geometry short course

8/8/2019 Geometry Short Course

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Principles f rom Pat t erns: Geomet ry - 1


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2 - Corners t one Curr iculum - David Quine - 20 0 2


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Chapt er 1: A Port ion of a Line

Observing t he Idea

1. Darken t he port ion of t he line from B and C.

2. Darken t he port ion of t he line from A and B.

3. Darken t he port ion of t he line from C and D.

4. Describe what you have done in problems 1 t hrough 3.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

. . . . --A B C D E.

. . . . --A B C D E.

. . . . --A B C D E.


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4 - Cornerst one Curr iculum - David Quine - 20 0 2

Naming t he Idea

The port ion of t he line t hat you darkened is called a

LINE SEGMENT.

It is simply a piece of t he line.It has t wo ENDPOINTS. The LINE SEGMENT is named by it s ENDPOINTS.

When t he symbol is writ t en above t he t wo let t ers naming t he line segment ,it refers t o t he endpoint s and all t he point s bet ween t he endpoint s.

If t he t wo endpoint s do not include t he line above, t hen it refers t o t he dist ance bet ween t het wo endpoint s.

For example, t he name of t he line segment in problem number 1 is BC.The dist ance from B t o C is writ t en BC.

5. Name t he line segment s for problems 2 and 3 .

a. Line Segment for 2: ____________

b. Line Segment for 3 : ____________

Expanding t he Idea Part 1

Finding t he Dist ance of a Line Segment

6 . The lengt h of AB is __________ .

7. The lengt h of AC is __________ .

8 . The lengt h of BE is __________ .

9 . The lengt h of CE is __________ .

. . . . --A B C D E.


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Principles fr om Pat t erns: Geomet ry - 5

10 . The lengt h of FH is found by 126 - 122 =

11. The lengt h of GJ is found by _______________________ .

12. The lengt h of FJ is found by _______________________ .

13. What mat hemat ical operat ion is used t o det ermine t he dist ance of a line segment ?

Addit ion? Subt ract ion? Mult iplicat ion? Division?

14. The diagram above shows a line segment AB. The numerical value of endpoint B is repre-sent ed by t he lower case let t er b. The numerical value of t he endpoint A is represent ed by t he lowercase let t er a. The equat ion used t o det ermine t he lengt h of line segment AB is

AB = ________________

Each point on a line has exact ly one real number associat ed wit h it .This number is called t heCOORDINATE.

The dist ance bet ween t wo point s is det ermined by subt ract ing t he smaller coordinat e from t helarger coordinat e. In t he above example, t he dist ance bet ween A and B is

AB = b - a

F G H I J

122 124 126 128 130

. . --A B

a b

. . . . -- .


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6 - Cornerst one Curriculum - David Quine - 2 0 0 2

10 15 20 25 30

. . . --A B C

. . . .--

.F G H I J

a b c


Finding t he Dist ance bet ween t wo or more Line Segment s

15. The lengt h of FH is equal t o FG + GH which is equal t o (15 - 10 ) + (20 - 15) =

16 . The lengt h of GI is equal t o _____ + _____ which is equal t o __________ .

17. The lengt h of HJ is equal t o _____ + _____ which is equal t o __________ .

18 . The lengt h of FI is equal t o _____ + _____ + _____ which is equal t o __________ .

19 . What mat hemat ical operat ion is used t o det ermine t he dist ance bet weent wo or more line segment ?

Addit ion? Subt ract ion? Mult iplicat ion? Division?

20 . The diagram above shows t hree line segment s: AB, BC, and AC.Writ e t he general equat ion expressing t he lengt h of line segment AC:

AC = _______ + _______


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Finding t he Dist ance Half Way Bet ween Endpoint s

21. What is t he point halfway bet ween t he line segment FJ? __________ .

The dist ance from F t o H is equal t o t he dist ance from H t o J.

22. What is t he point halfway bet ween t he line segment GI __________ .

23 . Explain how t he point halfway bet ween t he endpoint s can be found.

24 . B is halfway bet ween AC. That means t hat AB = BC.Writ e a mat h equat ion showing t he relat ionship bet ween AB and AC :

AB = ______________________

The point halfway bet ween t he t wo endpoint s of a line segment is called t he

MIDPOINT.

The midpoint is equal t o one-half t he dist ance of t he line segment .The midpoint can be det ermined by using eit her one of t wo met hods:METHOD 1:Find one - half of t he line segment and move t hat many spaces f rom eit her endpoint .METHOD 2:Find t he average of t he values for t he t wo endpoint s and divide by 2.

. . . --A B C

10 15 20 25 30. . . .--

.F G H I J


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CHAPTER 2: Shapes

Observing t he Idea

1. Describe each of t he following shapes.Give t hree t o four charact erist ics of each shape.


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Which shapes have t he same charact erist ics? How are t hey alike? How are t hese shapes dif ferencefr om t he ot hers? If you were t o place t hese shapes int o t hree diff erent groups, which shapes wouldyou place t oget her?

List t heCommon Charact erist ic. Make a sket ch of t he shapes wit h t he same charact erist ics.


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Naming t he Idea

The shapes on t he previous page have cert ain charact erist ics.

The first charact erist ic is t hat all t he object s are closed shapesSecondly, not ice t hat t he one group is made up of line segment s,

while some are cont inuous.

Closed shapes wit h t hree or more sides are called

POLYGONS.

The number of s ides is anot her charact erist ic of t he shapes. Someof t he shapes have 3 sides, while ot hers have 4 s ides.

Polygons wit h t hree sides are calledTRIANGLES.

Polygons wit h four s ides are calledQUADRILATERALS.

CLOSED SHAPES

Line Segment s Cont inuouswit h t hree or more sides A CIRCLE isare called POLYGONS one example of t his kind of shape.

Three - Sided Polygon Four - Sided Polygon TRIANGLE QUADRILATERAL


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Expanding t he Idea

1. Draw t hree examples of POLYGONS:

_____________________ _____________________ _____________________

2. Det ermine if t he following shapes are POLYGONS:

3. Give t he major charact erist ics of a POLYGON.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

____________________________________________________________________________

4. Why is a circle not a polygon?_______________________________________________________________________________________________________________________________________________________

5. What is t he major charact erist ic of a circle?_______________________________________________________________________________________________________________________________________________________

YES NO YES NO YES NO

YES NO YES NO YES NO


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Chapt er 3: Circles

Observing t he Idea

Mat erials:CompassPencilRuler

Set t he compass point at point C.Set t he pencil point of t he compass at point R.

Turn t he pencil around Point C unt il you make a complet e circle.

C R


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Set t he compass point at point C.Set t he pencil point of t he compass at point R.

Turn t he pencil around Point C unt il you make a complet e circle.

How is t his circle like t he fir st circle you drew?____________________________________________________________________________________________________________________________________________________________________________________________________________________________________

How is t his circle different from t he first circle you drew?________________________________________________________________________________________________________________________________________________________

______________________________________

St at e several charact erist ics of a circle._________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

C R


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R2

R3

R6

Make line segment s f rom Point C t o each of t he point s along t he circle.

Measure each line segment (t o t he nearest inch) and record your dat a in t he chart below.

LINE SEGMENT DISTANCE IN INCHES

CR1

CR2

CR2

CR4

CR5

CR6

Compare t he lengt h of each linesegment of t his circle..

Are t hey EQUAL or NOT EQUAL?

What does t his informat ion t ell youabout t he nat ure of a circle?

R1

R4

R5


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Naming the Concept

A CIRCLE is a closed figure wit h all point s t he same dist ance from one point .That one point is called

t he CENTER of t he CIRCLE.

Any line segment t hat goes from t he cent er of t he circle t o a point on t he circle is called

t he RADIUS of t he CIRCLE.

Any line segment which passes t hrough t he cent er and has it s endpoint s on t he circle is called

t he DIAMETER of t he CIRCLE.

The diamet er is t wice as long as t he radius.

