures of those two parts is equal to the sum of the whole angle.
1.1.3 Measuring and Constructing Angles (continued)
In the given figure, ∠ COD is one part of the larger angle ∠ COB , where the second part is ∠ DOB . So, by the Angle Addition Postulate, m ∠ COB = m ∠ COD + m ∠ DOB . Use the Angle Addition Postulate to find m ∠ COD . From the previous example we know that ∠ DOB = 65°. (Remember, ∠ DOB and ∠ BOD are the same angle.) Use the protractor to find m ∠ COB : m ∠ COB = 135°. Substitute the two degree measures into the equation and solve for m ∠ COD . Since 135° − 65° = 70°, m ∠ COD = 70°. This measure is less than 90°, so the angle is acute.
Example 3 Using the Angle Addition Postulate
The measure of the large angle, ∠ PMR , is given. m ∠ PMR = 65° And, the measure of one of the two angles that make up ∠ QMR is also given. m ∠ QMR = 38° Since ∠ QMR and ∠ PMQ make up ∠ PMR , use the Angle Addition Postulate to find m ∠ PMQ .
1.1.3 Measuring and Constructing Angles (continued) Example 4 Finding the Measure of an Angle
When angles have the same measure, they are called congruent angles. For example, if the measure of ∠ 1 is 100° and the measure of ∠ 2 is 100°, then the angle measures are equal. m ∠ 1 = m ∠ 2 But the angles themselves are congruent . ∠ 1 ≅ ∠ 2 When a ray is an angle bisector, it divides an angle into two congruent angles. For example, if AC bisects ∠ BAD , then the two angles created by this ray are congruent. ∠ BAC ≅ ∠ CAD In this example, the fact that ∠ XZW is bisected by a ray is given. Therefore, by the definition of angle bisector, ∠ XZW is divided into two congruent angles. ∠ XZY ≅ ∠ YZW So by the definition of congruent angles, m ∠ XZY = m ∠ YZW . Algebraic expressions are given for the measures of ∠ XZY and ∠ YZW . Substitute these expressions into the equation and solve for x . m ∠ XZY = m ∠ YZW (7 x + 1)° = (5 x + 11)° Thus, x = 5. Note that the value of x is not the answer to the question. The value of x must be used to find m ∠ XZW . First, substitute 5 for x into either expression given for the small angles. Then, double that result to get m ∠ XZW .
1.1.3 Measuring and Constructing Angles − Worksheet
Example 1: 1. Musicians use a metronome to keep time as they play. The metronome’s needle swings back and forth in a fixed amount of time. Name all of the angles in the diagram.
Example 2: Use the protractor to find the measure of each angle. Then classify each as acute, right, or obtuse. 2. ∠ VXW T
3. ∠ TXW
4. ∠ RXU
Example 3: L is in the interior of ∠ JKM. Find each of the following. 5. m ∠ JKM if m ∠ JKL = 42° and m ∠ LKM = 28°
6. m ∠ LKM if m ∠ JKL = 56.4° and m ∠ JKM = 82.5°
Example 4: BD bisects ∠ ABC . Find each of the following. 7. m ∠ ABD if m ∠ ABD = (6 x + 4)° and m ∠ DBC = (8 x − 4)°
8. m ∠ ABC if m ∠ ABD = (5 y − 3)° and m ∠ DBC = (3 y + 15)°
1.1.3 Measuring and Constructing Angles − Practice
1. Classify ∠ OMN as acute, right, straight, or obtuse.
2. Classify ∠ HIJ as acute, right, straight, or obtuse.
TV bisects ∠ STU , m ∠ STV =
⎛ ⎝ ⎜
⎞ ⎠ ⎟
3. D is in the interior of ∠ ABC , m ∠ ABD = 63°, and m ∠ DBC = 23°. Find m ∠ ABC .
+ 8 , and m ∠ UTV = ( x + 2)°. Find m ∠ STU . x
5. Identify the protractor that shows the measure of ∠ XYZ as 60°.
6. m ∠ DFG = (2 x + 6)° and m ∠ EFG = (3 x − 11)°. Find the value of x .
1.1.4 Pairs of Angles
Key Objectives • Identify adjacent, vertical, complementary, and supplementary angles. • Find measures of pairs of angles. Key Terms • Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. • A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays.
• Complementary angles are two angles whose measures have a sum of 90°. • Supplementary angles are two angles whose measures have a sum of 180°. • Vertical angles are two nonadjacent angles formed by two intersecting lines. Example 1 Identifying Angle Pairs
Two angles that have a common vertex and side, but do not overlap, are called adjacent angles. The figure to the left contains a pair of adjacent angles, ∠ BAC and ∠ CAD . Notice that ∠ BAD and ∠ BAC share a common vertex, point A , and a common side, ray AB , but those two angles are not adjacent angles because they overlap. A linear pair is a pair of adjacent angles that combine to form a straight angle. In a linear pair, the noncommon sides of the two adjacent angles are opposite rays. Five angles are named with numbers in this figure. For each given pair of angles, determine the type of pair by first considering whether they are adjacent. If they are adjacent, then consider whether they are a linear pair. ∠ 1 and ∠ 2 do not have a common vertex, so they are not adjacent. ∠ 4 and ∠ 5 have a common vertex and a common side. Furthermore, ∠ 4 and ∠ 5 do not overlap (i.e., they do not have any common interior points), so they must be adjacent. However, they are not a linear pair because their noncommon sides are not opposite rays. Therefore, ∠ 4 and ∠ 5 are adjacent only. ∠ 1 and ∠ 3 have a common vertex, a common side, and they do not overlap. So, they are adjacent. The noncommon sides of ∠ 1 and ∠ 3 are opposite rays. Therefore, ∠ 1 and ∠ 3 form a linear pair.
