Download 12.2 Conditions for Congruent Triangles

12
Congruence and Similarity
Case Study
12.1 Introduction to Congruence
12.2 Conditions for Congruent Triangles
12.3 Introduction to Similarity
12.4 Conditions for Similar Triangles
12.5 More about the Construction of Geometric Figures
Chapter Summary
Case Study
Alan took a photo and
printed it in 2 different
sizes, 2R and 5R.
2R photo
5R photo
Length
3.5 inches
7 inches
Width
2.5 inches
5 inches
If the corresponding lengths and widths of the photos are
proportional, then they are similar.
Length of a 2R photo 3.5 1
Width of a 2R photo 2.5 1

 and


Length of a 5R photo 7
2
Width of a 5R photo 5
2
Length of a 2R photo Width of a 2R photo

∵
Length of a 5R photo Width of a 5R photo
∴ The photos are similar.
P. 2
12.1 Introduction to Congruence
A. Congruent Figures
In our daily lives, we can often see figures of the same shape and size.
In mathematics, they are called congruent figures.
Any 2 congruent figures can overlap each other.
P. 3
12.1 Introduction to Congruence
A. Congruent Figures
Consider the following transformations:
1. Translation
Shapes: the same
Sizes: the same
∴ Congruent
2. Reflection
Shapes: the same
Sizes: the same
∴ Congruent
3. Rotation
Shapes: the same
Sizes: the same
∴ Congruent
4.
Enlargement or reduction
Shapes: the same
Sizes: different
∴ Not congruent
Congruent figures have the same shape and size.
They will remain congruent after translation,
reflection and rotation.
P. 4
12.1 Introduction to Congruence
B. Congruent Triangles
2 triangles are congruent only if their shapes and sizes are the same.
Consider the congruent triangles DABC and DXYZ.
A  X
B  Y
C  Z
and
AB  XY
BC  YZ
CA  ZX
We can denote their relationship as DABC  DXYZ.
For 2 congruent triangles, the sizes of their
corresponding angles and the lengths of their
corresponding sides are equal.
‘’ means ‘is congruent
to’.
We usually list the vertices of 2 congruent triangles in corresponding order.
1. (A, X), (B, Y) and (C, Z) are 3 pairs of corresponding angles.
2. (AB, XY), (BC, YZ) and (CA, ZX) are 3 pairs of corresponding sides.
P. 5
12.1 Introduction to Congruence
B. Congruent Triangles
Example 12.1T
In the figure, DABC  DXYZ, find x and y.
Solution:
∵ C  Z
∴ C  x
In DABC,
50  70  C  180
120  x  180
x  60
∵ XZ  AC
∴
y 4
P. 6
( sum of D)
12.1 Introduction to Congruence
B. Congruent Triangles
Example 12.2T
In the figure, DABC  DXYZ, find x and y.
Solution:
∵ Y  B
∴ 3r  45
r  15
∵
AB  XY
∴ 2s + 1  5
s2
P. 7
12.2 Conditions for Congruent Triangles
To verify whether 2 triangles are congruent, do we need to test all
the pairs of angles and sides?
In Chapter 8, we have learnt the construction of triangles with
(a) 3 sides given:
the sizes of the angles are not required
(b) 2 angles and 1 side given:
the size of the other angle and the lengths of
the other 2 sides are not required
(c) 2 sides and the included angle given:
the sizes of the other 2 angles and the length
of the other side are not required
Only 3 pieces of information are required in each case.
P. 8
12.2 Conditions for Congruent Triangles
A. Three Sides Equal (SSS)
If the 3 sides of a triangle are known, then we can always draw
another triangle with the same shape and size. Hence, we obtain
the following conclusion:
If the corresponding sides of 2 triangles
are
all equal, then they are congruent triangles.
If AB  XY, BC  YZ and CA  ZX,
then DABC  DXYZ.
(Reference: SSS)
P. 9
‘SSS’ stands for
‘Side-Side-Side’.
12.2 Conditions for Congruent Triangles
A. Three Sides Equal (SSS)
Example 12.3T
Name a pair of congruent triangles in the figure
and give the reason.
Solution:
DABC  DDBC (SSS)
P. 10
12.2 Conditions for Congruent Triangles
B. Two Angles and One Side Equal (ASA or AAS)
If 2 angles and 1 side of a triangle are known, then we can always
draw another triangle with the same shape and size. Hence, we
obtain the following conclusion:
If 2 pairs of corresponding angles and a pair
of included sides of 2 triangles are equal,
then they are congruent triangles.
If A  X, AC  XZ and C  Z,
then DABC  DXYZ.
(Reference: ASA)
P. 11
‘ASA’ stands for
‘Angle-Side-Angle’.
The ‘S’ should be written
between the 2 ‘A’s to
show that it is the
included side.
12.2 Conditions for Congruent Triangles
B. Two Angles and One Side Equal (ASA or AAS)
On the other hand, if the corresponding side is not included between
the 2 angles, we obtain the following conclusion:
If 2 pairs of corresponding angles and a pair
of corresponding sides of 2 triangles are equal,
then they are congruent triangles.
If A  X, C  Z and AB  XY,
then DABC  DXYZ.
(Reference: AAS)
P. 12
‘AAS’ stands for
‘Angle-Angle-Side’.
The ‘S’ is not between
the 2 ‘A’s.
12.2 Conditions for Congruent Triangles
B. Two Angles and One Side Equal (ASA or AAS)
Example 12.4T
Name a pair of congruent triangles in the figure
and give the reason.
Solution:
DABC  DCDA (ASA)
P. 13
12.2 Conditions for Congruent Triangles
C. Two Sides and One Included Angle Equal (SAS)
If 2 sides and the included angle of a triangle are known, then we
can always draw another triangle with the same shape and size.
Hence, we obtain the following conclusion:
If 2 pairs of corresponding sides and a pair
of included angles of 2 triangles are equal,
then they are congruent triangles.
If AB  XY, B  Y and BC  YZ,
then DABC  DXYZ.
(Reference: SAS)
SSA is not a condition for congruence.
P. 14
‘SAS’ stands for
‘Side-Angle-Side’.
The ‘A’ should be written
between the 2 ‘S’s to
show that it is the
included angle.
12.2 Conditions for Congruent Triangles
C. Two Sides and One Included Angle Equal (SAS)
Example 12.5T
Name a pair of congruent triangles in the figure
and give the reason.
Solution:
DABC  DADC (SAS)
P. 15
12.2 Conditions for Congruent Triangles
D. One Right Angle, One Hypotenuse and One Side
Equal (RHS)
In a right-angled triangle, DABC, with C  90, the
longest side AB is called the hypotenuse (the side
opposite to the right angle) of the triangle.
If the hypotenuses and a pair of corresponding
sides of 2 right-angled triangles are equal, then
they are congruent triangles.
If B  Y  90, AB  XY and AC  XZ,
then DABC  DXYZ.
(Reference: RHS)
‘RHS’ stands for ‘Right
angle-Hypotenuse-Side’.
If both pairs of corresponding sides are not hypotenuse, ‘SAS’ should be used.
P. 16
12.2 Conditions for Congruent Triangles
D. One Right Angle, One Hypotenuse and One Side
Equal (RHS)
Example 12.6T
Name a pair of congruent triangles in the
figure and give the reason.
Solution:
DABE  DCDE (RHS)
P. 17
12.3 Introduction to Similarity
A. Similar Figures
In our daily lives, we can often see figures with
the same shape but with different sizes.
In mathematics, they are called similar figures.
We can obtain similar figures by enlargement or
reduction.
Similar figures have the same shape but can be different in size. They
will overlap each other after suitable enlargement or reduction.
1. Congruent figures are also similar figures.
2. 2 similar figures are still similar after translation, rotation and reflection.
P. 18
12.3 Introduction to Similarity
B. Similar Triangles
2 triangles are similar if their shapes are the same.
Consider the similar triangles DABC and DXYZ.
A  X
B  Y
C  Z
and
AB BC CA


