practice and problem solving topic 1 foundations of geometry answers

Snapsolve any problem by taking a picture. Try it in the Numerade app?

Edward Burger, David J. Chard,Earlene J. Hall

Foundations for geometry - all with video answers.

Understanding Points, Lines, and Planes

Apply the vocabulary from this lesson to answer each question. Give an example from your classroom of three collinear points.

Apply the vocabulary from this lesson to answer each question. Make use of the fact that endpoint is a compound of end and point and name the endpoint of $\overrightarrow{S T}$.

Use the figure to name each of the following. five points

Use the figure to name each of the following. FIGURE CAN'T COPY. two lines

Use the figure to name each of the following. FIGURE CAN'T COPY. two planes

Use the figure to name each of the following. FIGURE CAN'T COPY. point on $\overrightarrow{B D}$

Draw and label each of the following. a segment with endpoints $M$ and $N$

Draw and label each of the following. a ray with endpoint $F$ that passes through $G$

Use the figure to name each of the following. FIGURE CAN'T COPY. a line that contains $A$ and $C$

Use the figure to name each of the following. FIGURE CAN'T COPY. a plane that contains $A, D,$ and $C$

Sketch a figure that shows each of the following. three coplanar lines that intersect in a common point

Sketch a figure that shows each of the following. two lines that do not intersect

Use the figure to name each of the following. FIGURE CAN'T COPY three collinear points

Use the figure to name each of the following. FIGURE CAN'T COPY four coplanar points

Use the figure to name each of the following. FIGURE CAN'T COPY a plane containing $E$

Draw and label each of the following. a line containing $X$ and $Y$

Draw and label each of the following. a pair of opposite rays that both contain $R$

Use the figure to name each of the following. FIGURE CAN'T COPY. two points and a line that lie in plane $\mathcal{T}$

Use the figure to name each of the following. FIGURE CAN'T COPY. two planes that contain $\ell$

Sketch a figure that shows each of the following. a line that intersects two non-intersecting planes

Sketch a figure that shows each of the following. three coplanar lines that intersect in three different points

This problem will prepare you for the Concept Connection on page $34 .$ Name an object at the archaeological site shown that is represented by each of the following. a. a point b. a segment c. a plane

Draw each of the following. plane $\mathcal{H}$ containing two lines that intersect at $M$

Draw each of the following. $\overrightarrow{S T}$ intersecting plane $\mathcal{M}$ at $R$

Use the figure to name each of the following. the intersection of $\overleftrightarrow{T V}$ and $\overrightarrow{U S}$

Use the figure to name each of the following. FIGURE CAN'T COPY. the intersection of $\overrightarrow{U S}$ and plane $\mathcal{R}$

Use the figure to name each of the following. FIGURE CAN'T COPY. the intersection of $\overline{T U}$ and $\overline{U V}$

Write the postulate that justifies each statement. The line connecting two dots on a sheet of paper lies on the same sheet of paper as the dots.

Write the postulate that justifies each statement. If two ants are walking in straight lines but in different directions, their paths cannot cross more than once.

Write the postulate that justifies each statement. Critical Thinking Is it possible to draw three points that are noncoplanar? Explain.

Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. If two planes intersect, they intersect in a straight line..

Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. If two lines intersect, they intersect at two different points.

Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. $\overleftarrow{A B}$ is another name for $\overleftarrow{B A}$

Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. If two rays share a common endpoint, then they form a line.

Art Pointillism is a technique in which tiny dots of complementary colors are combined to form a picture. Which postulate ensures that a line connecting two of these points also lies in the plane containing the points?

Probability Three of the labeled points are chosen at random. What is the probability that they are collinear? FIGURE CAN'T COPY.

Campers often use a cooking stove with three legs. Which postulate explains why they might prefer this design to a stove that has four legs? FIGURE CAN'T COPY.

Write About It Explain why three coplanar lines may have zero, one, two, or three points of intersection. Support your answer with a sketch. FIGURE CAN'T COPY.