CR1

r = radius

R2

R3

d=

diam

-

eter


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Expanding t he Concept

Find t he RADIUS, and DIAMETER for each circle below.Make your measurement t o t he nearest one-half inch.

FIGURE 1 FIGURE 2

RADIUS DIAMETER

FIGURE 1

FIGURE 2

FIGURE 3

FIGURE 3

Explain t he relat ionship bet ween t he radius and diamet er of a circle.

C

C

C


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20 - Cornerst one Curriculum - David Quine - 20 0 2

Using your pencil and ruler form t he line segment s: CR1and CR2.

Set t he compass point at point C.

Set t he pencil point of t he compassat Point R1.

Turn t he pencil count erclockwisearound Point C unt il you reachPoint R2.

Using your pencil shade t he areabet ween CR1 and CR2.

Using your pencil and ruler form t he linesegment s: CR1 and CR2.


Set t he pencil point of t he com-pass at Point R1.



C R1

R2

C R1

R2


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Using your pencil and ruler form t he linesegment s: CR1 and CR2.


Set t he pencil point of t he com-pass at Point R1.



C R1

R2

C R1

R2


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Naming t he Concept

An ANGLEis f ormed when t wo line segment s or rays have a common endpoint .

The common endpoint is called t he

VERTEX.

An angle is measured in degrees. A prot ract or is t he inst rument used t o measure t he size oft he angle. In t he drawing above t he VERTEX is at A. AB and AC are t he sides of t he angle.

The symbol is used t o represent an angle.The not at ion m means t he measure of t he angle.

C R1R2


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Expanding t he Idea - Part 1Measuring an Angle

An angle is represent ed in one of t hree ways: By t he let t er of t he vert ex, By t he number or let t er locat ed in t he int erior of t he angle,

By t he t hree point s t hat make t he t riangle.

The angle in t he diagrams above could be represent ed in any of t he following ways:

SYMBOL BY The C Let t er of t he vert ex.

read angle C.

1 or c Number or let t er locat ed in t he int er ior of t he angleread angle 1 or angle c.

ACB or BCA Three point s t hat make t he angle,read angle ACB or angle BCA.

An angle is measured in degrees. A PROTRACTOR is used t o measure t he size of t he angle.Reading a Prot ract or:The numbers along t he each of t he prot ract or are t he degrees.

To measure an angle place t hecross-hair at t he vert ex .

Align t he st raight edge of t heprot ract or t o one of t he rays.

Not ice where t he ot her ray crossest he prot ract or. The number t hataligns wit h t his ray is t he measurementof t he angle in degrees.

c1

40

20

90

6 0 8 0

120

140

160

0 18 0

Cross-hair at Vert exof t he angle.

Align t his edge alongone side of t he angle.

C B C B

A A


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Measuring an Angle

6 . A = __________

Using your pencil shade Angle A.Next , using your prot ract or,measure Angle a.

7. 1 = __________

Using your pencil shade Angle 1.Next , using your prot ract or,measure Angle 1.

a

1


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Measuring an Angle

Using your prot ract or measure t he following angles:

8 . ACB = __________

Using your pencil shade Angle ACB.Next , using your prot ract or,measure Angle ACB.

9 . a = __________

Using your pencil shade Angle a.Next , using your prot ract or,measure Angle a.

C

A

B

a


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Measuring an Angle

8 . A = __________

Using your pencil shade Angle A.Next , using your prot ract or,measure Angle a.

9 . 1 = __________

Using your pencil shade Angle 1.Next , using your prot ract or,measure Angle 1.

a

1


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It is import ant t hat you become very familiar wit h t he size of cert ain angles.You must be able t o recognize t he following angles.

30 angle 45 angle 6 0 angle 90 angle 120 angle 18 0 angle

10 . Wit hout using your prot ract or, make a rough sket ch of each of t he following angles:

A. 30 angle

B. 45 angle

C. 6 0 angle

D. 90 angle

E. 120 angle

F. 18 0 angle

11. Using your prot ract or check how close your drawings are t o t he correct angle. If you aremore t han 5 degrees off, redraw t he angle.


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Expanding t he Idea - Part 2

Using your prot ract or measure ABC in Figure 1.

You should have found t hat ABC was equal t o 90 .

What is one-half of 90 ? __________

In Figure 2 const ruct DBC = 45.

This new angle split s ABC in half.

Several relat ionship result from Figure 2.

How does ABD compare t o DBC?Writ e = or in t he box.

ABD DBC

How do t he sum of t he t wo smaller angles compare wit h t he larger angle?

ABD + DBC = ___ ___ ___

The line segment DB is said t o BISECT ABC.

What are t hree charact erist ics of a line segment t hat BISECTS an angle?

B C

AFigure 1 Figure 2

B C

A


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Measure each angle.Using your prot ract or and ruler draw t he angle bisect or.Label t he bisect or CD.

Figure 3

ABC = _____

DCB = _____

Figure 4

ABC = _____

DCB = _____

C

A

B

C

A

B


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CA

B

.

C

A

B

Figure 5

ABC = _____

DCB = _____

Figure 6

ABC = _____

DCB = _____


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Principles fr om Pat t erns: Geomet ry - 3 1

Chapt er 5: More Angles

Observing t he Idea

Mat erials:PencilProt ract or

Using your pencil make line segment AB.and line segment CD.How many angles were formed?

St art ing wit h t he t op angle and moving clockwise, label t he angles 1, 2, 3 , and 4.

Measure each of t he angles. Be very precise in your measurement s.

1 = __________ Look at t he relat ionships bet ween t he four angles.What do you observe? _____________________ 2 = __________

3 = __________ Shade t he equal angles wit h t he same color.

4 = __________

.

.

..

A

C D

B

Figure 1


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32 - Corners t one Curriculum - David Quine - 2 0 0 2


St ar t ing wit h t he t op angle and moving clockwise, label t he angles 1, 2, 3, and 4.


1 = __________ Look at t he relat ionships bet ween t he four angles.

2 = __________ What do you observe? _____________________


4 = __________

.

.

..

A

C D

B

Figure 2


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St art ing wit h t he t op angle and moving clockwise, label t he angles 1, 2, 3 , and 4.


1 = __________ Look at t he relat ionships bet ween t he four angles.

2 = __________ What do you observe? _____________________


4 = __________

When t wo lines int ersect , four angles are formed.St at e t he relat ionship of angles t hat are across f rom each ot her.

.

.

.

.AC

D

B

Figure 3


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Naming t he Idea

When t wo lines int ersect , four angles are formed.The angles t hat are across f rom each ot her are called

VERTICAL ANGLES.

All ver t ical angles are EQUAL.


Place t he dat a from t he previous act ivit y on t he chart below.

Figure 1 TOTAL

1

2

3

4

Add 1 t o 2 and place t he t ot al in t he t op box labeled TOTAL.

Add 2 t o 3 and place t hat t ot al in t he middle box labeled TOTAL.

What was t he TOTAL of t he t wo angles? ___________________

Add all four angles and place t hat value in t he bot t om box labeled TOTAL.


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Figure 2 TOTAL

1

2

3

4




Add all four angles and place t hat value in t he bot t om box labeled TOTAL.

What pat t erns do you see?


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Figure 3 TOTAL

1

2

3

4




Add all four angles and place t hat value in t he bot t om box labelled TOTAL.

What pat t erns do you see?


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If you know one of t he angles, is it possible t o det ermine t he ot her t hree angles?

IF Angle 1 is equal t o 125 , predict each of t he ot her angles.

1 = 125

2 = __________

3 = __________

4 = __________

Carefully measure each of t he angles.

1 = 125

2 = __________

3 = __________

4 = __________

Was your predict ion correct ? Yes No

If not , why not ?

A

C

D

B

Figure 4

1

23

4


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38 - Corners t one Curriculum - David Quine - 20 0 2

Conclusions

When t wo lines int ersect

Four angles are formed.

Vert ical Angles are equal.

1 = 3

2 = 4

The sum of t he angles t hatare adjacent is equal t o 18 0 .

