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Transcript
SIMILARITIES AND CONGRUENCY
Similarity ( all lengths in examples and in exercises are in cm)
Two figures having the same shape are known as similar figures. The figures A and B
show is a example of similar figure. So in short similarity is a relation between angles
and sides.
A
B
Two congruent figures must also be similar.
When a figure is enlarged or reduced, the new figure is similar to the original one.
e.g.
find out the figures similar to figure A:
A
B
C
D
SOLUTION:
D only.
Similar Triangles
If two triangles are similar, then
i. their corresponding angles are equal;
ii. their corresponding sides are proportional.
B
A
Y
C
X
Z
In the figure, if
ABC =
XYZ,
then ‹A = ‹X, ‹B = ‹Y, ‹C = ‹Z and AB = BC = AC
XY YZ XZ
Example
In the figure,
ABC is similar to
RPQ. Find the value of the unknowns.
B
39
x
A
52
z
C
Solution:
Since
ABC =
RPQ,
‹B = ‹P
So x = 90o.
Also,
AB = BC
RP PQ
Also,
39 = 58
y 48
so
AC = BC
RQ PQ
z = 52
60 48
48 * 39
52
=y
so
Y = 36
60 * 52
48
=z
z = 65
CONDITIONS FOR TRIANGLES TO BE SIMILAR
A)
THREE ANGLES EQUAL.
If two triangles have three pairs of equal corresponding angles, then they must be similar.
(Reference: AAA)
Example:
Show that
ABC and
PQR in the figures are similar.
R
A
P
75o
35o
35o
75o
B
C
Q
Solution:
In
ABC and
PQR as shown, ‹B = ‹Q, ‹C = ‹R,
‹A = 180o – 35o – 75o = 70o.
‹P = 180o – 35o – 75o = 70o
so ‹A = ‹P
therefore ABC is similar to PQR (AAA).
B)
THREE SIDES PROPORTIONAL
if the three pairs of side of two triangles are proportional, then the two triangles must be
similar.
(Reference: 3 sides proportional)
a
b
d
e
c
f
Example:
Show that
PQR and
LMN in the figure are similar.
L
3
R
Q
16
2
4
8
P
N
12
M
In
PQR and
LMN as shown, LM = 8 = 4, MN = 12 = 4, LN = 16 = 4
PQ 2
QR
3
PR 4
So LM = MN = LN
PQ QR
PR
Therefore
PQR is similar to LMN (3 sides proportional)
Example 2:
Are the two triangles in the figure similar? Give reasons.
A
X
6
2
12
4
Y
4.5
Z
C
B
9
Solution:
AB = 4 =2, AC = 12 = 2, BC = 9 = 2
XZ 2
XY 6
YZ 4.5
Yes,
ABC is proportional to
XZY as their 3 sides are proportional.
C)
TWO SIDES PROPORTIONAL AND THEIR INCLUDED
ANGLE EQUAL.
If two pairs of sides of two triangles are proportional and their includes angles are equal,
then the two triangles are similar.
(Reference: ratio of two sides, and the angle between two sides equal)
p
r
P = q, x = y
R s
s
q
Example:
Show that
ABC and
FED in the figure are similar.
D
A
4.5
E
45o
4
450
2
B
9
C
F
Solution:
‹B = ‹E‹‹‹
AB = 4 = 2, BC = 9 = 2
EF 2
DE 4.5
Therefore, AB = BC
EF DE
Therefore, ABC is similar to
sides equal)
FED (ratio of two sides, the angle between two are
Example 2:
Are the two triangles ZYX and CBA similar? Give reasons.
Solution
‹ZYX = 180o – 78o – 40o = 62o, ‹ZYX = ‹CBA = 62o.
BC = 6 = 2, AB = 4 = 2
YZ 3
XY 2
Yes,
ZYX is similar to
equal)
CBA (ratio of two sides, the angle between two are sides
Similar polygons
Two polygons are said to be similar if there is correspondence between their vertices such
that:
a. Corresponding angles are congruent
b. Lengths of corresponding sides are in proportion.