1.1.4 Pairs of Angles (continued) Example 2 Finding the Measure of Complements and Supplements The relationship between adjacent angles is based on the position of the angles. There are also special relationships between angles based on their measures. Complementary and supplementary angles are pairs of angles that are related to each other by their measures. Complementary angles are two angles whose measures have a sum of 90°. So, the complement of an angle can be found by subtracting its measure from 90°. Supplementary angles are two angles whose measures have a sum of 180°. So, the supplement of an angle can be found by subtracting its measure from 180°.
The measure of ∠ M is given in the figure. m ∠ M = 37.2°
The complement of an angle is found by subtracting the measure of the angle from 90°. So, the measure of the complement of ∠ M is 90° − 37.2°, or 52.8°. You can check your work by adding the angles. If the sum of two angles is equal to 90°, then the angles are complementary. 37.2° + 52.8° = 90° So, the angles are complementary and 52.8° is the complement of 37.2°. An expression for the measure of ∠ N is given in the figure. m ∠ N = (5 x + 49)° The supplement of an angle is found by subtracting the measure of the angle from 180°. So, the measure of the supplement of ∠ N is 180° − (5 x + 49)°, or (131 − 5 x )°. Add the expressions for the two angles to confirm that they are supplementary angles. The sum is 180°, so the angles are supplementary.
Example 3 Applying Complements and Supplements If x ° is the measure of an angle, then the measure of that angle’s complement can be represented by the expression (90 − x )°. Similarly, the angle’s supplement can be represented by the expression (180 − x )°.
1.1.4 Pairs of Angles (continued)
Translate this sentence into an equation to find the measure of the complement. Dividing the sentence into parts can make the translation easier. First, identify a variable for the unknown. If the unknown is represented by x , then “twice an angle” can be translated to “2 x ,” because “twice” a number means to multiply the number by 2.The next word in the sentence, “measures,” can be translated to “ = ” because if an angle “measures” some number of degrees, that means the angle’s measure is equal to some number of degrees. The next part of the sentence is “9° more than.” The phrase “9 more than” some number is equivalent to 9 plus some number. So, “9° more than” can be translated to “9° + .” The last part of the sentence, “the measure of its complement,” can be translated to (90 − x )°. Put these parts together to form an equation and then solve the equation for x . Remember, x is the measure of the unknown angle, but the answer to this example is the “measure of its complement,” or the measure of the complement of x . So, subtract the value of x , 33°, from 90° to find the answer.
1.1.4 Pairs of Angles (continued) Example 4 Science Application
Four angles are named in this figure. Angles may appear to be congruent in a figure, but it cannot be assumed that angles (or any figures) are congruent unless that fact is given information. Begin by listing the given information. Then note the given information in the figure. Since ∠ 2 and ∠ 3 are congruent, each of those angles can be marked in the figure with a single curve to denote their congruency. It is also given that there are two pairs of complementary angles. ∠ 1 and ∠ 2 are complementary and ∠ 3 and ∠ 4 are complementary. Remember, the sum of two complementary angles is 90° and 90° is the measure of a right angle. So, in the figure, the angle composed of ∠ 1 and ∠ 2 can be marked as a right angle, as can the angle composed of ∠ 3 and ∠ 4. The last piece of given information is the measure of an angle. m ∠ 4 = 52° So, in the figure ∠ 4 can be labeled with 52°. Now use all of this information to find m ∠ 1, m ∠ 2, and m ∠ 3. Since ∠ 3 and ∠ 4 are complementary and m ∠ 4 = 52°, m ∠ 3 = 90° − 52° = 38°. Now that m ∠ 3 is known, m ∠ 2 can easily be found since ∠ 3 and ∠ 2 are congruent. So, m ∠ 2 = 38°. Now m ∠ 1 can be found by using the fact that ∠ 1 and ∠ 2 are complementary. So, m ∠ 1 = 90° − 38° = 52°.
1.1.4 Pairs of Angles (continued) Example 5 Identifying Vertical Angles
When two lines intersect, four angles are formed where the point of intersection is the vertex for all four angles. The angles that are opposite from one another are called vertical angles. Vertical angles have a common vertex, but no common sides. In this figure there are three lines. Two of these lines intersect the third line, creating two points of intersection. So, there are four pairs of vertical angles in this figure.
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