Math 416Math 416Math 416Math 416

Geometry Isometries Geometry Isometries

Topics Covered• 1) Congruent Orientation – Parallel Path• 2) Isometry • 3) Congruent Relation• 4) Geometric Characteristic of Isometry • 5) Composite • 6)Geometry Properties • 7) Pythagoras – 30 - 60

Congruent Figures• Any two figures that are equal in every

aspect are said to be congruent • Equal is every aspect means…

– All corresponding angles– All corresponding side lengths– Areas – Perimeters

Congruent Figures• We also note that we are talking

about any figures in the plane not just triangles

• However, it seems in most geometry settings, we deal with triangles

• We hope this section will allow you to look at all shapes but…

Orientation• One of the most important

characteristics of shapes in the plane is its orientation

• How the shape is oriented means the order that corresponding point appear

Orientation • Consider

#1 #2

A A’

B C B’ C’

Orientation• To establish the order of the points, we

need two things;• #1) A starting point – that is a

corresponding point • #2) A direction – to establish order• “Consistency is the core of mathematics” • I will choose A and A’ as my starting points

Orientation • I will choose

counterclockwise as my direction

• Hence in triangle #1 we have A -> B > C.

• In Triangle #2 we have A’ -> B’ -> C’

Orientation • Since the corresponding points

match, we say the two figures have the same orientation.

• Consider A A’

#1 #2

B C C’ B’

Orientation Vocabulary• These figures do not have the

same orientation• Same orientation can be phrased

as follows; – orientation is preserved

- orientation is unchanged- orientation is constant

Orientation Vocabulary• Different Orientation can be stated

- orientation is not preserved- orientation is changed- orientation is not constant

Parallel Path• We are interested how one congruent

figure gets to the other• We are interested how one congruent

figure is transformed into another• We call the line joining corresponding points its path

• i.e. A A’ is the path• If we look at all the paths between corresponding

points, we can determine if all the paths are parallel.

Examples

These are a parallel path

A

C’

C

B’

B

A’

Examples

A

BC

B’

A’

C’

These are not parallel paths

It is called Intersecting Paths

Types of Isometries• There are 4 Isometries

1) Translation2) Rotation3) Reflection4) Glide Reflection

Translation• Translation – moving points of a

figure represented by the letter t.

• As you may recall t (-2,4) (x – 2, y + 4) You move on the x axis minus 2 and on the y axis you move plus 4.

Rotations• Rotations: Rotations can be either

90, 180, 270 or 360 degrees. • Rotations can be clockwise or

counter-clockwise• Represented by the letter r

Reflection• You can have reflections of x• You can have a reflections of y

Glide Reflection• Glide reflection occurs when the

orientation is not preserved AND does not have a parallel path.

• Can be best seen with examples…

Tree Diagram• We can define the four isometries

by the way of these two characteristics

Orientation Same? Parallel Path?

YES

YES

YES

No

No

No

TRANSLATION

ROTATION

REFLECTION

GLIDE REFLECTION

Table RepresentationOrientation Same (maintained)

Orientation Different (changed)

With Parallel Path

Translation Reflection

Without Parallel Path

Rotation Glide Reflection

Notes• The biggest problem is establishing

corresponding points.

• It is easy when they tell you AA’, BB’ but it is usually not the case

• Let’s try two examples… what kind of isometric figures are these…

• You may choose to cut up the figure on a piece of paper which can help locate the points…

Example #1• Consider (we assume they are

congruent)

• We need to establish the points. Look for clues (bigger, 90 and smaller angle).

90°

90°Bigger Angle

Bigger Angle

Smaller Angle

Smaller Angle

Which Isometric Figure?

• Hence orientation ABC A’C’B’ are NOT the same…

• Parallel paths… No!A

C’

C

B’

B

A’

GLIDE REFLECTION

ORIENTATION? PARALLEL PATH?

Example #2 A

B’

CB

C’

A’

ABC and A’B’C’ – Orientation the same

ORIENTATION? PARALLEL PATHS?

Not Parallel Paths

ROTATION

Other Figures • When the figure is NOT a triangle,

you can usually get away with just checking three points. The hard part is finding them. Let’s take a look at two more examples

Example with a Square

°

°

B

C

B’

C’

A’

A

Orientation / Parallel Paths?

Orientation Changed, Not Parallel

Glide Reflectio

n

Practice

°°

Orientation? Parallel?

Orientation Same; Not Parallel Rotation

90o counter clockwi

se rotatio

n

The Congruency Relation

• When we know two shapes are congruent (equal), we use the symbol.