XY YZ ZX
We say that DABC is similar to DXYZ:
DABC  DXYZ
‘’ means ‘is similar to’.
For 2 similar triangles, the sizes of the
corresponding angles are equal and the
corresponding sides are proportional.
If
AB BC CA


 1 , then the 2 triangles are congruent.
XY YZ ZX
P. 19
12.3 Introduction to Similarity
B. Similar Triangles
Example 12.7T
In the figure, DABC  DXYZ, find r and s.
Solution:
∵ Z  C
∴
r  60
YZ ZX

BC CA
s 1.5

4 3
1.5  4
s
3
s2
P. 20
12.3 Introduction to Similarity
B. Similar Triangles
Example 12.8T
In the figure, DABC  DXYZ, find r, s and f.
Solution:
In DABC,
5r  3r  20  180 ( sum of D)
8r  160
r  20
∵ Y  B
∴
s  3r
 3  20
 60
P. 21
XY
AB
8f
5
2
8f
5
f

ZX
CA
4
5
8

5
1

12.4 Conditions for Similar Triangles
If 2 triangles have 3 equal pairs of corresponding angles and
3 proportional pairs of corresponding sides, then the 2 triangles
are similar.
Like congruent triangles, we can identify 2 similar triangles with
different conditions.
3 pieces of information are required in each case.
P. 22
12.4 Conditions for Similar Triangles
A. Three Angles Equal (AAA)
If the corresponding angles of 2 triangles are
all equal, then they are similar triangles.
If A  X, B  Y and C  Z,
then DABC  DXYZ.
(Reference: AAA)
‘AAA’ stands for
‘Angle-Angle-Angle’.
Since the angle sum of any triangle is 180, 2 pairs of corresponding angles is
a sufficient condition.
For example: Consider DABC and DXYZ.
Given that A  X  60 and B  Y  50.
Then one can easily deduced that C  Z  70.
∴ DABC  DXYZ
(Reference: AAA)
P. 23
12.4 Conditions for Similar Triangles
B. Three Sides Proportional
If the corresponding sides of 2 triangles are all
proportional, then they are similar triangles.
AB BC CA