Which of the following is a set of noncollinear points? (A) $P, R, T$ (B) $Q, R, S$ (C) $P, Q, R$ (D) $S, T, U$

What is the greatest number of intersection points four coplanar lines can have? F. 6 G. 4 H. 2 J. 0

Two flat walls meet in the corner of a classroom. Which postulate best describes this situation? A. Through any three noncollinear points there is exactly one plane. B. If two points lie in a plane, then the line containing them lies in the plane. C. If two lines intersect, then they intersect in exactly one point. D.If two planes intersect, then they intersect in exactly one line.

What is the greatest number of planes determined by four noncollinear points?

Use the table for Exercises. FIGURE CAN'T COPY. What is the maximum number of segments determined by 4 points?

Use the table for Exercises. TABLE CAN'T COPY. What is the maximum number of segments determined by 4 points?

Multi-Step Extend the table. What is the maximum number of segments determined by 10 points?

Use the table for Exercises. TABLE CAN'T COPY. Write a formula for the maximum number of segments determined by $n$ points.

Critical Thinking Explain how rescue teams could use two of the postulates from this lesson to locate a distress signal.

The combined age of a mother and her twin daughters is 58 years. The mother was 25 years old when the twins were born. Write and solve an equation to find the age of each of the three people. (Previous course).

Determine whether each set of ordered pairs is a function. $$\{(0,1),(1,-1),(5,-1),(-1,2)\}$$

Determine whether each set of ordered pairs is a function. $$\{(3,8),(10,6),(9,8),(10,-6)\}$$

Find the mean, median, and mode for each set of data. (Previous course) $$0,6,1,3,5,2,7,10$$

Find the mean, median, and mode for each set of data. (Previous course) $$0.47,0.44,0.4,0.46,0.44$$

Topic 1: Foundations of Geometry

Table of contents, 1.1: measuring segments and angles.

• Segment Addition Postulate
• Angle Addition Postulate

1.3: Midpoint and Distance

• Midpoint Formula
• Distance Formula

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Basic Geometry Practice Questions with Full Answer Key – Area, Perimeter, Volume & Angles

• Posted by Brian Stocker MA
• Date April 3, 2014
• Comments 17 comments

High School geometry questions similar to what you will find on a standardized test.

Geometry practice test.

1. What is measurement of the indicated angle assuming the figure is a square?

a. 45 o b. 90 o c. 60 o d. 30 o

2. What is the sum of all the angles in the rectangle above?

a. 180 o b. 360 o c. 90 o d. 120 o

3. What is the measurement of the indicated angle?

a. 45 o b. 90 o c. 60 o d. 50 o

4. If the line m is parallel to the side AB of ? ABC, what is angle a ?

a. 130 o b. 25 o c. 65 o d. 50 o

5. What is perimeter of the above shape?

a. 12 cm b. 16 cm c. 6 cm d. 20 cm

6. What is (area of large circle) – (area of small circle) in the figure above?

a. 8 п cm 2 b. 10 п cm 2 c. 12 п cm 2 d. 16 п cm 2

7. What is perimeter of ? ABC in the above shape?

8. What is the volume of the figure above?

a. 125 cm 3 b. 875 cm 3 c. 1000 cm 3 d. 500 cm 3

9. What is the volume of the above solid made by a hollow cylinder with half in size of the larger cylinder?

a. 1440 п in 3 b. 1260 п in 3 c. 1040 п in 3 d. 960 п in 3

1. B The diagonals of a square intersect perpendicularly with each other so each angle measures 90 o x =90 o

2. B a+b+c+d = ? The sum of angles around a point is 360 o    a+b+c+d = 360 o

Video Solution

3. C The sum of angles around a point is 360 o d+300 = 360 o d = 60 o

4. D Two parallel lines( m & side AB) intersected by side AC a= 50 o (interior angles)

5. B The square with 2 cm side common to the rectangle apart from 4cm side Perimeter = 2+2+2+4+2+4 = 16 cm

6. C In the figure, we are given a large circle and a small circle inside it; with the diameter equal to the radius of the large one. The diameter of the small circle is 4 cm. This means that its radius is 2 cm. Since the diameter of the small circle is the radius of the large circle, the radius of the large circle is 4 cm. The area of a circle is calculated by: пr 2 where r is the radius.  Area of the small circle: п(2) 2 = 4п Area of the large circle: п(4) 2 2 = 16п

