geometry short course
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Principles f rom Pat t erns: Geomet ry - 1
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Principles f rom Pat t erns: Geomet ry - 3
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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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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10 15 20 25 30
. . . --A B C
. . . .--
.F G H I J
a b c
Expanding t he Idea Part 2
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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Principles fr om Pat t erns: Geomet ry - 7
Expanding t he Idea Part 3
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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Principles f rom Pat t erns: Geomet ry - 9
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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Principles f rom Pat t erns: Geomet ry - 11
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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Principles f rom Pat t erns: Geomet ry - 13
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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Principles f rom Pat t erns: Geomet ry - 17
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
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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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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 compass point at point C.
Set t he pencil point of t he com-pass at 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.
C R1
R2
C R1
R2
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Principles fr om Pat t erns: Geomet ry - 21
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 compass point at point C.
Set t he pencil point of t he com-pass at 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.
C R1
R2
C R1
R2
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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.
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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Principles fr om Pat t erns: Geomet ry - 23
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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Principles fr om Pat t erns: Geomet ry - 25
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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Principles fr om Pat t erns: Geomet ry - 27
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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Principles fr om Pat t erns: Geomet ry - 29
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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Using your pencil make line segment AB.and line segment CD.How many angles were formed?
St ar t 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.
2 = __________ What do you observe? _____________________
3 = __________ Shade t he equal angles wit h t he same color.
4 = __________
.
.
..
A
C D
B
Figure 2
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Principles fr om Pat t erns: Geomet ry - 3 3
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.
2 = __________ What do you observe? _____________________
3 = __________ Shade t he equal angles wit h t he same color.
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.
Expanding t he Idea - Part 1
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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Principles fr om Pat t erns: Geomet ry - 3 5
Place t he dat a from t he previous act ivit y on t he chart below.
Figure 2 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.
What pat t erns do you see?
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Place t he dat a from t he previous act ivit y on t he chart below.
Figure 3 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 labelled TOTAL.
What pat t erns do you see?
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Principles fr om Pat t erns: Geomet ry - 3 7
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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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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Principles fr om Pat t erns: Geomet ry - 3 9
Expanding t he Idea - Part 2
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
Did your predict ions mat ch your measurement s? If not , why not ?
How many degrees are t here in a complet e circle? ___________
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Principles fr om Pat t erns: Geomet ry - 41
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 = _________
Did your predict ions mat ch your measurement s? If not , why not ?
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Principles fr om Pat t erns: Geomet ry - 43
Expanding t he Idea - Part 3
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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Principles fr om Pat t erns: Geomet ry - 45
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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46 - Cornerst one Curriculum - David Quine - 20 0 2
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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Principles fr om Pat t erns: Geomet ry - 47
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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48 - Cornerst one Curriculum - David Quine - 20 0 2
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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Principles fr om Pat t erns: Geomet ry - 49
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 = __________
Was your predict ion correct ?
YES NO
d
a b = 80
c
f
hg
e
l1
l2
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50 - Cornerst one Curriculum - David Quine - 20 0 2
PREDICTSINCE b and c are small angles (or VERTICAL ANGLES),and since b = 8 0
THEN c = __________
MEASURE
c = __________
Was your predict ion correct ?
YES NO
PREDICTSINCE c and f are small angles (or VERTICAL ANGLES),and since c = 8 0
THEN f = __________
MEASURE
f = __________
Was your predict ion correct ?
YES NO
PREDICTSINCE f and g are small angles (or VERTICAL ANGLES),and since f = 8 0
THEN g = __________
MEASURE
g = __________
Was your predict ion correct ?
YES NO
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Principles fr om Pat t erns: Geomet ry - 5 1
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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52 - Cornerst one Curriculum - David Quine - 20 0 2
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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Principles fr om Pat t erns: Geomet ry - 53
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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54 - Cornerst one Curriculum - David Quine - 2 0 0 2
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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Principles fr om Pat t erns: Geomet ry - 55
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
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Principles fr om Pat t erns: Geomet ry - 57
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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Principles fr om Pat t erns: Geomet ry - 5 9
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 .