A
D
V
Y
W
B
X
C
Angle A is similar to angle V
Angle B is similar to angel W
Angle C is similar to angle X
Angel D is similar to angel Y
AB = BC = CD = DA
VW WX XY VY
If the above interpretation is accepted the polygons are similar.
Perimeter of similar polygons:
If two polygons are similar, the ratio of their perimeter equals to the ratio of the lengths
of corresponding sides.
So for the above diagram the perimeter can be found like this.
P = AB
P2 VW
AREA
A
B
W
Z
a
C
Ka
b
D
X
Kb
Y
The two figures are similar the ratio of corresponding sides as k.
Area of ABCD = ab
Area of WXYZ = Ka * Kb
Therefore K2ab = K2
ab
This a general rule for finding area for similar figures.
Exercise for similarities.
Find the sides marked with letters in the questions.
A
1)
5
D
E
a
4
C
b
6
F
B
2)
3)
16
y
x
6
3
6
L
4)
A
m
a
M
15
N
6
m
B
9
5)
X
4cm
9cm2 8cm
8cm
1.5cm
6)
Given triangle XYZ with line VW parallel to it find rations of :
Y
a) YW to WZ
7
b) ZY to WY
V
W
c) ZY to ZW
4
X
Z
CONGRUENCE
Two figures having the same shape and size are called congruent figures.
Example of congruent shapes:
C
If two figures are congruent they will fit exactly on each other
e.g.
find out the congruent figures among the following:
.
D
B
C
A
E
F
G
G
In the above diagram only figure D, E and A, G are similar.
Congruent Triangles
when two triangles are congruent, all their corresponding sides and corresponding angles
are equal.
C
E.g.
A
X
B
Z
In the figure, if
‹A = ‹X
‹B = ‹Y
‹C = ‹Z
ABC is congruent to
XYZ then,
AB = XY
and BC = YZ
CA = ZX
e.g.
name the pair of congruent triangles in the figure.
B
12
13
P
5
R
C
12
13
5
A
Q
So automatically ‹C =‹P
From the figure we see that
e.g. 2 :
given that
ABC is congruent to
RPQ.
Y
ABC is congruent to
XYZ find the unknowns p, q and r.
q
r
50o
Z
6
X
Solution
For two congruent triangles, for their corresponding sides and angles are equal.
Therefore p = 6, q = 5 and r = 50o.
CONDITINS FOR TRIANGLES TO BE
CONGRUENT
A)
THREE SIDES EQUAL.
If AB = XY, BC = YZ and CA = ZX then the triangles are congruent.
Y
A
X
B
C
Z
Reference: SSS. It means when three sides of one triangle is equal to the three sides of
the other triangle, the two triangles are congruent.
E.g.
5
Determine which of the pairs of triangles in the following triangles.
8
3
4
5
7
(i)
(ii)
7
5
4
5
3
8
(iv)
(iii)
Solution:
(i)
(ii)
and (iv) are a pair of congruent triangles,
and (iii) are a pair of congruent triangles.
Example 2
Each of these pairs of triangles are congruent. Which of them are congruent because
of SSS?
Solution
B
B)
TWO SIDES EQUAL AND THEIR INCLUDED ANGLE EQUAL
If AB = XY, ‹B = ‹Y and BC = YC then
B
ABC is congruent to
XYZ
Y
A
C
X
Z
Reference: SAS. This means that if two sides of one triangle are equal to two sides of
the other triangle and the angle between this pair of sides is the same in both
triangles, then the two triangles are congruent.
Example 1)
Determine which of the following pairs are congruent
(i)
(ii)
(iii)
7
6
50o
5
50o
6
3
7
1200
(iv)
3
5
120o
Solution
In the figure because of SAS,
(i)
and (iii) are congruent triangles
(ii)
and (iv) are congruent triangles
example 2
in each of the following, equal sides and equal angles are indicated with the same
markings. Write down a pair of congruent triangles, and give reasons.
