Math presentationMath presentation

About

“CONGRUENCE”

A B

C

D

K L

M

N

Look at the figure !From the figure we know if :

AB = KL A = K BC = LM and B = LCD = MN C = MDA = NK D = N

The conclusion is“If there two plane which are perfectly coincident are called two congruent figures”

Congruence of two figures

The conditions for the congruence of two figures are:1. All the corresponding sides are equal in length, and2. All the corresponding angles are equal in measure

Similarity of two figures

A B

CD

e f

ghLook at figure below,!

x

y

2x

2y

Thus, the ratios of the corresponding sides are equal :EF : AB = GH : CD = EH : AD = FG : CB = 2 : 1

From the figure

we know if :A = E = 900

B = F = 900 C = G = 900 D = H = 900

Thus, the rectangles ABCD and EFGH are similar and have

the following properties :1. All the corresponding sides are

proportional.2. All the corresponding angles are equal in

measure.

Because ABCD and EFGH are similar, we can conclude if

point 1 and 2 is “The conditions for similarity of two figures”

Corresponding parts Corresponding parts of congruent trianglesof congruent triangles

Triangles that are the same size and shape are congruent triangles.

Each triangle has three angles and three sides. If all six corresponding parts are congruent, then the triangles are congruent.

Corresponding parts of congruent triangles

A

C

B

X

Z

Y

If ΔABC is congruent to ΔXYZ , then vertices of the two triangles correspond in the same order as the

letter naming the triangles.

ΔABC = ΔXYZ~

Corresponding parts of congruent triangles

A

C

B

X

Z

Y

This correspondence of vertices can be used to name the corresponding congruent sides and angles of the two triangles.

ΔABC = ΔXYZ~

Properties of Properties of Triangle CongruenceTriangle CongruenceCongruence of triangles is reflexive, symmetric, and transitive.

REFLEXIVEREFLEXIVEK

J

L

K

J

LΔJKL = ΔJKL~~

Properties of Properties of Triangle CongruenceTriangle CongruenceCongruence of triangles is reflexive, symmetric, and transitive.

SYMMETRICSYMMETRICK

J

L

Q

P

R

If If ΔΔJKL = JKL = ΔΔPQR,PQR,

then then ΔΔPQR =PQR = ΔΔJKL.JKL.

~~

~~

Properties of Properties of Triangle CongruenceTriangle CongruenceCongruence of triangles is reflexive, symmetric, and transitive.

TRANSITIVETRANSITIVEK

J

L

Q

P

R

If If ΔΔJKL = JKL = ΔΔPQR, andPQR, and

ΔΔPQR = PQR = ΔΔXYZ, thenXYZ, then

ΔΔJKL =JKL = ΔΔXYZ.XYZ.

~~

~~

~~

Y

X

Z

Side-Side-Side (SSS)Side-Side-Side (SSS)

1. AB DE

2. BC EF

3. AC DF

ABC DEF

B

A

C

E

D

F

Side-Angle-Side (SAS)Side-Angle-Side (SAS)

1. AB DE

2. A D

3. AC DF

ABC DEF

B

A

C

E

D

F

included angle

The angle between two sides

Included AngleIncluded Angle

G I H

Name the included angle:

YE and ES

ES and YS

YS and YE

Included AngleIncluded Angle

SY

E

E

S

Y

Angle-Side-Angle (ASA)Angle-Side-Angle (ASA)

1. A D

2. AB DE

3. B E

ABC DEF

B

A

C

E

D

F

included side

The side between two angles

Included SideIncluded Side

GI HI GH

Name the included angle:

Y and E

E and S

S and Y

Included SideIncluded Side

SY

E

YE

ES

SY

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)

1. A D

2. B E

3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

side

Warning:Warning: No SSA Postulate No SSA Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

postulate!

Warning:Warning: No AAA Postulate No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT

All the corresponding sides of the two triangles are

PROPORTIONAL

A

C

B

P

R

Q

ABPQ

BCQR

ACPR

= =

Two angles of one triangle are equal in measure to

two corresponding angles of the other triangle.

A

b

c

g

h

i

A

b

c

D

E

F

An angle of one triangle is equal in measure to an angle of the other triangle, and the sides which include the equal angle of both triangles are proportional

The formulas for a right triangle with altitude on the hypotenuse

A B

C

AD2 = BD X CDAB2 = BD X BC

D

AC2 = CD X CB

The formulas for a triangleThe formulas for a triangle containing a line parallelcontaining a line parallel

to one of its sidesto one of its sides

>

>A B

C

D E

a

b

c

d

e

f

Cdca

Cecb

Deab

= =

aa+b

= Cc+d

= ef

ab

= =cd

ac

bd