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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 !
Expanding t he Idea - Part 2
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 = _____
Was your predict ion correct ? Yes No
If your predict ion was not correct , explain why?
a b c
ca
50 30
x =
l1
l2
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Principles fr om Pat t erns: Geomet ry - 6 3
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 = _____
Was your predict ion correct ? Yes No
If your predict ion was not correct , explain why?
ANGLE
Predict ions
3. Predict t he sum of
x + y + z = __________
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.
ANGLEMeasurement
x + y + z = __________
Was your predict ion correct ? Yes No
If your predict ion was not correct , explainwhy?
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6 4 - Corners t one Curriculum - David Quine - 20 0 2
ANGLEPREDICTION
4. x = _____
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
x = _____
Was your predict ion correct ? Yes No
If your predict ion was not correct , explain why?
Lines l1
and l2are parallel.
ANGLE PREDICTIONS
5. x = _____
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.
ANGLEMeasurement
x = _____
30
10 0
x
130
x
150
l1
l2
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Principles fr om Pat t erns: Geomet ry - 6 5
Lines l1
and l2are parallel.
ANGLE PREDICTION
6 . Find 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.
ANGLEMEASUREMENTS
x = _____ y = _____ x + y = _____
Was your predict ion correct ? Yes No
If your predict ion was not correct , explain why?
ANGLE
Predict ion
7. x + y less t han, equal t o, or great er t han 90 ?
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.
ANGLEMeasurement s
x = _____ y = _____ x + y = _____
Was your predict ion correct ? Yes No
70
135 x
y
x
y
l1
l2
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6 6 - Corners t one Curriculum - David Quine - 20 0 2
Lines l1
and l2are parallel.
ANGLE PREDICTION
8 . 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.
ANGLEMEASUREMENT
x = _____ y = _____
Was your predict ion correct ? Yes No
If your predict ion was not correct , explain why?
110 y
xx
l1
l2
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Principles fr om Pat t erns: Geomet ry - 6 7
Expanding t he Idea - Part 3
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 4 Triangle 5 Triangle 6
Triangle 7 Triangle 8
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6 8 - Corners t one Curriculum - David Quine - 20 0 2
Which t riangles were alike?
Triangle 1 was like _____________________________________
How were t hese t riangles alike?
Triangle 2 was like _____________________________________
How were t hese t riangles alike?
Triangle 3 was like _____________________________________
How were t hese t riangles alike?
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Principles fr om Pat t erns: Geomet ry - 6 9
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.
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70 - Corners t one Curriculum - David Quine - 20 0 2
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
_________ t riangle.
Triangle 3 is an
_________ t riangle.
Triangle 4 is an
_________ t riangle.
Triangle 5 is an
_________ t riangle.
Triangle 6 is an
_________ t riangle.
Triangle 7 is an
_________ t riangle.
Triangle 8 is an
_________ t riangle.
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Principles fr om Pat t erns: Geomet ry - 71
Draw t hree examples of each of t he t hree kinds of t riangles.
EQUILATERAL TRIANGLES
ISOSCELES TRIANGLES
SCALENE TRIANGLES
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72 - Cornerst one Curr iculum - David Quine - 20 0 2
Expanding t he Idea - Part 4
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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Principles f rom Pat t erns: Geomet ry - 73
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
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Principles f rom Pat t erns: Geomet ry - 75
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
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76 - Cornerst one Curriculum - David Quine - 20 0 2
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
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Principles f rom Pat t erns: Geomet ry - 77
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
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78 - Cornerst one Curriculum - David Quine - 20 0 2
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
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Principles fr om Pat t erns: Geomet ry - 79
Rest at e t he major charact erist ic of each of t he t hree t riangles:
EQUILATERAL TRIANGLES
ISOSCELES TRIANGLES
SCALENE TRIANGLES
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Expanding t he Idea - Part 5
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.