A)
A
B
D
A
C
B
C
Solution
D
E
ABC is congruent to
ACB iss congruent to
CDA (SSS)
ECD (SAS)
C)
TWO ANGLES AND ONE SIDE EQUAL.
If ‹A = ‹X, AB = XY and ‹B= ‹Y, then
ABC is congruent to
XYZ
Reference: ASA. It means that if two angles of one triangle are equal to two
corresponding angles of another triangles and the side between each pair of angles is the
same length in both triangles, then the two triangles are congruent.
And even if
‹A= ‹X, ‹B = ‹Y and BC = YZ then,
ABC is congruent to
XYZ.
Reference: AAS
Determine which pairs of triangles in the following are congruent:
(i)
(ii)
(iii)
30o
30o
7
(iv)
1350
70o
7
8
45o
8
1350
70
0
45o
Solution
In this figure because of ASA
(i)
and (iv) are congruent triangles.
(ii)
and (iii) are congruent triangles.
Example 2
In the figure, equal angles are indicated with the same markings. Write down a pair of
congruent triangles, and give reasons.
Solution
ABD is congruent to
ACD.
Example 2
Determine which pairs of triangles in the following are congruent
(i)
(ii)
(iii)
(iv)
45o
4
3
4
o
45
3
60o
100o
100o
55o
Solution
In the figure, because AAS
(i) and (ii) are a pair of congruent triangles
(iii) and (iv) are another pair of congruent triangles.
60o
55o
Example 2
In the figure, equal angles are indicated with the same marking. Write down a pair of
congruent triangles, and give reasons.
Solution
ABD is congruent to
CBD (AAS)
D)
TWO RIGHT ANGLED TRIANGLES WITH EQUAL HYPOTHESIS AND
ANOTHER PAIR OF EQUAL SIZES.
A
C
X
B
Z
‹C = ‹Z = 90o, AB = XY and BC = YZ, then
Y
ACB is congruent to
XZY.
Reference: RHS
Example 1:
In the figure, ‹DAB and ‹BCD are both right angles and AD = BC. Judge whether
ABD and
CBD are congruent, and give reasons for your answer.
A
B
Solution
Yes,
ABD is congruent to
CBD (RHS)
Congruency questions
1)Say whether the triangles are congruent or not. If they are give a reason.
a)
3cm
85o
b)
450
3cm
45o
85o
5cm
500
3cm
50o
3cm
5cm
c)
3cm
d)
3cm
80o
SIMILAR 3D SHAPES
o
60similar
Volume of
3d objects.
7cm
3 cm
o
40
When objects are similar, and is an enlargement of another. If the two are similar and the
3. 3cm
ratio of corresponding
side is k, the ratio
4cm
4cmof volume will be k
e.g.
6
V=70cm3
8
v= X
7cm
70 = 6
V2
8
3
70 = 216
V2 512
V2= 512 * 70
216
V2 = 166 cm3
Exercise
1) Find out the missing volume
V
200
5
10
Answers
Similarities
1) a = 2.5, b = 3
2) c = 16
3) x = 12, y = 8
4) m = 10, a = 6.6.
5) x = 64cm2
6) a) 7
4
b) 11
7
C) 11
4
Congruency
1) a as AA corresponding S
c as SAS
d as RHS
3D shapes
1) 62.5
ASSESSMENT - SIMILARITY AND CONGRUENCY
Writing Drawin Formatti Polygo
skill
g skill
ng skill
n
Triangle
3
2
3
2
2
creativit picture
page
always- theore
y flow
s
paragrap similar
ms
neatnes labelin
hs math
polygo
applns
s
g
equation
ns
s
appln
1
0
1
1
2
3D
shapes
3
perimet
er
vol, SA
2
Question
s
3
3D
shapes
congruen
cy
variety
2
Exampl
es
2
ex+ans
1
20
10