The difference area is found by: Area of the large circle – Area of the small circle = 16п – 4п = 12п

7. D Perimeter of a shape with two squares and triangle ABC. Perimeter = 8.5+8.5+6+6 Perimeter = 29 cm.

8. C Large cube is made up of 8 smaller cubes of 5 cm sides. Volume = Volume of small cube x 8 Volume = (5 x 5 x 5) x 8, 125 x 8 Volume = 1000cm 3

9. B Volume= Volume of large cylinder – Volume of small cylinder (Volume of cylinder = area of  base x height) We know the small cylinder is 1/2 the size of the large cylinder So, Volume of the small cylinder = Large Cylinder (п 12 2 x 10) – Small Cylinder(п 6 2 x 5) = 1440п – 180п Volume of small cylinder = 1260п in 3

See also our tutorial on Complex Shapes

Most Popular Geometry Questions

Common geometry questions on on standardized tests :

• Solve for the missing angle or side
• Finding the area or perimeter of different shapes (e.g. triangles, rectangles, circles)
• Problems using the Pythagorean Theorem
• Calculate properties of geometric shapes such as angles, right angles or parallel sides
• Calculating volume or surface area of complex shapes for example spheres, cylinders or cones
• Solve geometric transformations such as rotation, translation or reflections

Most Common Geometry Mistakes on a Test

• Not clearly labeling or identifying the given and unknown information in a problem
• Not understanding the properties and definitions of basic geometric figures (e.g. line, angle, triangle, etc.)
• Incorrectly using basic formulas (e.g. area of a triangle, Pythagorean theorem)
• Incorrectly interpreting geometric diagrams
• Not understanding the relationship between parallel lines and transversals
• Not understanding the relationship between angles and their degree measures
• Not understanding the relationship between perimeter and area

Tag: Basic Math , Geometry , Practice Questions

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Basic linear inequalities have one of the following forms: ax + b > 0 ax + b < 0 ax + b > 0 ax + b < 0 where a and b are some real numbers. Our solution to …

17 Comments

answer to question five should be 18, you forgot to include the 2cm outer side of the rectangle

Divide the figure into a square and a rectangle – perimeter of the square – 3 X 2 = 6 note 3 sides are included in the perimeter. Upper rectangle 2 X 4 = 8 + 2 = 10 total perimeter = 10 + 6 = 16

Try again – the question asks for the PERIMITER – 3 sides of the square – 2 + 2 + 2 = 6; 2 sides of the rectangle – 4 + 4 = 8; 1 side of the rectangle (top) = 2 6 + 8 + 2 = 16

The line between the square and the rectangle isn’t counted for perimeter, perimeter is the amount of units around the OUTSIDE. Since the square and rectangle are connected, try thinking of them as one big rectangle and forget about the line in between.

Correct – the bottom square has 2 cm each side on 3 sides (2 + 2 + 2 = 6). The top rectangle has 2, 4 cm sides (4 + 4 = 8) and 1, 2 cm side. So the total is 6 + 8 + 2 = 16.

Please in no 6 ,42 -22 is 20,so how comes about 16 and how did u get 122 in no 10

The answers are correct – I have updated and expanded the solutions – also the п symbol wasn’t displaying properly – thanks!

102°-2x +78°=180°

Love this. Super helpful!

In #9, since it is a cylinder; shouldn’t you use Pi in calculating the area of the base??

hi – yes the answer does use pi – the web rendering is a little funny tho – looks like an ‘n’ but is is pi – e. g. Volume= (п 122x 10) – (п 62x 5), 1440? – 180п

in question #9 what is the formula of a cylinder if the diameter is given and not the radius?

I understand how you got the answer in question 5, however the way it is displayed is unclear. It is not clear that the number 4 is attached to just the rectangle and not to the line as a whole. This means there are two interpretations of the figure.

