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Name ____________________ Date ____ _

• Check for Understanding

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Explore the Building Blocks of Geometry: Investigation 1

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Use the clues to create a geometric diagram in y

the coordinate plane.

• PQ intersects line II at R. l l

• M, R, and rare collinear. f X

• T lies on 0R. - 4 t 4

, Q is coplanar, but not coUinear, with R and T

• R( I I½)

Which of the following is another name for LVW.X? X

Select all that apply.

A. LW B. LWVX

C. LXVW D. L2

E. LXWV F. LU\fWX V

Identify each statement as either true or false.

A. Through any three collinear points, there is exactly one plane. __

B. 'Through any two points, there is exactly one line. __

C. A point is an exact location that has no size or dimension. __

D. A circle is the set of all points in a plane that are the same distance from a given point called the center of the circle. __

4. Why is precision in notation, descriptions, and drawings important in the preparation of a mathematical argument?

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@'~l!l Check for Understanding -~ Explore the Building Blocks of Geometry: Investigation 2 ~~; ·1he figures below show the first three figures in a pattern.

3 3 ~a I. What is the perimeter of the fifth figure in this pattern?

2. Make a conjecture about the perimeter of the 11111 figure in the pattern.

3. Give a counterexample for each statement.

A. All numbers that eml in 3 are prime numbers.

B. lf three points are in the same plane, they are collinear.

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1

1

3

.,__ ___ ...

4. How would you explain the difference between a postulate, a conjecture, and a theorem?

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• 4 I Foundation~ of Geometry

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Explore the Building Blocks of Geometry

Practice Exercises .R

. . wl,at you've learned using these practice problems For t· cv1ew • • · prac 1cc roblems with feedback, try the Coach and Play items m the Practice section onli.ie.

Which statements accurately describe Euclidean l. gt'Omctry? Select all that apply.

A. Parallel lines stay the ~amc distance apart and never intersect.

B. A p0int has lt·ngth and width.

C. A line segment is determined by its end

points.

D. An angle consists of two rays that share an end point.

2. In the diagram, C is a point on AB.

I A

I C

I B

If AC= 2.x- 13 centimeters, BC= 3x + 4 centimeters, and AB = 36 centimeters, what is the

• length of BC?

BC= _____ cm

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3. If LGFH measures 35° and LGFJ measures 85°, what is the measure of LHFJ?

F J

A. 35°

B. 40°

C. 45°

D. 50°

E. 120°

4. Exammc the angl , r I h . . . cs 1orn1e< y mtersectmg rays 111 the diagram Wh· h · 1c statement is not I rul' ahout the relat · •h· . h . ions 1ps among the angles in t c diagram?

p w s

A. mLRWT + mLTWS = mL RWS

B. mLTWQ - mLTWR = mLRWQ

C. mLPWR + mLRWS = mLPWS

D. mLPWT - mLQTW = mLPWQ

5. If Dis between Band C, which statement is true?

A. BC+DB = DC

B. CD-DB =CB

C. CB - CD= DB

D. CD+CB=DB

6. Three points are shown.

He

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In total, how many lines can oe drawn through any two or more of the given points?

·1 here are _____ line!, that (an oe drawn through two or more of the point,//, /, and/.

Explore the Building Blocks of Geometry I 9

txp1ore the Building Blocks of G eometry

. e Exercises (continued) pract1c

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11 show~ two 111tt'rsccting lmt>s. I ht' thaµr .11 •

Given that L I, L2, L3, and L4 are formed by the intcrsection of two lines, what conclusions can you draw and why? Select all that apply.

A. mL l+mL2+mL3+mL4 =~~ because LI and L2 are a linear pair and L3 and L4 are a linear pair.

B. mL I = mL2 , because the sum of mL I + mL2 equals the sum of mL3 + mL4. It then follows by the subtraction property of equality that mLI = mL2.

C. mL I = mL3, because the sum of mL I + mL2 equals the sum of mL3 + mL2. It then follows by the subtraction property of equality that mLI = rnL3.