1 + 2 = 180

2 + 3 = 180

3 + 4 = 180

4 + 1 = 180

The sum of all four angles is equal t o 36 0

1 + 2 + 3 + 4 = 3 6 0

1

23

4


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Measure 1 _______________ .

Predict

2 = __________

3 = __________

4 = __________

Measure angles 2, 3, and 4.

2 = __________

3 = __________

4 = __________

Did your predict ions mat ch yourmeasurement s?If not , why not ?

A 90 angle is called a

RIGHT ANGLE.

A RIGHT ANGLE is designat ed as follows:

The t wo lines are said t o be

PERPENDICULAR.

Complet e t he chart below:

ANGLE PREDICTION ACTUAL MEASUREMENT

1 + 2 =

2 + 3 =

3 + 4 =

4 + 1 =

1 + 2 + 3 + 4 =

1

23

4


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1

2

34

Measure each angle as precisely as possible.

1 = __________

2 =

__________

3 = __________

4 = __________

5 = __________

6 = __________

7 = __________

8 = __________

PREDICT t he sum of

1 + 2 + 3 + 4 = _________

ADD t he sum of

1 + 2 + 3 + 4 = _________

Did your predict ions mat ch your measurement s? If not , why not ?

PREDICT t he sum of

1 + 2 + 3 + 4 5 + 6 + 7 + 8 = _________

ADD t he sum of

1 + 2 + 3 + 4 5 + 6 + 7 + 8 = _________

56

7

8


How many degrees are t here in a complet e circle? ___________


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C

Using your compass:Place t he compass point at Point C.Place t he pencil at Point R.

Rot at e t he pencil around Point C one complet e t urn.

What shape did you make? __________

How many degrees did your pencil t ravel? __________

Draw several line segment s t hat pass t hrough Point C and have endpoint s on t he circle.

Measure t he angles t hat are formed.

Add all t he angles t oget her.

The sum of t he angles is equal t o how many degrees? __________

The sum should have been 36 0 .

If your measurement s are not precisely 36 0 , at least t hey should have been very, very close.

The t ot al number of degrees in a circle is

3 6 0

R. .


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12

3

4

Using your knowledge of t he number of degrees in a circle, andusing your knowledge of t he symbol for a r ight angle,

PREDICT t he sum of

1 + 2 + 3 + 4 = _________

Explain your reasoning:

Measure each angle:

1 = __________

2 = __________

3 = __________

4 = __________

CALCULATE t he sum of

1 + 2 + 3 + 4 = _________



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Measure Which angles (1, 2, 3 , 4) are VERTICAL ANGLES

1 = __________ 1 and _____ 2 and _____

2 = __________ What do you know about VERTICAL ANGLES?

3 = __________

4 = __________

MeasureLine Segment AC __________

Line Segment BD __________

Since AC = BD,

THEN AB is parallel t o CD.

Measure

5 = __________

6 = __________

7 = __________

8 = __________

1 2

34

5 6

78

A B

C D

.

.

.

.

Which angles (5, 6 , 7, 8 ) are VERTICAL ANGLES?

5 and _____ 6 and _____

Connect t he t wo diagonal lines in t he above problem.


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In t he figure on t he previous page, line segment s AB and CD are parallel, andt here is a diagonal line t hat cut s across t he parallel lines.

Shade all t he angles t hat are equal t o t he BIG ANGLE, 1 , wit h one color.

Shade all t he angles t hat are equal t o t he small angle, 2 , wit h a different color.

Summarize t he relat ionships among t he BIG ANGLES:

1 = _________ = _________ = _________

Summarize t he relat ionships among t he small angles:

2 = _________ = _________ = _________

Does t his relat ionship always exist ?

Make t he following measurement s

SINCE AC = _______ BD = _______ , THEN AB is parallel t o CD.

Shade all t he angles t hat are equal t ot he BIG angle, 1 , wit h one color.

1 = ___ = ___ = ___

Shade all t he angles t hat are equal t ot he small angle, 2 , wit h a different color.

2 = ___ = ___ = ___

1 2

34

5 6

78

A B

C D

.

.

.

.


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When a diagonal line cut s across t wo parallel lines,what is t he relat ionship bet ween t he BIG ANGLES?

When a diagonal line cut s across t wo parallel lines,what is t he relat ionship bet ween t he SMALL ANGLES?

In t he figure below one angle is marked B (for BIG angle)and t he ot her angle s (for t he small angle.)

Label t he ot her angles B or s based upon t he pat t ern.

B s

_____

A B

C D

.

.

_____

_____ _____

_____ _____


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A B

C D

.

.

Line segment s AB and CD are parallel and are cut by a diagonal line.

Shade t he BIG ANGLES in one color.

Shade t he small angles in anot her color.

Wit hout measuring How do you t hink t he BIG ANGLES are relat ed? Equal or Not Equal

How do you t hink t he small angles are relat ed? Equal or Not Equal

Using your prot ract or, measure each angle.

Were t he BIG ANGLES equal? YES NO

Were t he small angles equal? YES NO


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A B

C D

.

.

Shade t he BIG ANGLES in one color.

Shade t he small angles in anot her color.

Wit hout measuring How do you t hink t he BIG ANGLES are relat ed? Equal or Not Equal

How do you t hink t he small angles are relat ed? Equal or Not Equal

Using your prot ract or, measure each angle.

Were t he BIG ANGLES equal? YES NO

Were t he small angles equal? YES NO

.

.


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In t he diagram below lines l1and l

2are parallel and are cut by a diagonal line.

Of t he eight angles formed only one angle is known.Is it possible t o det ermine all t he angles knowing t he one angle?

Use what you know about BIG ANGLES and small angles , about t he degrees of a st raightline, and t he degrees in a circle t o predict all t he unknown angles.

PREDICT each angle:

a = __________

b = __________

c = __________

d = 45

e = __________

f = __________

g = __________

h = __________

MEASURE each angle:

a = __________

b = __________

c = __________

d = 45

e = __________

f = __________

g = __________

h = __________

Were your predict ions correct or incorrect ?

d = 45

a b

c

f

hg

e

l1

l2


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In t he diagram below lines l1and l

2are parallel and are cut by a diagonal line.

Of t he eight angles formed only one angle is known.Is it possible t o det ermine all t he angles knowing t he one angle?

Use what you know about BIG ANGLES and small angles , about t he degrees of a st raightline, and t he degrees in a circle t o predict all t he unknown angles.

PREDICT each angle:

SINCE a + b = __________

and since b = 80

THEN a = __________

MEASURE

a = __________

Was your predict ion correct ?

YES NO

How are b and c relat ed? _____________

SINCE b = 80 ,

THEN c = __________ .

PREDICTSINCE b + d = __________

and since b = 80

THEN d = __________

MEASURE

d = __________


YES NO

d

a b = 80

c

f

hg

e

l1

l2


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PREDICTSINCE b and c are small angles (or VERTICAL ANGLES),and since b = 8 0

THEN c = __________

MEASURE

c = __________


YES NO

PREDICTSINCE c and f are small angles (or VERTICAL ANGLES),and since c = 8 0

THEN f = __________

MEASURE

f = __________


YES NO

PREDICTSINCE f and g are small angles (or VERTICAL ANGLES),and since f = 8 0

THEN g = __________

MEASURE

g = __________


YES NO


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The diagram below is of t wo parallel lines cut by a diagonal line.Of t he eight angles formed only one angle is known.Is it possible t o det ermine all t he angles knowing t he one angle?

PREDICT

h = __________

Explain how you made t his predict ion.On what basis do you t hink so?

MEASURE

h = __________

d

a = 70 b

c

f

hg

e

The diagram below is of t wo parallel lines cut by a diagonal line.Of t he eight angles formed only one angle is known.

Is it possible t o det ermine all t he angles knowing t he one angle?

PREDICT

b = __________

Explain how you made t his predict ion.On what basis do you t hink so?

MEASURE

b = __________

a = 120

b = _______

l1

l2

l1


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Shade all t he angles in t his diagram which are equal t o 35 .