1) 4*2+2*2 = 12 2) 2(4+2) + 2*2 = 16

Thanks! this was super helpful!

Can you explain the question, the question isn’t asked properly. They are asking the volume made by the small cylinder, but the answer is solved for the remaining volume removing the small cylinder volume.

Not quite – the small cylinder is 1/2 the size of the larger cylinder, so Large Cylinder – Small Cylinder = Size of Small Cylinder. I have updated the answer explanation.

Tahnk you for all the wonderful explanations and answers

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Strengthen your foundations in geometry with angles, triangles, and polygons.

Triangles and Hexagons

Get started by thinking about patterns that combine triangles and hexagons.

Strategic Geometry

Work out strategies to solve these area challenges.

Driving on a Polygon

Derive a fundamental theorem of geometry!

End of Unit 1

Complete all lessons above to reach this milestone.

0 of 3 lessons complete

Angle Hunting Axioms

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Advanced Mental Shortcuts

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Internal Angles in a Polygon

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Advanced Angle Hunts

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Bass Fishing

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Uh... Where did the missing square go?

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Advanced Angle Hunting

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End of Unit 6

Course description.

In this course, you'll solve delightful geometry puzzles and build a solid foundation of skills for problem-solving with angles, triangles, and polygons. You'll also improve your visual intuition and learn how to come up with clever, creative solutions to tough challenges. This course is the perfect place to start (or continue) your exploration of geometry if you know how to measure angles and calculate the areas of rectangles, circles, and triangles; and want to learn the next level of geometric problem-solving techniques. Additionally, this is a great course to take if you want to strengthen your geometric intuition in preparation for taking a geometry or design course in school. You'll also need to use a little bit of fundamentals-level algebra in this course, but nothing more advanced than two-variable equations, squares, and square roots.

Topics covered

• Angle Axioms
• Angle Hunting Shortcuts
• Composite Area
• Polygon Angles
• Regular Polygons
• Triangle Congruence
• Triangle Similarity

Prerequisites

• Geometry II
• Beautiful Geometry

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• Student Edition - Interactive Student Edition—available in digital or print write-in format.
• Student Companion - A write-in student worktext that actively engages students with lessons in the classroom or at home and fosters conceptual understanding with Habits of Mind questions. This workbook helps students solidify their understanding and record their thoughts, strategies, and understanding
• enVision A|G|A Digital - enVision digital courseware on Savvas Realize™ includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. All program resources, including personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation Students will use technology to interact with text and activities, and they can write directly in their digital Student Edition to make interaction with text more meaningful. Students will engage in activities that will inspire conceptual understanding, classroom discourse, and build their mathematical thinking skills, while learning to formulate and defend their own opinions.

Take an Interactive Tour of enVision A|G|A .

The learning model in the enVision program—problem-based learning, visual learning, and data-driven differentiated instruction—has been researched and verified as effective. Core instruction used for every lesson has been shown to be effective for developing conceptual understanding.

enVision A|G|A features comprehensive differentiated instruction and intervention support to allow access for all students. The program’s balanced instructional model provides appropriate scaffolding, differentiation, intervention, and support for a broad range of learners, and is designed to facilitate conceptual understanding of mathematics for students at a range of learning levels.

Comprehensive, built-in differentiation resources support all levels of learners, including those with learning disabilities and ELLs, through personalized, adaptive learning.

The program meets a variety of student needs and provides Response to Intervention (RtI) during each lesson, at the end of each lesson, at the end of each Topic, and any time as indicated in the Teacher’s Edition. A description of RtI tiered instructional resources for the program is included in the Teacher’s Program Overview for each grade. The following are examples of tiered instructional support found online for each lesson.

Tier 1 ongoing Intervention includes the following resources that can be used during the lesson:

• Prevent Misconceptions. During the Understand & Apply, a remediation strategy is included to address a common misconception about the lesson concept.
• Error Intervention (If... Then...). During Practice & Problem Solving, error intervention identifies a common error and provides remediation strategy
• Reteaching Set. This set is provided before independent practice to develop understanding prior to practice.
• MathXL for School: Practice & Problem Solving, during the lesson, includes personalized practice for the Practice & Problem Solving portion of the lesson, along with Additional Practice or Enrichment; auto‐scored with on‐screen help, including Help Me Solve This and View an Example tools, tutorial videos, Math Tools, and one‐click animated glossary access.

