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Congruence
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CONGRUENCE
By: Afida Zahara Adzkiya (IX. 4)
• All the corresponding sides are proportional, and
• All the corresponding angles are equal in measure.
Question : Are the trapezoids congruent?
Answer : 1) AB = WX, BC = XY, CD = YZ, DA = ZW, and
2) A = W, B = X, C = Y, D = Z.
Based on description above we can concluded that, both of
trapezoids are congruent.
D
A B
C
W X
YZ
A picture or scale model has the same form as theoriginal form. Corresponding parts of the pictures or scalemodel will have the same comparison with the original build.Since the corresponding sides equal then the comparison canbe made as follows.
Length of the model = Width of the model = Height of the modelLength of the original Width of the original Height of the original
• The Properties of Two Triangles
1) The three Corresponding Sides are Equal (SSS property / side, side,
side)
• Explanation:
AB = ED
BC = DF
AC = FE
A B
C
D E
F
2) The Three Corresponding Angles are Equal (AAA Property / angle, angle, angle)
Explanation:
H = K
I = L
J = M
H I
J
K L
M
3) The Two Sides and The Included Angles are Equal (SAS / side, angle, side)
Explanation:
OP = PQ (given)
OPR = QPR = 90
PR = RP (coincident)
OP
Q
R
4) One Side, Two Angles Property (angle, angle, side), (angle, side, angle), (side, angle, angle).
Explanation:
UST = SUV (alternate interior angle)
SU = US (coincident)
VSU = TUS (alternate interior angle)
S T
UV
1)
or
a
b
c
d
e
f
a / a+b = c / c+d = e / f
a / b = c / d
2)
a
b
c
d
e
e = (a . d) + (b . c)
a + b
3)
A B
D
C
AD² = BD . CD
AB² = BD. BC
AC² = CD. CB
AB. AC = BC. AD
1)
Exercise!
If AC // DE, with AC = 12 cm, DE = 6 cm,
BE = 4 cm, so the length of CE is …
A D B
C
E
Answer:
EB + CE / EB = CA / ED4 + CE / 4 = 12 / 64 + CE / 4 = 2 / 14 + CE = 8CE = 8 – 4CE = 4 cm So the length of CE is 4 cm
2) Height and the width of a tower in the photograph is 24 cm and 6 cm. If the actual width of tower is 2 m, so the actual height is?
Answer:
6 cm
24 cm
2 m
x
x / 24 = 200 / 6
x / 24 = 100 / 3
3 x = 100 . 24
3 x = 2400
x = 800 cm
x = 8 m
3)
a b
c
d
e
If bac = edc, with
ae = 12 cm, ce = 6 cm
and cd = 8 cm,
so the length of db is?
Answer:
cd / ca = ce / cb
8 / 16 = 6 / cb
1 / 2 = 6 / cb
cb = 6 . 2
cb = 12 cm
db = cb – cd
db = 12 – 8
db = 4 cm
4)
A B
CD
E
F
9 cm
15
cmAsked: Area of
ABCD…?
A E
FD 15
cm
9 cm9 cm
?
F E
BC
Answer:
CF / DA = CB / DFCF / 9 = 9 / 15CF / 9 = 3 / 55 CF = 9 . 35 CF = 27CF = 5,4 cm
Area ABCD = DC. DA
= (15 + 5.4) x 9
= 20.4 x 9
= 183.6 cm²
5)
A B
CD
EF
Known:
DF = 3
FA = 9
DC = 12
AB = 18
Asked: The length of FE?
Answer:
FE = (DF . AB) + (FA . DC)
DF + FA
FE = (3 . 18) + (9 . 12)
3 + 9
FE = 54 + 108
12
FE = 162
12
= 13. 5 cm
6)
L U
NA
K
I
12 cm
Asked: The length of AI
?
Answer:
AU² = AL² + LU²= 12² + 12²= 144 + 144= 288
AU = √288 = √144 x 2= 12 √2 cm
AI = AU – IU
AI = 12 √2 – 12
AI = (12 √2 – 2) cm
7)
A D B
C
At the figure above known
the length of AD = 9, CD =
12, the length of CB is?
I.CD² = AD . DB
12² = 9 . DB
144 = 9 . DB
DB = 144 / 9
DB = 16
II.CB² = BD . BA
= 16 . (16 + 9)= 16 . 25= 400
CB = √400= 20
8)
A B
CD (10 x + 2) cm
(18 x + 6) cm
12 cm
6 cm
In the picture above,
∆AEB congruent with
∆DEC. The value o f x
is?
E
AB / DC =
EB / ED
18 x + 6 / 10 x + 2 =
12 / 6
18 x + 6 / 10 x + 2 = 2
/ 1
18 x + 6 = 20 x + 4
18 x – 20 x = 4 – 6
-2 x = -2
x = 1
So, the value of x is 1.
9)
a
b 12
69
21
Determine the value
of a and b !
a / 6 = 21 / 9
a / 6 = 7 / 3
3 a = 7. 6
3 a = 42
a = 14
b / 21 = 12 / 14
b / 21 = 6 / 7
7 b = 6 . 21
7 b = 126
b = 18
10) A photograph measuring 24 cm and 30 cm placed in a carton. At the top, left, and
right images it remaining 3 cm. If the images and carton congruent. Find the area of
carton that is not covered by the photograph!
24
cm
30
cm
3 cm
3 cm 3 cm
24
cm
30
cm
30
cm
x
x / 30 = 30 / 24x / 30 = 5 / 44 x = 5 . 304 x = 150x = 37.5 cm
Area carton not covered:= A carton – A photograph= (30 x 37.5) – (24 x 30)= 1025 – 720= 305
THANK YOU
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