Explain how t hese angles are relat ed t o one anot heron t he diagram in t erms are t heir relat ive posit ions.

Shade all t he angles in t his diagram which are equal t o 145.

Explain how t hese angles are relat ed t o one anot heron t he diagram in t erms are t heir relat ive posit ions.

35

145

l1

l2

l1

l2


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When t wo angles are across from one anot her like t his t hey are equal.Angles which are opposit e one anot her in t his way are called

VERTICAL ANGLES.All vert ical angles are equal.

Since t hese t wo angles are opposit e on anot her also

Then t hey are also VERTICAL ANGLES.

Therefore t hey t oo must be ___________ .

3535

35

35

l1

l2


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35

35

The t wo angles shaded in t his diagram are locat ed inside t he t wo parallel linesand on opposit e sides of t he diagonal line. These angles are called

OPPOSITE INTERIOR ANGLES.

All opposit e int erior angles are equal.

The t wo angles shaded in t his diagram are locat ed out side t he t wo parallel linesand on opposit e sides of t he diagonal line. These angles are called

OPPOSITE EXTERIOR ANGLES.

All opposit e ext erior angles are equal.

35

35

l1

l2

l1

l2


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How do you t hink a is relat ed t o d ?

THE RELATIONSHIP THE REASON

SINCE a = b because VERTICAL ANGLES ARE EQUAL .

SINCE b = c because OPPOSITE INTERIOR ANGLES ARE EQUAL .

THEN a = c because bot h of t hese angles are equal t o b .

SINCE c = d because VERTICAL ANGLES ARE EQUAL .

THEREFORE a = d because bot h angles are equal t o c .

MEASUREUse your prot ract or t o verify t hese relat ionships.

a = __________

b = __________

c = __________

d = __________

Do t hese measurement s confirm t hese relat ionships? YES NO

a

b

c

d

l1

l2


56/121


57/121


Chapt er 6 : Three-Sided Shapes

Observing t he Idea

Remember, closed shapes wit h t hree or more sides are called

POLYGONS.

Polygons wit h t hree sides are calledTRIANGLES.

What are some charact erist ics of t riangles?

MEASURE

Mat erials:PencilProt ract or

Using your prot ract or precisely measure t he angles of each t riangle.

Triangle 1

a = __________

b = __________

c = __________

a

b

c


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Triangle 4

a = __________

b = __________

c = __________

Triangle 5

a = __________

b = __________

c = __________

a

b

c

a

b c


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6 0 - Cornerst one Curriculum - David Quine - 20 0 2

Transfer your measurement s t o t he chart below:

Triangles

1 2 3 4 5

a

b

c

a + b + c

Charact erist ics of Triangles

A t riangle has how many sides? __________

A t riangle has how many int erior angles? __________

The some of t he int erior angles of a t riangle is __________ ?

Naming t he Idea

The word TRIANGLE means THREE ANGLES.

Every t riangle has t hree int erior angles.

Consequent ly, every t riangle has t hree sides.

The sum of t he int erior angles of a t riangle is always 18 0 .

The symbol used t o represent a t riangle is .


61/121


62/121

6 2 - Corners t one Curriculum - David Quine - 20 0 2

Lines l1

and l2are parallel.

SINCE a = d,

we can replace d wit h a, and

SINCE c = e,

we can replace e wit h c,

THEN bce becomes abc .

SINCE a + b + c = 18 0

THEREFORE t he sum of t he int erior angles of a t riangle is equal t o 18 0 !


Using what you know about angles of t riangles and st raight lines, predict t he missing angles.

ANGLEPredict ion

1. x = _____

St at e t he principle t hat youused t o make your predict ion:

Now act ually measure t he angle using your prot ract or.

ANGLE

Measurement

x = _____


If your predict ion was not correct , explain why?

a b c

ca

50 30

x =

l1

l2


63/121


30

45

y =

z

50 x =

yx

ANGLEPredict ion

2. x = _____

y = _____

St at e t he principles t hat youused t o make your predict ion:

Now act ually measure t he angles using your prot ract or.

ANGLE

Measurement s

x = _____ y = _____



ANGLE

Predict ions

3. Predict t he sum of

x + y + z = __________


Now, act ually measure t he angles using your prot ract or.

ANGLEMeasurement

x + y + z = __________


If your predict ion was not correct , explainwhy?


64/121


ANGLEPREDICTION

4. x = _____



ANGLE

Measurement

x = _____



Lines l1

and l2are parallel.

ANGLE PREDICTIONS

5. x = _____


Now, act ually measure t he angles using your prot ract or.

ANGLEMeasurement

x = _____

30

10 0

x

130

x

150

l1

l2


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Lines l1

and l2are parallel.

ANGLE PREDICTION

6 . Find x + y = _____



ANGLEMEASUREMENTS

x = _____ y = _____ x + y = _____



ANGLE

Predict ion

7. x + y less t han, equal t o, or great er t han 90 ?



ANGLEMeasurement s

x = _____ y = _____ x + y = _____


70

135 x

y

x

y

l1

l2


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Lines l1

and l2are parallel.

ANGLE PREDICTION

8 . x = _____

y = _____



ANGLEMEASUREMENT

x = _____ y = _____



110 y

xx

l1

l2


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Observe each t riangle belowWit h a ruler measure t he sides of each t riangle.The first t riangle is done for you.

1 inches

1

2

1

inch

es

12

1inches

12

Triangle 1 Triangle 2 Triangle 3


Triangle 7 Triangle 8


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Which t riangles were alike?

Triangle 1 was like _____________________________________

How were t hese t riangles alike?






69/121


Triangles can be grouped according t o t heir s ides.

A t riangle wit h all t hree sides equal in measurement is called an

EQUILATERAL TRIANGLE.

The t riangle is marked t o show t hat all t hree sides are equal.

A t riangle wit h t wo sides equal in measurement is called a

ISOSCELES TRIANGLE.

The t riangle is marked t o show t hat all t hree sides are equal.

A t riangle wit h no sides equal in measurement is called a

SCALENE TRIANGLE.


70/121


Mark t he sides which are equal in lengt h wit h t he slash marking.Ident ify each t riangle as Equilat eral, Isosceles, or Scalene.

1 inches12

1

inch

es

12

1inches

12

Triangle 1 is an

_________ t riangle.

Triangle 2 is an


Triangle 3 is an


Triangle 4 is an


Triangle 5 is an


Triangle 6 is an


Triangle 7 is an


Triangle 8 is an



71/121


Draw t hree examples of each of t he t hree kinds of t riangles.

EQUILATERAL TRIANGLES

ISOSCELES TRIANGLES

SCALENE TRIANGLES


72/121



Did you find it difficult t o const ruct any of t he t riangles on t he previous page?Const ruct ing a specific t riangle requires t hat you use your knowledge of t he charact erist ics of

each t riangle.

Rest at e t he major charact erist ic of an equilat eral t riangle:

The problem is t o const ruct an equilat eral t riangle using only a st raight edge (if you use a rulert urn it over so t hat you are unable t o read t he markings) and a compass.

Connect Point A t o Point B using your st raight edge.

Is it possible t o make line segment XY exact ly t he same lengt h as line segment AB wit houtusing a ruler t o measure t he dist ance? You may use your compass and s t raight edge.

A B

X


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Do you need some help?Using your compass measure t he dist ance bet ween Point A and Point B by placing t he endpointof t he compass on Point A and t he pencil point on POINT B.

Next , wit hout expanding t he dist ance of t he compass place t he end point of t he compass onPoint X. Now draw an arc (a par t of a circle). Every point along t he arc is equal dist ance fromPoint X which is t he same dist ance from Point X as Point B is away from Point A. You can nowuse your st raight edge t o connect Point X any where along t he arc. Mark t his as Point Y.

Next , using your ruler measure t he line segment AB and line segment XY.Are t hey equal or not equal?