Tier 2 strategic intervention includes the following resources that can be used at the end of the Lesson:

• Reteach to Build Understanding. This provides guided reteaching as a follow‐up to the intervention activity.

Tier 3 intensive intervention instruction is delivered daily outside of the core math instruction, often in a one‐to‐one situation. Skills Review & Practice scaffolded instruction can be used for this purpose, for example. 

• Variety of Instructional Strategies
• Multisensory instruction is provided in online Explore & Reason/Model & Discuss/Critique & Explain activities that include audio and visual learning
• Virtual Nerd videos, interactive MathXL for School: Practice & Problem Solving, Additional Practice, Mixed Review, Reteach to Build Understanding, and Enrichment, online digital math tools, and embedded Desmos interactivities. example.

To learn more about the enVision program, take a look at the Overview Brochure .

The authorship team is made up of respected educational experts and researchers whose experiences working with students and study of instructional best practices have positively influenced education. Contributing to enVision with a mind to the evolving role of the teacher and with insights on how students learn in a digital age, these authors bring new ideas, innovations, and strategies that transform teaching and learning in today’s competitive and interconnected world.

Explore the enVision A|G|A authors:

• Eric Milou is a Professor in the Department of Mathematics at Rowan University in Glassboro, NJ. He is an author of Teaching Mathematics to Middle School Students. Recently, his focus has been on approaches to mathematical content and the use of technology in middle grades classrooms.
• Dan Kennedy, Ph.D is a classroom teacher and the Lupton Distinguished Professor of Mathematics at the Baylor School in Chattanooga, Tennessee. A frequent speaker at professional meetings on the subject of mathematics education reform, Dr. Kennedy has conducted more than 50 workshops and institutes for high school teachers.
• Christine D. Thomas, Ph.D is a professor of mathematics education in the Department of Middle and Secondary Education at Georgia State University. Thomas is a former high school Geometry teacher and taught for 14 years. Her research is grounded in developing, enhancing and retaining effective teachers of mathematics in urban high-need schools.
• Rose Mary Zbiek, Ph. D is a Professor of Mathematics Education at The Pennsylvania State University, College Park, PA. She is a former Pennsylvania mathematics and computer science teacher. Recent work includes theory-building research in the area of representation and models of mathematics teachers' incorporation of technology in classroom practice. She is the series editor for the National Council of Teachers of Mathematics Essential Understanding project.
• How do I sign up for an enVision digital demo? enVision digital courseware on Savvas Realize®  includes robust digital tools that give teachers flexibility to use a digital, print, or blended format in their classrooms. Teachers can customize the program to rearrange content, upload their own content, add links to online media, and edit resources and assessments. Program resources, personalized practice, remediation, and assessments are available in one location for easy lesson planning and presentation. Click here to sign up for a demo .

enVision A|G|A is designed to achieve a coherent progression of mathematical content within each course and across the program, building lesson to lesson. Every lesson includes online practice instructional examples as the progression of topics builds, allowing students additional practice with these skills and to develop a deeper conceptual understanding.

At the beginning of every topic, teachers are provided with support for the focus of the topic, how the topic fits into an overall coherence of the grade and across the grades, the balance of rigor in the topic, and how the practices enrich the mathematics in the topic. Carefully designed learning progressions achieve coherence across grades.

Coherence is supported by common elements across grades, such as Thinking Habits questions for math practices and diagrams for representing quantities in a problem. Coherence across topics within a grade is the result of developing mathematics as a body of interconnected concepts and skills. Across lessons and standards, coherence is achieved when new content is taught as an extension of prior learning–developmentally and mathematically. (For example, Model & Discuss/Explore & Reason/Critique & Explain at the start of lessons engages students in a problem-based learning experience that connects prior knowledge to new ideas and sets them up for the new concepts they will encounter in Step 2 of the lesson: Understand & Apply.)