D. 180 - mL4 = mL2 + mL3, because L2 and L3 are a linear pair and, therefore, supplementary.

8. If Mis the midpoint of AB, what is AB?

A M B

(3• • 8) cm (5x - 1) cm

AB = _____ cm

10 I Foundations of Geometry

9. How many Ii . b . . nt:s can c drawn through pomt K, perpendicular to line HJ?

A. 0

B.

C. 2

H

K

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D. any number oflincs

J

10. If Mis the midpoint of AB, AB = 6x + 40, MB = 2x + 30, find x, AB, and MB.

X= -----AR = ____ _

MB= ____ _

11. In the diagram, Mis the midpoint of AB.

A M B

Support the following argument for AM= MB by writing the reason that validates each statement.

A. Definition of midpoint

B. Definition of congruent segments

C. Given information

We know that Mis the midpoint of AB. __

This means that AM= MB. __

Therefore, AM= MB. --

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'-i~\TH •:>iscovery TECHBOOK f OUCAT IO N Explore the Building Blocks of Geometry

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l UNIT 1: Foundations of Geometry

1.1 Explore the Building Blocks of Geometry

Lesson Objectives • Interpret the axioms and

undefined terms of geometry.

• Use undefined terms and axioms to formally define ray, angle, circle, and line segment.

• Differentiate between conjectures, theorems, and axioms.

• Verify that a statement is false using a counterexample.

Essential Question • What are the fundamental parts

of geometry, and how can they be used to create valid arguments?

Investigations The Case for Geometry What makes a strong argument? Geometric proofs are the mathematician's version of making a case.

Research and Analyze Evidence Take on the role of a geometry lawyer. Collect pertinent evidence and present your findings.

An Educated Guess Look for patterns and use them to make conjectures. Do you have enough evidence to build a theorem?

Explore Axioms What past rulings can you use to build your case? Create Exhibit C to show the evidence.

Theorems on Trial Prove your first case! Analyze and defend court briefs. Present evidence to prove a theorem.

Key Vocabulary angle, axiom, circle, collinear, complementary angles, congruent, conjecture, counterexample, distance, intersection, line, line segment, linear pair, midpoint, parallel, perpendicular, plane, point, postulate, radius, ray, skew lines, supplementary angles, theorem, undefined term

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Discover As rou ..:ompletc Engage and the investigations, recor<l the most important ideas you've lcarn<.'<l. •

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[!)~[!) Investigation 1 g Investigation 2

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• Ii Investigation 3 a Investigation 4

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Explore the Building Blocks of Geometry

Name ______________________ Date _____ _

• Check for Understanding Explore the Building Blocks of Geometry: Investigation 3

You are given the following clue~:

• M, N, and Pare three collinear points .

• MP >MNand MP>NP .

• MP = 18 - 3.Sx, MN = 7.74, and NP=- l.9x.

1. Draw a diagram to represent this situation and label each segment with the given expressions .

• 2. Determine the values of x and MP.

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3. Miguela is creating a geometric design by combining different polygons. How does the angle addition postulate relate to Miguela's design? Include at least one specific example in your explanation .

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•)1scovery •.• ~ · · ◄ , D u C A T 1 0 N TECHBOOK

Explore the Building Blocks of Geometry

Name _____________________ _ Date ____ ~

• Check for Understanding Explore the Building Blocks of Geometry: Investigation 4

I . r n the figurr, mLAOC = mLflOlJ. Lisa claimed that hascJ on thr given information, mLAOB = mLCOD. Rrmemhcr that to prove a con;ccturc is true hcyond a doubt, you can only use evidence, or n .. -.1~011~. in the form of givens, definitions, postulates, properties, and previously proved theorems. Examine the argument Lisa presented. Cirdl! the reason Lisa used for each statement of her defense.

Statement Reason

I. I know that mLAOC = mL BOD. A. ang le a<l<lit 1011 postulate 8. given C. definition of LAOD

A le addition postulate 2. I can rewrite the angle measures as . ang 8. substitution mLAOC = mLAOB + mLBOC and C. definition of LAOD mLBOD = m L BOC + mLCOD.