Triangle 1 Triangle 2 Triangle 3
Triangle 4 Triangle 5 Triangle 6
Triangle 7 Triangle 8
a
bc
c
a ba
bc
a
b
c
a
bc
a
b
c
a
cb
a
bc
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Principles fr om Pat t erns: Geomet ry - 8 1
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:
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8 2 - Cornerst one Curriculum - David Quine - 20 0 2
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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Principles fr om Pat t erns: Geomet ry - 8 3
Expanding t he Idea - Part 6
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
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Principles fr om Pat t erns: Geomet ry - 8 5
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.
Triangle 1 Triangle 2
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
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8 6 - Corners t one Curriculum - David Quine - 20 0 2
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.
Triangle 3 Triangle 4
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
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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 ___________________________________________
=
=
Triangle 3 Triangle 4
c
B
A D
a
FC
b f
d
e
E
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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
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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
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Expanding the Idea Part 2
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
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Principles fr om Pat t erns: Geomet ry - 9 1
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?
Equal Not Equal I Do Not Know
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
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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?
Equal Not Equal I Do Not Know
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
THEREFORE it can be concluded t hat vvABC v DEF.
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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Principles fr om Pat t erns: Geomet ry - 9 3
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
Equal Not Equal I Do Not Know
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.
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 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
Triangle 1 Triangle 2
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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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Principles fr om Pat t erns: Geomet ry - 9 5
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
Triangle 1 Triangle 2
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Principles fr om Pat t erns: Geomet ry - 9 7
Expanding the Idea Part 3
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?
Equal Not Equal I Do Not Know
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
THEREFORE it can be concluded t hat vvABC v DEF.
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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Principles fr om Pat t erns: Geomet ry - 9 9
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
THEREFORE it can be concluded t hat vvABC v DEF.
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.
Triangle 1 Triangle 2
A
B C
D
E F
e fb c
a d
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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
Equal Not Equal I Do Not Know
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.
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 ABC DEF
Mark t he corresponding sides t hat are equal as on t he previous page.
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
Triangle 1 Triangle 2
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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.
Triangle 1 Triangle 2
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Principles f rom Pat t erns: Geomet ry - 10 3
Expanding the Idea Part 4
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?
Equal Not Equal I Do Not Know
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.
Triangle 1 Triangle 2
b
a
c
d
B
A D
FC E
c c
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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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Principles f rom Pat t erns: Geomet ry - 10 5
Expanding the Idea Part 5
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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Expanding the Idea Part 6
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
=
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Principles f rom Pat t erns: Geomet ry - 10 7
RULE 3: vv ABC vv DEF by t he__________ - __________ - __________ - RULE
RULE 4: vv ABC vv DEF by t he__________ - __________ - __________ - RULE
b
a
c
d
B
A D
FC E
c c
a
A
B
b cc
d
D
E
e cf
FC
=
=
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10 8 - Corners t one Curriculum - David Quine - 2 0 0 2
Expanding the Idea Part 7
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
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Principles f rom Pat t erns: Geomet ry - 10 9
Expanding the Idea Part 8
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
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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
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Principles f rom Pat t erns: Geomet ry - 111
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
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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
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Principles f rom Pat t erns: Geomet ry - 113
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
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Principles f rom Pat t erns: Geomet ry - 115
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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116 - Corners t one Curr iculum - David Quine - 20 0 2
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
Triangle 3 Triangle 4
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
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Principles f rom Pat t erns: Geomet ry - 117
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
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8/8/2019 Geometry Short Course
118/121
118 - Corners t one Curr iculum - David Quine - 20 0 2
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
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8/8/2019 Geometry Short Course
119/121
Principles f rom Pat t erns: Geomet ry - 119
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=
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8/8/2019 Geometry Short Course
120/121
120 - Cornerst one Curr iculum - David Quine - 20 0 2
= = =
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
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8/8/2019 Geometry Short Course
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
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