Now you know how t o make a line of equal dist ance t o anot her line.Because an equilat eral t riangle is a t riangle wit h all t hree sides equal, it is now possible t o drawan equilat eral t riangle wit h only using a st raight edge and a compass. Complet e t he diagrambelow by making an equilat eral t riangle ABC.

Do you need some help?See t he direct ions on t he following page.

A B


74/121


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Given t he Point X and t he Point Y const ruct anot her equilat eral t riangle XYZ

X Y

Given t he Point D and t he Point E const ruct anot her equilat eral t riangle DEF.

X

Y


76/121


Rest at e t he major charact erist ic of an isosceles t riangle:

The problem is t o const ruct an isosceles t riangle using only a st raight edge (if you use a rulert urn it over so t hat you are unable t o read t he markings) and a compass.

Connect Point A t o Point B using your st raight edge.Let t his be t he side of t he t riangle t hat is not equal t o t he ot her t wo sidesand let t his side of t he t riangle be short er t han t he ot her t wo sides.Using your compass find Point C.

Then connect line segment s AC and CB.

A B


77/121


Do you need some help?Line segment AB is t o be short er t han line segment s AC and BCLine segment s AC and BC must be equal.Using your compass measure a dist ance great er t han line segment AB by placing t he endpointof t he compass on Point A and t he pencil point some dist ance beyond POINT B.

Moving count erclockwise draw an arc (less t han a half circle).Next , wit hout expanding or short ening t he dist ance of t he compass place t he end point of t hecompass at Point B and t he pencil point beyond Point A.Moving clockwise draw an arc (again less t han a half a circle).

Where t he t wo arc meet is Point C.Connect Point A t o Point C and connect Point C t o Point B.Wit h a ruler measure each side of t he t riangle.

Is line segment AC and line segment CBequal or not equal?

Is line segment AB less t han, great er t han, or equal t o line segment s AC and CB?

Given t he Point X and t he Point Y const ruct anot her isosceles t r iangle XYZ.Let t his be t he side of t he t riangle t hat is not equal t o t he ot her t wo sidesand let t his side of t he t riangle be longer t han t he ot her t wo sides.

Wit h a ruler measure each side of t he t riangle.

Is line segment XZ and line segment YZ equal or not equal?

Is line segment XY less t han, great er t han, or equal t o line segment s XZ and YZ?

X Y


78/121


Rest at e t he major charact erist ic of a scalene t riangle:

The problem is t o const ruct a scalene t riangle using only a st raight edge (if you use a rulert urn it over so t hat you are unable t o read t he markings) and a compass.

Connect Point A t o Point B using your st raight edge.Let line segment AB be t he longest side of t he t riangle.Using your compass f ind a Point C such t hat AC is not equal t o AB, AC is not equal t o CB, andCB is not equal t o AB.Then connect line segment s AC and CB.

Wit h a ruler measure each side of t he t riangle.

Is line segment A B and line segment A C equal or not equal?

Is line segment A B and line segment CB equal or not equal?

Is line segment AC and line segment CB equal or not equal?

A B


79/121


Rest at e t he major charact erist ic of each of t he t hree t riangles:

EQUILATERAL TRIANGLES

ISOSCELES TRIANGLES

SCALENE TRIANGLES


80/121



In addit ion t o grouping t riangles according t o t he measurement of t he lengt h of t he sides,t riangles are also grouped according t o t he measure of t heir angles.

Observe each t riangle below.Wit h a prot ract or measure t he angles of each t riangle.Record your dat a in t he chart on t he next page.




a

bc

c

a ba

bc

a

b

c

a

bc

a

b

c

a

cb

a

bc


81/121


Angle a Angle b Angle c

Triangle 1

Triangle 2

Triangle 3

Triangle 4

Triangle 5

Triangle 6

Triangle 7

Triangle 8

Place t he t riangles int o four groups based upon similar angles.

List Triangles Describe t he charact erist ic you used t ot hat go t oget her: group t he t r iangles. (HINT: Wat ch for angles t hat are

equal t o 90 , less t han 90 , or great er t han 90 .)

GROUP 1:

GROUP 2:

GROUP 3:

GROUP 4:


82/121


Just as t riangles can be grouped based upon t he lengt h of t he sides, t hey can also beclassif ied based upon t he int erior angles of t he t riangle.

A t riangle t hat has a right angle (90 ) in it s int erior is called a

RIGHT TRIANGLE.

A t riangle having an int erior angle great er t han 90 but less t han 18 0 is called an

OBTUSE TRIANGLE.

A t riangle having all int erior angles less t han 90 is called an

ACUTE TRIANGLE.

A t riangle having all int erior angles of equal measurement is called an

EQUIANGULAR TRIANGLE.

In t he chart on t he previous page, name each of t he 8 t riangles based upon t he int erior angles

of each t riangle.

Triangles may be grouped or class ified in one of t wo ways They may be grouped according t o

1 - _________________________________________________ , or

2 - _________________________________________________ .


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Special Names for Sides of Triangles

Triangle 1 is an ISOSCELES TRIANGLE.

The short side is called t he BASE.

The t wo equal sides are called t he LEGS.

Triangle 2 is a RIGHT TRIANGLE.

In a right t riangle, t he side opposit e t he right angleis called t he HYPOTENUSE.

The ot her t wo sides are called LEGS

Triangle 1

a

b c

Triangle 2

c

a b

BASE

LEG

LEG

{

}}

}

{

{LEG

LEG

HYPO

TENU

SE


84/121



85/121


Chapt er 7: The Same

Observing t he Idea

Mat erialsprot ract orruler

Using your ruler and prot ract or precisely measure t he sides and angles of t he t wo t rianglest hat are drawn below.


AB = __________ DE = __________

BC = __________ EF = __________

AC = __________ EF = __________

a = __________ d = __________

b = __________ e = __________

c = __________ f = __________

Using t racing paper, t race vvABC . Then lay it over v DEF.How are Triangles 1 and 2 alike?

Triangle 1 a

cb

Triangle 2 d

fe

B EC F

A D


86/121


Using your r uler and prot ract or precisely measure t he sides and angles of t hese t wo t rianglest hat are drawn below.


AB = __________ DE = __________

BC = __________ EF = __________

AC = __________ EF = __________

a = __________ d = __________

b = __________ e = __________

c = __________ f = __________

Make t he following comparisons: Writ e = or in each box.

AB DE

BC EF

AC EF

Using t racing paper, t race vvABC . Then lay it over v DEF.How are Triangles 3 and 4 alike?

Triangle 3 Triangle 4a

cb

d

fe

B C E F

A D

a d

b

c f


87/121


Naming t he IdeaSome t riangles are exact ly t he same size and shape. When t he t hree sides and t he t hreeangles of one t riangle have t he same measurement s as t he t hree sides and t hree angles of asecond t riangle, t he t wo t riangles are called

CONGRUENT TRIANGLES.

The symbol t o show t hat t wo t riangles are congruent is .

In t he previous problem vABC v DEF.

The part s of t he t wo t riangles t hat have t he same measurement s are called

CORRESPONDING PARTS.

It nat urally follows t hat Corresponding Part s of Congruent Triangles are equal.

To indicat e which sides are t he same , small markings are placed upon t he corresponding sides.St raight lines are used t o show congruent sides.Curved lines indicat e congruent angles.This is demonst rat ed below.

The single line crossing AB indicat es t hat it is congruent t o DE.

The double line crossing AC indicat es ______________________________________ .

The t riple line crossing BC indicat es _______________________________________ .

The single curved line indicat es t hat a is congruent t o d

The double curved line indicat es __________________________________________ .

The t riple curved line indicat es ___________________________________________

=

=


c

B

A D

a

FC

b f

d

e

E


88/121


Expanding the Idea Part 1

Is it possible t o const ruct a t riangle congruent t o a given t riangle wit hin t racing it ?

Using only a st raight edge and a compass const ruct vXYZ in such a way t hat it is

congruent t o v ABC.