CongruentSymbol

             

Congruency Relation• Hence if we say HGIJ KLMN• We note • H corresponds to K• G corresponds to L• I corresponds to M• J corresponds to N

Congruency Relation• From this we state the following

equalities. • Line length• HG = KL (1st two)• GI = LM (second two)• IJ = MN (last two)• HJ = KN (outside two)

Congruency Relation• Angles• < HGI = < KLM (1st two)• < GIJ = < LMN (second two)• < IJH = < MNK (last two first)• < JHG = < NKL (last one 1st two)• We have established all this

without seeing the figure!

Exam Question• State the single isometry.

State the congruency relation and the resulting equalities.

A

DC

B K

L N

M

Hence BACD KMNL

Exam Question• We can also can note that…• B K• D L• C N• A M Clockwise• Orientation / Parallel Path?

Exam Solution• Orientation Changed• Parallel Path• Reflection

Other Findings• Line Length• BA = KM• AC = MN• CD = NL• DB = LK

• Angles• < BAC = < KMN• < ACD = < MNL• < CDB = < NLK• < DBA = < LKM

Test QuestionGiven ABCDE FGHIJ

True or False?

•You should draw a diagram to clarify…

False

A

D C

B

IGJ

H

F

< ABC = HIJ

E

< ABC = HGF True

BC = HI False

Two Isometries – Double the fun!

• At certain points, we may impose more than one isometry.

• Consider 1 2

We say 1 2 is a reflection of s

3

Math

#

1Math

#1

Math #1

2 3 is a rotation r

Notes• We would say that the composite

is

r ° s after

We can say there is a rotation after a

reflection. So you should read from right to left

Notes• We also note that 1 – 3 is a glide

reflection (gr)

• Hence r ° s = gr

Practice• Consider

1 2 3

1 2 t

2 r

Thus r ° t = r

Math is fun

Math

is

fu

n

Math is fun

Geometry RemindersComplimentary Angles

• Here are some reminders of things you should know.

ba

Complimentary angles add up to 90o. Thus <a +

<b = 90o

Supplementary Angles

a b

Supplementary angles add up to 180o. All straight lines form an angles of 180o. Thus

<a + < b = 180o

Vertically Opposite Angles

a

bd

c

Vertically opposite

angles are equal. Thus <a = <c and

<b = <d

Isoscelles Triangles

The angles opposite the equal sides are equal or vice versa

x x

Angles in a Triangle

a

b c

Angles in a triangle

add up to 180o. Thus <a + <b + <c = 180o.

Parallel Lines

a bc d

xw

y z

When a line (transversal) crosses two

parallel lines, four angles are created at each

line

Transversal Line

Parallel Lines• The following relationship between

each group is created.• Alternate Angles

- both inside (between lines) & the opposite side of tranversal are EQUAL.

Thus, < c = < x < d = < w

a bc d

xw

y z

Corresponding Angles• Both same side of tranversal one

between parallel lines the other outside parallel lines are EQUAL

• <a = <w• <c <y• <b = < x• <d = <z

a bc d

xw

y z

e

<b & <e are called alternate interior

angle

Supplemental Angles• Both same side of transversal • Both between parallel lines• Add up to 180°• Therefore, <c + <w = 180° • <d + <x = 180°

Practice

5x+35

2x + 92

A

D F

C

G

H

B

We note < DEB = < ABG

(corresponding)

<DEB = <HEF (vertical)

E

5x+35=2x+92

3x = 57

X = 19

130

130

A

D F

C

G

H

B

50130

Solution

Replace x = 19 into 5x+355(19) +

35

= 130

Test Question• What is the angle < ABC?• 5x + 3 + 2x - 20 + x + 5 = 180• 8x -12 = 180• 8x = 192• x = 24• Replace x = 24 into 2x – 20• 2 (24) – 20• = 28°

5x+3

2x-20

x+5

A

C

B

Pythagoras Theorem• The most famous and most used

theorem or geometric / algebraic relationship is Pythagoras Theorum

• In words – the square of the hypotenuse is equal to the sum of the square on the of the other two sides

Pythagoras Example• Which of these numbers (3,4,5) mustbe the hypotenuse? Establish 90°

5 3 4• Does the placement of the 3, 4 or 5 make a

difference? • Formula c2 = a2 + b2

• Have one unknown. Solve and switch for practice

Pythagoras in Geometry

• If we have a right angle triangle with a 30° (or a 60°)

• The side opposite the 30° angle is half the hypotenuse

• Or.. the hypotenuse is twice the

side opposite the 30° angle

Practice

½x x

30°

Hence if the hypotenuse is 8, x

= ?x = 4

or 2x

x30°

Practice

5 x60° x = ?

x = 10y

y = ?

102 = 52 + y2

100 = 25 + y2

75 = y2

y =8.66