Congruent triangles

Congruent triangles

Season 01

Episode AP09 Time frame 1 period

Prerequisites :

^{Voabularyandnotionsabouttrianglesandangles.}

Objectives :

•

^{Disover the onept of ongruent triangles.}

•

^{Apply out the three ongruene ases toprove that some trianglesare ongruent.}

Materials :

•

^Slideshow.

•

^Lesson.

•

^Exerises.

•

^{Slideshow with the orretion of the exerises.}

1 – Lesson 15 mins

Using a slideshow, the teaher explains the onept of ongruent triangles and the three

ongruene ases. Studentsshould haveanative part tothe lesson. Atthe end, a lesson

sheet is handed out.

2 – Exercises Remaining time

Working ingroups, students have to work ona few exerisesabout ongruent triangles.

Congruent triangles

Season 01 Episode AP09 Document Lesson

Definition 1 Equal sides

Two triangles are congruent if they have three pairs of equal sides.

Proposition 1 Equal angles

Two congruent triangles have three pairs of equal angles.

b A

b

B

b C

b E

b D

b F

Toprovethat two trianglesare ongruent, one an use one of the following methods :

SSS : Prove that the triangles have three

pairs of equal sides.

SAS : Prove that the triangles have two

pairs of equal sides and the angles

between these sides equal.

ASA : Prove that the triangles have two

pairsofequalanglesandthesidesbe-

tween these sides equal.

Congruent triangles

Document Exercises

Exercise 1

Let

ABCD

be aparallelogram.

1

^{. Prove that triangles}

ABC

and

CDA

are ongruent.

2

^{. Find another pair of ongruent trianglesonthis gure.}

3

^{. Let}

O

be the intersetion point of the diagonals of

ABCD

. Find two pairs of on- gruent triangles involvingthe point

O

.Prove these properties.

Exercise 2

Let

ABC

be an isoseles triangle with main vertex

B

, and

H

the foot of the altitude through

B

. Findtwo ongruent trianglesin the gure and prove the property.

Exercise 3

Let

ABCD

be anisoseles trapezium,where

•

^{the sides}

AD

and

BC

are parallel;

•

^{the sides}

AB

and

CD

are equal;

•

^{the angles}

ABC

and

BCD

are equal.

Let

I

be the midpointof the side

BC

.

1

^{. Prove that the triangles}

ABI

and

CDI

are ongruent.

2

^{. Dedue a property about triangle}

ADI

.

Exercise 4

Let

ABC

bean equilateraltriangle,with

M

,

N

,

P

three pointsrespetively onthe sides

BC

,

CA

and

AB

suh that

BM = CN = AP

.

1

^{. Prove that the triangles}

BM P

,

CN M

and

AP N

are ongruent toone another.

2

^{. Dedue a property about triangle}

M N P

.

Congruent triangles

Season 01 Episode AP09 Document Answers

Exercise 1

1

^.

• AB = DC

beause

ABCD

isa parallelogram.

• BC = DA

beause

ABCD

isa parallelogram.

• AC = AC

beause it's a ommun side.

so the triangles

ABC

and

CDA

have three pairs of equal sides, so they areongruent.

b

A

b B

b C

b

D

b O

2

^{. The triangles}

ABD

and

BCD

areongruent too.

3

^.

• AO = OC

beause

O

is the midpoint of the diagonals.

• DO = OB

beause

O

is the midpoint of the diagonals.

• AD = BC

beause

ABCD

isa parallelogram.

so the triangles

AOD

and

COB

have three pairs ofequal sides, so they areongruent.

• AO = OC

beause

O

is the midpoint of the diagonals.

• DO = OB

beause

O

is the midpoint of the diagonals.

• AB = DC

beause

ABCD

isa parallelogram.

so the triangles

AOB

and

COD

have three pairs ofequal sides, so they areongruent.

Exercise 2

• BH = BH

beause it's a ommon side.

• AB = CB

beause

△ ABC

is isoseles.

• ∠ BAH = ∠ BCH

beause

△ ABC

is isoseles.

b

A

b

C

b B

b

H

So the triangles

ABC

and

CDA

have two pairs of equal sides and the angles between these sides equal, so they are ongruent.

Exercise 3

1

^.

• CI = BI

beause

I

isthe midpoint of

BC

.

• AB = CD

beause

ABCD

isanisoseles trapez- ium.

• ∠ ABI = ∠ I CD

beause

△ ABC

is isoseles.

So

△ ABC

and

△ CDA

havetwopairsofequalsides andthe anglesbetweenthese sidesequal,so theyare

ongruent.

b

A

b

B

b

C

b

D

b

I

2

^{. Sine the triangles}

ABC

and

CDA

are ongruent, the third sides are also equal so

AI = I D

.Consequently

△ ADI

is isoseles.

Exercise 4

1

^.

• BM = CN

aordingto the instrutions.

• BP = CM

beause

BP = side − AP = side − BM = CM

.

• ∠ P BM = ∠ M CN

beause the triangle

ABC

is equilateral.

So

△ BM P

and

△ CN M

have two pairs of equal sides and the angles between these sides equal, so

they are ongruent.

b

A

b

B

b

C

b

M

b

N

b

P 2

^{. In the same way, we provethat}

△ CN M

and

△ AP N

are ongruent.

3

^{. Wean dedue thatthe third sidesof thetriangles areequal, so}

△ M N P

is equilateral.