Begin by const ruct ing line segment XY t hat is congruent t o line segment AB.Using your knowledge of const ruct ing lines, cont inue const ruct ing vXYZ .

Measure t he corresponding sides and angles of v ABC and vXYZ.Are t he t wo t riangles congruent ? Yes NoHow do you know?Using t racing paper, t race v ABC . Set t hat image on t op of vXYZ. Are t hey ident ical?

Do you need help making your const ruct ion?Place t he endpoint of t he compass at Point A and t he pencil point at Point B.Move t he compass point t o Point X. St rike an arc across t he line segment .

Where t he arc crosses t he line segment becomes Point Y.Move t he compass point back t o Point A and t he pencil point t o Point C.Move t he compass point back t o Point X. St rike an arc above t he line segment XY.Move t he compass point t o Point B and t he pencil point t o Point C.Move t he compass point t o Point Y. St rike an arc so t hat t his arc crosses t he ot her arc.The place where t he t wo arcs int ersect becomes Point Z.Connect Point s X and Z. Connect Point s Z and Y.Measure t he t hree corresponding sides and angles of v ABC and vXYZ.Are t he t wo t riangles congruent ? Yes No

A B

C

X


89/121


Using only a st raight edge and a compass const ruct vDEF in such a way t hat it iscongruent t o v ABC.

Begin by const ruct ing line segment XY t hat is congruent t o line segment AB.Using your knowledge of const ruct ing lines, cont inue const ruct ing vDEF .

Measure t he corresponding sides of v ABC and vDEF.Measure t he corresponding angles of v ABC and vDEF.Are t he t wo t riangles congruent ? Yes No

How do you know?Using t racing paper, t race v ABC . Set t hat image on t op of vDEF. Are t hey ident ical?

Do you need help making your const ruct ion?On t he line wit h endpoint D const ruct a segment congruent t o line segment AB.Label t he ot her endpoint E.Draw an arc wit h cent er D and radius congruent t o line segment AC.Draw an arc wit h cent er E and radius congruent t o line segment BC.

Make t he t wo arcs int ersect . Label t he point where t hey int ersect Point F.Connect DF and EF.

You now have const ruct ed vDEF t hat is congruent wit hv ABC.

A C

B


90/121



Is it always necessar y t o measure all six pairs of corresponding part s of a t riangle t odet ermine if t he t riangles are congruent ?

PREDICTIONIF each side of one t riangle is congruent t o t he corresponding side of anot her t riangle,THEN what would you predict regarding t he corresponding angles?

Equal Not Equal I Do Not Know

MEASUREUsing a ruler measure t he sides of each t riangle t o det ermine if t hey are equal.Put = or in each box below.

AB DE BC EF AC DF

The single line perpendicular t o AB and DE indicat es t hese corresponding segment s are equal.The double line perpendicular t o BC and EF indicat es t hese corresponding segment s are equal.The t riple line perpendicular t o AC and DF indicat es t hese corresponding segment s are equal.

For t he t riangles t o be congruent t he t hree corresponding angles must also be equal.

MEASURE

Measure t he corresponding angles of each t riangle t o det ermine if t hey are equal.Put = or in each box below.

a d b e c f

THEREFORE it can be concluded t hat vvABC is congruent t o v DEF.

This may be writ t en as vABC v DEF.

Triangle 1is a scalenet riangle.

c

a b

Triangle 2is a scalenet riangle.

f

d e

A B D E

F

=

C


91/121


PREDICTIONIs t his relat ionship t rue for t he ot her t ypes of t riangles?IF each side of one t riangle is congruent t o t he corresponding side of anot her t riangle,THEN what would you predict regarding t he corresponding angles?


MEASUREMeasure t he sides of each t riangle t o det ermine if t he corresponding sides are equal.Put = or in each box below.

AB DE BC EF AC DF

Mark t he corresponding sides t hat are equal as on t he previous page.

Triangle 1 is an _____________ t riangle. Triangle 2 is an _____________ t riangle.

MEASUREMeasure t he corresponding angles of each t riangle t o det ermine if t hey are equal.

a d b e c f

SINCE t he corresponding sides are equal,THEN t he corresponding angles are also equal,

THEREFORE it can be concluded t hat vvABC v DEF.

From t hese t wo examples, what pat t ern do you see?

IF each side of one t riangle is ___________________________________________

THEN t he corresponding angles are _____________________________________

THEREFORE, t he t riangles are _________________________________________ .

Triangle 1

a c

Triangle 2

A

B

C

d f

D

E

F

b e


92/121

9 2 - Corners t one Curriculum - David Quine - 2 0 0 2

TESTING YOUR IDEAIF each side of one equilat eral t riangle is congruent t o t he corresponding side of anot herequilat eral t riangle, THEN what would you predict regarding t he corresponding angles?



AB DE BC EF AC DF


Triangle 1 is an _____________ t riangle. Triangle 2 is an _____________ t riangle.

MEASUREMeasure t he corresponding angles of each t riangle t o det ermine if t hey are equal.

a d b e c f


THE IDEAIF each side of one t riangle is congruent t o t he corresponding side of anot her t riangle,THEN t he corresponding angles are also equal,THEREFORE, t he t riangles are CONGRUENT.

This is known as t heSIDE - SIDE - SIDE RULE.

Triangle 1

a c

Triangle 2

A

B

C

d f

D

E

F

b e


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APPLYING t he IDEATo det ermine if t wo t riangles are congruent is it always necessar y t o measure all t he sides andall t he angles t o see if t here are equal? Yes No I Do Not Know

IF each side of one t riangle is congruent t o t he corresponding side of anot her t riangle,THEN corresponding angles are


THEREFOR t he t wo t riangles are ____________________________ .

Det ermine if t he t wo t riangles below are congruent by measur ing t he sides.


AB DE BC EF AC DF


Triangle ABC is _______________ t o Triangle DEF

based upon t he ____________ - ____________ - ____________ Rule.

If you are uncert ain as t o whet her t he t wo t riangles are congruent ,Then measure t he t hree corresponding angles.

B C

A

E F

D



94/121


Triangle 1

A

B

C

Using t he SIDE- SIDE - SIDE Rule det ermine which t riangles below are congruent t o vv ABC.

v ABC is congruent t o ____________________________________________________ .

Triangle 4

Triangle 5

Triangle 3Triangle 2

Triangle ABC


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Measure each side of Triangle 1.Try t o const ruct a second t riangle wit h t he sides equal in lengt h t o Triangle 1,but not congruent t o Triangle 1.In ot her words, t ry t o const ruct a second t riangle in such a way t hat t he angles are not equal.

Are t he sides of Triangle 2 equal t o t he corresponding sides of Triangle 1? YES NOMeasure t he angles of Triangle 2 and Triangle 1.Are t he corresponding angles EQUAL or NOT EQUAL?

Were you able t o const ruct a second t riangle wit h corresponding sides equal t o t he first ,but t he corresponding angles not equal t o t he first ? YES NO

IF you measure t he sides of t wo t riangles, andIF t he corresponding sides are equal,THEN how are t he cor responding angles relat ed? EQUAL NOT EQUALTHEREFORE t he t wo t r iangles are _________________________ .

Bc

A

c

a

C

b



96/121


97/121



From t he previous act ivit y you saw t hat is it not always necessar y t o measure all six pairs ofcorresponding part s of t he t wo t riangles t o det ermine if t he t riangles are congruent .

PREDICTIONIF t wo sides and t he angle bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN what would you predict regarding t he t hird side and t he ot her t wo angles?


MEASUREUsing a ruler measure t he sides AB and AC t o det ermine if t hey are equal t o sides DE and DF.

Also, measure a and d t o see if t hey are equal. Mark t he t riangles appropriat ely.

Put = or in each box below.

AB DE AC DF a d

MEASUREUsing a prot ract or measure t he remaining angles of each t riangle and using a ruler measuret he lengt h of t he ot her s ide of each t riangle t o det ermine if t hey are equal.

b e c f BC EF


What t ype of t riangle is Triangle 1? _________ Triangle 2? _________

What appears t o be t he relat ionship in t his sit uat ion?

