cogruence
TRANSCRIPT
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