3. In other words, my original statement becomes A. definition of LAOV B. symmetry property mLAOB + mLBOC : mL BOC + mLCOD. c. substitution

4. Notice that both sides of the equation mclude A. substitution 8. reflexive property mLBOC. I know that mL BOC = mLBOC. C. symmetric property

5. I can conclude that mLAOB = mLCOD. A. given 8. angk addition postulate c. subtraction property

2. In the figure, Mis the midpoint of AC and BD, _andhAfiC = :eD. Make a con1·ecture about the different segments mt _e gu .

' · tur s correct? How would you convince someone your conJec e I •

A

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C

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' • r • ~ . J,scovery T. E .. CHBOOK l:0uC AT t ON Explore the Building Blocks of Geometry

Summary (continued)

Postulates

al A p~stulate or 3;Xiom is a statement that 1s accepted m geometry without proof. In the same way that you can use undefined terms to help define other terms, you can use postulates to help prove other statements.

Unique Line Postulate

Through any two points, there is exactly one line.

Unique Plane Postulate

Through any three points not on the same line, there is exactly one plane.

Line Intersection Postulate

If two lines intersect, then their intersection is exactly one point.

Parallel Line Postulate

If there is a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

My Notes

8 I Foundations of Geometry

Plane Intersection Postulate

If two planes intersect, then their intersection is a lint'.

Segment Addition Postulate

If A, B, and Care collinear and R is between A and C. then AB+ BC= AC.

Angle Addition Postulate

If R is in the interior of L.PQS, then mL.PQR + mL.RQS = mL.PQS.

Theorems

i) In geometry, a theorem is a . statement that has been proved usmg definitions, postulates, properties, previously proven theorems, and logical reasoning.

Congruent Complements Theorem

If two angles are complementary to the same ang1e, then the angles are congruent.

Midpoint Theorem

If Mis the midpoint of AB, then - -AM~MB.

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then the angles are congruent.

Linear Pair Theorem

If two angles form a linear pair, then the angles are supplementary.

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Summary • Before you attempt the Practice Exercises, review what you've learned.

Undefined Terms For <.·xamplc. triangular numbt•rs form the pattern:

Un<ldined tcrm5 an' the most basic building blocks 1, 3, 6, I 0, 15, ... After careful ob~erva11on anJ of geometry.

• A point 1s an exact l<Xat10n that has no SIU or dimension.

• A line is a one-dimensional path that extends mfinitely in both directions.

• A plane is a flat two-duncnsional surface that extends fon.·ver.

Defined Terms

Ii} A definition in geometry gives the meaning of a term based only on commonly understood words, undefined terms, or other previously defined terms.

gc.:ncraJizations. a conJCcturl' might be that the

nth term m this pattern is l'qual 10 ,,(,~d ) . ·1 he

statement remains a conJecture until it i~ proved

true using defini tions, properties, ,lXloms, or

previously proved thrnrems .

al A countettxample is an example that is used to disprove a statement; it contradicts a statement or argument.

For example, someone.: might exam me the following partial list of primes:

3,5,7. 11 , 13

-~----------- After examining this partial list, the person might conjecture that aJI prime numbers are od<l. A

counterexample to this conjecture is the prime number 2. A single counterexample is enough to prove that a conjecture is not true.

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• A line segment is a part of a line between two points on the line called endpoints. (Notice that this definition makes use of the undefined terms point and line.)

• A distance along a line is measured between two points on the line.

• A ray is a part of a line that extends from one endpoint infinitely in one direction .

• An angle is formed by two rays with the same endpoint. The two rays are called the sides of the angle, and the shared endpoint is called the vertex.

• A circle is the set of atl points in a plane that are the same distance from a given point called the center of the circle.

Conjecture and Counterexample

Ii} A conjecture is statement based on known information or observation that is believed to be true but not yet proven.

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My Notes

Explore the Building Blocks of Geometry I 7