Triangle 1

c

a Triangle 2

A

B Cb

f

d

D

E F

e


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99/121


TESTING YOUR IDEAIF t wo sides and t he angle bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN what would you predict regarding t he t hird side and t he ot her t wo angles?

Equal Not Equal Not Possible t o Tell

MEASUREUsing a ruler measure t he sides AB and AC t o det ermine if t hey are equal t o sides DE and DF.Also, measure a and d t o see if t hey are equal. Mark t he t riangles appropriat ely t oindicat e t he equalit y of corresponding angles and sides.Put = or in each box below.

AB DE AC DF a d

MEASUREUsing a prot ract or measure t he remaining angles of each t riangle and using a ruler measuret he lengt h of t he ot her s ide of each t riangle t o det ermine if t hey are equal.

b e c f BC EF


THE IDEAIF t wo sides and t he angle bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN t he corresponding angles and s ide are also equal,THEREFORE, t he t riangles are CONGRUENT.

This is known as t heSIDE - ANGLE - SIDE RULE.


A

B C

D

E F

e fb c

a d


100/121

10 0 - Cornerst one Curriculum - David Quine - 2 0 0 2

APPLYING t he IDEATo det ermine if t wo t riangles are congruent is it always necessar y t o measure all t he sides andall t he angles t o see if t here are equal? Yes No I Do Not Know

IF t wo sides and t he angle bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN corresponding angles and corresponding side are


THEREFOR t he t wo t riangles are ____________________________ .

Det ermine if t he t wo t riangles below are congruent by using your knowledge of t heSIDE - ANGLE - SIDE RULE.


AB DE BC EF ABC DEF


Triangle ABC is _______________ t o Triangle DEF

based upon t he ____________ - ____________ - ____________ Rule.

If you are uncert ain as t o whet her t he t wo t riangles are congruent ,Then measure t he each corresponding side and each corresponding angle.

B C

A

E F

D



101/121


102/121

10 2 - Corners t one Curr iculum - David Quine - 20 0 2

a

A

B

b cc

C

IS IT POSSIBLE?Try t o const ruct a second t riangle wit h sides congruent t o AB and AC and t he angle bet weent he t wo sides equal t o a, but t he t hird side and t he t wo ot her angles not equal t o t hose ofTriangle 1.

Measure AB, AC and

a of Triangle 1.

Were you able t o const ruct a second t riangle not congruent t o Triangle 1?

YES NO

IF t wo sides and t he angle bet ween t hem in one t riangle are congruent t o t he corresponding

part s in anot her t riangle,THEN t he corresponding angles and s ide are also equal,THEREFORE, t he t riangles are CONGRUENT.

This is known as t he __________ - __________ - __________ RULE.



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Principles f rom Pat t erns: Geomet ry - 10 3


Is it always necessar y t o measure all six pairs of corresponding part s of a t riangle t odet ermine if t he t riangles are congruent ?

PREDICTIONIF t wo angles and t he side bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN what would you predict regarding t he ot her t wo sides and t he ot her angle?


MEASUREMeasure t he side AC and b and c t o det ermine if t hey are equal t o side DF and d and f. If t hey are equal, markv DEF t o show t hat t hey are equal.

AC DF a d c f

MEASUREUsing a prot ract or measure t he remaining angle of each t riangle and using a ruler measure t helengt h of t he ot her s ides of each t riangle t o det ermine if t hey are equal.

b e AB DE BC EF

THEREFORE it can be concluded t hat vvABC v DEF.From t hese t wo t riangles..... you can begin to formulat e a relat ionship....IF t wo angles and t he side bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN ________________________________________________ .THEREFORE, t he t riangles _______________________________ .This relat ionship would be descr ibed as

ANGLE - __________ - __________ RULE.


b

a

c

d

B

A D

FC E

c c


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10 4 - Corners t one Curr iculum - David Quine - 20 0 2

B

Triangle 1

From t he previous observat ions you saw t hat ....

IF t wo angles and t he side bet ween t hem in one t riangle are congruent t o t he correspondingpart s in anot her t riangle,THEN t he remaining sides and angles are also congruent ;THEREFORE, t he t riangles are congruent .

This relat ionship would be descr ibed asANGLE - SIDE - ANGLE RULE.

IS IT POSSIBLE?Using your compass, st raight edge and your knowledge of making const ruct ions, is it possiblet o const ruct a second t riangle (Triangle 2: vv DEF) in which t wo angles and t he side bet weent hem are congruent t o t hose of Triangle 1, but t he remaining angle and sides are notcongruent ?

YES NO I DONT KNOW

Triangle 1 Triangle 2 a = d =AC = DF = c = f =

b = e =AB = DE =BC = EF =

In my at t empt t o const ruct a second t riangle (vv DEF) in which

d = a and f = c and AC = DF

I found t hat it was (possible, or not possible) t o const ruct t he second t riangle in such a wayt hat t he t wo t riangles were not congruent . Thought t his does not prove t he relat ionship, onecan conclude t hat t he ANGLE - SIDE - ANGLE RULE will result in congruent t riangles.

c

A

c

a

Triangle 2

.

.C

D

F

b


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If a person t old you t hat If t wo angles and a s ide not bet ween t hem in one t riangle are congruent t o t hecorresponding part s in anot her t riangle, t hen t he t riangles are congruent

would you believe him or not ?

YES NO I DONT KNOW

CONSTRUCTIONUsing your compass, st raight edge and your knowledge of making const ruct ions, make a secondt riangle (Triangle 2: vv DEF) in which t wo angles and a side not bet ween t hem are congruentt o t he first t riangle.

a = d c = fAB = DE

Then det ermine if t he t wo t riangles are congruent or not .Measure t he ot her angles and sides t o det ermine if t hey are = or . Writ e = or in t he box.

b e

AC DF vv ABC is (congruent , not congruent ) t o vv DEF

BC EF

If t hey are congruent , t hen at t empt t o change t he second t riangle in such a way as t o maket hem not congruent . (However, t he original condit ion - t hat is, ANGLE - ANGLE - SIDE - must

remain congruent .) Is it possible t o change t he second t riangle, t o make it not congruent ?YES NO

Triangle 1 a

A

B

b

Triangle 2

cc .C

.D

E


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From t he diagrams below det ermine why t he pairs of t riangles are congruent .St at e t he corresponding rule regarding congruent t riangles:

RULE 1: vv ABC vv DEF by t he SIDE - SIDE - SIDE RULE

RULE 2: vv ABC vv DEF by t he__________ - __________ - __________ - RULE

=

B

c

A

c

a

C

b

E

c

D

f

d

F

e

a

A

B

b cc

d

D

cf

C E F

=


107/121




b

a

c

d

B

A D

FC E

c c

a

A

B

b cc

d

D

E

e cf

FC

=

=


108/121



From t he diagrams below det ermine why t he pairs of t riangles are congruent .St at e t he corresponding rule regarding congruent t riangles:

RULE 1: vv ABC vv DEF by t he SIDE - SIDE - SIDE RULE



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vv ABC is an isosceles t riangle.Bisect vv ABC by drawing a line segment AD such t hat BAD = DAC.

Compare vv ADB t o vv ADC .

RELATIONSHIP REASON

Wit hout measuring any angles or dist ances, On t he basis of what principle do youwould you say t hat t hese t wo t r iangles give your answer?are congruent ? YES NO

Measure t he legs and angles of each of t he t riangles.

Writ e t he mat hemat ical relat ionship

vv ADB vv ADC .

A

CB D


110/121


Two ot her relat ionships follow from vv ADB being congruent t o vv ADC .See if you can see deduce t hese relat ionships:

Number 1:If t wo sides of a t riangle are equal, t hen how are t he angles opposit e t hose sides relat ed.

Number 2:If t wo angles of a t riangle are equal, t hen how are t he sides opposit e t hese angles relat ed?