Look Back! and Look Ahead! connections are highlighted in the Coherence part of Topic Overview pages in the Teacher’s Edition.

The Topic Background: Rigor page shows teachers how areas of rigor will be addressed in the topic, and details how conceptual understanding, procedural skill and fluency, and application builds within each topic to provide the rigor required.

On the first page of every lesson, the Lesson Overview includes sections titled Focus, Coherence, and Rigor. The Rigor section highlights the element or elements of rigor emphasized in the lesson, which may be one, two, or all three. Features in every lesson support each element, but the emphasis will vary depending on the standard being developed for conceptual understanding, procedural fluency, and application during both instruction and practice, as described below:

• Explore Step 1 Explore supports coherence by helping students connect what they already know to a problem in which new math ideas are embedded. When students make these connections, conceptual understanding emerges. Problem-based learning provides students with opportunities for productive struggle, time to make connections to the mathematical ideas and conceptual understandings. They can choose to represent their thinking and learning in a variety of ways. Online tools and manipulatives are available.
• Understand and Apply Step 2 Understand and Apply is designed to connect students’ thinking about the opening activity to the new ideas of the lesson. These concepts are presented through a series of visually rich example types purposefully designed to promote understanding. Conceptual Understanding examples present a key mathematical concept in the lesson to help students develop deep understanding of the mathematical content. Skill examples focus on helping students build fluency with skills. Finally, Application examples show students how mathematics can be used to solve real-world problems.
• Practice and Problem Solving Step 3 Practice & Problem Solving offers robust and balanced practice to solidify understanding. Students embark on a series of carefully sequenced and crafted exercises to apply what they just learned and to practice towards mastery. The design of the Practice & Problem Solving section is intentionally sequenced into four parts: Understand, Practice, Apply, and Assessment Practice.
• Assess & Differentiate Step 4 Assess & Differentiate features a Lesson Quiz and a comprehensive array of intervention, on-level, and advanced resources for all learners, with the goal that all students have the opportunity for extensive work in the state standards. Leveled practice with scaffolding is included at times. Varied problems are provided and math practices are identified as appropriate. Higher Order Thinking problems offer more challenge. Students have ample opportunity to focus on conceptual understanding and procedural skills and to apply the mathematics they just learned to solve a range of problems.

enVision A|G|A diagnostic assessments include the course readiness assessment given at the beginning of the year and the topic readiness assessment given at the beginning of a topic. Teachers use the results to develop personalized study plans for each student that are designed to address gaps in prerequisite knowledge. Students use their study plans to catch up so teachers can focus on grade-level material.

Additionally, the program offers Progress Monitoring Assessments-Form A, B, C-for each course in print and digitally.

• How does the relationship between enVision A|G|A and Desmos benefit students? Exclusive integration of Desmos into Savvas Realize® offers a groundbreaking interactive experience designed to foster conceptual understanding through highly visual interactives that bring mathematical concepts to life. Embedded interactives powered by Desmos and animated examples engage students and deepen conceptual understanding. Allowing students to manipulate data and see an immediate effect on graphs, number lines, etc. clarifies concepts as students are learning new content. Unique to enVision , the Desmos best-in-class graphing calculator and brand new geometry tools are available to middle and high school enVision students anytime, anywhere.

enVision A|G|A portrays diverse individuals and groups in a variety of settings and backgrounds. The program has been reviewed and approved for unbiased and fair representation. Our educational materials feature a fair and balanced representation of members of various cultural groups, including racial, ethnic, and religious groups; males and females; older people; and people with disabilities. The program integrates social diversity throughout all of its lessons, and includes a balanced representation of cultures and groups in multiple settings, occupations, careers, and lifestyles.

We strive to accurately portray diverse groups within our society as well as diversity within groups. Our programs use language that is appropriate to and respectful of our cultural diversity. We involve members of diverse ethnic and cultural groups in the concept development of our products as well as in the writing, editing, illustration, and design.

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