Using t he above t riangle, a rule and prot ract or det ermine if you st at e t he above t worelat ionship correct ly. If not , rest at e t he proper relat ionships.

A

CB D


111/121


vv ABC is an equilat eral t riangle.Bisect vv ABC by drawing a line segment AD such t hat BAD = DAC.

Compare vv ADB t o vv ADC .

RELATIONSHIP REASON

Wit hout measuring any angles or dist ances, On t he basis of what principle do youwould you say t hat t hese t wo t r iangles give your answer?are congruent ? YES NO

Measure t he legs and angles of each of t he t riangles.

Writ e t he mat hemat ical relat ionship

vv ADB vv ADC .

A

CB


112/121

112 - Cornerst one Curr iculum - David Quine - 2 0 0 2

Two ot her relat ionships follow from vv ADB being congruent t o vv ADC .See if you can see deduce t hese relat ionships:

Number 1:If a t riangle is equilat eral, t hen how are t he angles relat ed?

Number 2:If a t riangle is equiangular, t hen how are t he sides relat ed?

Using t he above t riangle, a rule and prot ract or det ermine if you st at e t he above t worelat ionship correct ly. If not , rest at e t he proper relat ionships.

A

CB D


113/121


Wit hout using a ruler or prot ract or det ermine t he missing sides and angles of vv ABC below:

Since

A +

B +

C = _____

and in t his t riangle

A = B = C

Then

A = AC = 6 inches

B = AB = _____ inches

C = BC = _____ inches

Therefore Therefore

vv ABC is what vv ABC is whatt ype of t riangle? t ype of t riangle?

____________ _______________ .

A

CB

6in

ches

_____

___

inch

es

________ inches


114/121

114 - Corners t one Curr iculum - David Quine - 2 0 0 2


115/121


A

B

C

D

E

Chapt er 8 : Similar But Not t he Same

Observing t he Idea

Look at t he diagram below.

Suppose t his is a map view overlooking a land and pond area.Line segment DE represent s a bridge t hat t he Boy Scout s need t oconst ruct over a small pond. Line segment s AB and DE are parallel line segment s.

However, t he Scout s do not know t he dist ance from D t o E.All t he point s are on land. Except for t he dist ance from D t o E, you have a measuring device t o

det ermine all ot her dist ances.

How could t he dist ance for t he bridge be calculat ed?Give your suggest ions:

As you look at t he diagram what shapes t o you recognize?How are t he shapes alike?

How are t he shapes dif ferent ?

BOY SCOUT BRIDGE PROBLEM


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Observe each t riangle below.Using your knowledge of t he sum of t he angles wit hin a t riangle, det ermine missing angles.Using a ruler measure t he sides of each t riangle.Look for similarit ies and dif ferences among t he four t riangles.

Descr ibe how t he four t riangles are alike:

Descr ibe how t he four t riangles are not alike:

Are t hese four t riangles congruent t r iangles? YES NO

Triangle 1

Triangle 2


90 26

26

6 4

1 inch

1/2inch

11/8 i

nches

4 inches

_______inches

______

____inc

hes

___

inch _

______inches

_____ inches

__________ inches

_____

inches

______

____inc

hes

90 26

26

6 4


117/121


Naming t he IdeaSome t riangles are exact ly t he same size and shape. When t he t hree sides and t he t hreeangles of one t riangle have t he same measurement s as t he t hree sides and t hree angles of asecond t riangle, t he t wo t riangles are called CONGRUENT TRIANGLES.

In t he four t riangles on t he previous page t he t hree angles are t he same,however, t he sides of t he t riangles are diff erent . They are similar in shape, but not exact ly alike.These t riangles are called

SIMILAR TRIANGLES.

The symbol t o show t hat t riangles are SIMILAR is .

In t he previous problem v 1 v 2 v 3 v 4 and is read

Triangle 1is SIMILAR t o Triangle 2 which is SIMILAR t o Triangle 3 which is SIMILAR t o Triangle 4.

Expanding t he IdeaLook at t he sides of each of t he four t riangles on t he previous page.Record t he measurement s of t he sides in t he chart below:

BASE SIDE HYPOTENUSETriangle 1 1inch 1/2 inch 11/8 inches

Triangle 2 4 inchesTriangle 3Triangle 4Triangle 5

Do you not ice any mat hemat ical relat ionship bet ween t he sides of t he four t riangles?If so, explain t hat relat ionship:

Look at Tr iangle 5.Is it similar t o t he ot her four t riangles? YES NOPredict t he measurement s of t he sides.

Finally, measure each s ide.Was your predict ion correct ?

YES NO

___

inches

3 inches

_____

inches

26

6 4

90

Triangle 5


118/121


Mat erialsprot ract orruler

Observe Triangle 1.Precisely measure t he sides and angles.

TRIANGLE 1:

a = __________ b = __________ c = __________

AB = __________AC = __________BC = __________

Observe Triangle 2.The dashed lines indicat e Triangle 1. DF is parallel t o AC of Triangle 1.

TRIANGLE 2:

d = __________ e = __________ f = __________

DE = __________DF = __________EF = __________

Is Triangle 1 CONGRUENTt o Triangle 2?

YES NO

Is Triangle 1 SIMILAR t o Triangle 2? YES NO

Express t he mat hemat ical relat ionship bet ween Triangle 1 and Triangle 2:

v 1 v 2

Complet e t he chart on t he following page.

Triangle 1a

cb

B C

A

Triangle 2 d

fe

EF

D


119/121


Comparing Triangle 1 and Triangle 2 on t he Charact erist ic of Angles:

TRIANGLE 1: TRIANGLE 2:

Angle Measurement Angle Measurement

a d

b e

c f

What mat hemat ical relat ionship exist s bet ween t he corresponding anglesof Triangle 1 compare wit h Triangle 2?

a d b e c f

Comparing Triangle 1 and Triangle 2 on t he Charact erist ic of Line Segment s:

TRIANGLE 1: TRIANGLE 2:Line Segment Measurement Line Segment Measurement

AB DE

AC DF

BC EF

What mat hemat ical relat ionship exist s bet ween

AB and DE? _____________ AC and DF? _____________ BC and EF? _____________

How is Triangle 2 similar t o Triangle 1?

How are t he t riangles not alike?

Which mat hemat ical relat ionship best describes t hese t wo t riangles? Circle your answer.

v 1 vv 2 v 1 v 2=


120/121


= = =

Perform t he following mat hemat ical operat ions:

AB AC BCDE DF EF

Then compare t he answers by writ ing = or in t he box.

AB AC BCDE DF EF

In general, if t wo t riangles are similar, t hen1 - all pairs of corresponding angles are equal,2 - t he corresponding sides are not equal, but2 - all rat ios of each pair of corresponding sides are equal.

Look at t he diagram below.How does Triangle ABC compare wit h Triangle DEC?AB is parallel t o DE.Use your knowledge of t riangles t o come t o your conclusion. Do not use any measur ing devices.

RELATIONSHIP REASON

1. ACB = DCE

2. CAB = __ __ __ alt ernate int er ior angles

3. ABC = __ __ __

4. The corresponding sides by observat ion.of t he t wo t rianglesare not equal.

5. Therefore,

vv ABC vv ECD

A B

C

D E


121/121

A

B

C

D

Let s ret urn t o t he Boy ScoutBridge problem.

vv ABC vv ECD

This is a map view overlooking aland and pond area. Line segment DErepresent s a bridge t hat t he BoyScout s need t o const ruct over asmall pond.

Line segment s AB and DEare parallel line segment s.

However, t he Scout s do not know

t he dist ance from D t o E. All t hepoint s are on land.

All t he dist ances can bemeasured except t he dist ance from Dt o E. You have a measuring device t odet ermine dist ances.

Knowing t he relat ionshipsbet ween similar t riangles, det erminet he lengt h of t he bridge (t he line

segment DE).

HINT In Similar Triangles

E

2miles

Boy Scout Bridge Problem

1.4miles

2.8miles

1.4miles

2.8

mile

s

geometry short course

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