If
, then DABC  DXYZ.
XY YZ ZX
(Reference: 3 sides proportional)
If
AB BC CA


 1 , then the 2 triangles are congruent. (Reference: SSS)
XY YZ ZX
P. 24
12.4 Conditions for Similar Triangles
B. Three Sides Proportional
Example 12.9T
Name a pair of similar triangles in the figure
and give the reason.
Solution:
∵
AB BC CA


2
CD DE EC
∴
DABC ~ DCDE
P. 25
(3 sides proportional)
12.4 Conditions for Similar Triangles
C. Two Sides Proportional and One Included Angle Equal
(ratio of 2 sides, inc.  )
If 2 pairs of corresponding sides are proportional
and a pair of included angles of 2 triangles are
equal, then they are similar triangles.
If
AB CA

and A  X, then DABC  DXYZ.
XY ZX
(Reference: ratio of 2 sides, inc.  )
If
AB CA

 1, then the 2 triangles are congruent. (Reference: SAS)
XY ZX
P. 26
12.4 Conditions for Similar Triangles
C. Two Sides Proportional and One Included Angle Equal
(ratio of 2 sides, inc.  )
Example 12.10T
In the figure, AE  DE  3.5, AD  3, AB  AC  7
and AED  BAC.
(a) Name a pair of similar triangles in the figure and
give the reason.
(b) Find BC.
Solution:
(a) ∵
∴
AB CA

 2and CAB  DEA
EA DE
DABC ~ DEAD (ratio of 2 sides, inc.  )
(b)
BC AB

AD EA
BC 7

3 3.5
BC  6
P. 27
12.5 More about the Construction of
Geometric Figures
There are many ways to construct plane figures.
It is a good practice to use minimal tools to do the construction and
investigation.
In this section, we will construct geometric figures with a pair of
compasses and a straightedge.
A straightedge is a ruler
without markings.
P. 28
12.5 More about the Construction of
Geometric Figures
A. Angle Bisector
An angle bisector is a line segment which cuts an
angle into 2 equal halves, such as OP.
To bisect AOB:
Step 1: Draw an arc with O as the centre and an arbitrary radius
which cuts OA at X and OB at Y (OX  OY) respectively.
Step 2: Draw 2 arcs with X and Y as centres and an arbitrary radius.
Mark the point of intersection as ‘Z’ (XZ  YZ).
Step 3: Use a straightedge to join OZ. OZ is the angle
bisector of AOB (AOZ  BOZ).
P. 29
12.5 More about the Construction of
Geometric Figures
B. Perpendicular Bisector
A perpendicular bisector is a line segment which passes
through the mid-point of a line segment and is perpendicular
to it, such as PM.
To construct a perpendicular bisector:
Step 1: Draw 2 arcs with A as the centre and an
arbitrary radius on both sides of line AB.
Step 2: Draw 2 arcs with B as the centre and the
same radius as in Step 1 on both sides of
line AB, such that they meet the arcs
constructed in Step 1. Mark the points of
intersection as ‘P’ and ‘Q’ (AP  AQ  BP  BQ) separately.
Step 3: Use a straightedge to join PQ. PQ is the perpendicular
bisector of AB (AM  MB and AB  PM).
P. 30
12.5 More about the Construction of
Geometric Figures
C. Special Angles
(a) Angles 90 and 45
An angle of 90can be obtained by bisecting a straight angle (180).
Step 1: Draw a horizontal line and mark an arbitrary point O on it.
Draw 2 arcs with O as the centre and an arbitrary radius on
both sides of point O on the line. Mark the points of
intersection as ‘X’ and ‘Y’ (OX  OY) respectively.
Step 2: Draw 2 arcs with X and Y as centres and an arbitrary
radius. Mark the point of intersection as ‘Z’ (XZ  YZ).
Step 3: Use a straightedge to join OZ.
OZ is the angle bisector of XOY
(XOZ  YOZ  90).
P. 31
12.5 More about the Construction of
Geometric Figures
C. Special Angles
Angle 45 can be obtained by bisecting an angle of 90.
Step 4: Draw 2 arcs with O as the centre and an arbitrary radius
on lines OY and OZ. Mark the points of intersections as
‘P’ and ‘Q’ (OP  OQ) respectively.
Step 5: Draw 2 arcs with P and Q as centres and an arbitrary radius.
Mark the point of intersection as ‘R’ (PR  QR).
Step 6: Use a straightedge to join OR.
OR is the angle bisector of ZOY
90
(ZOR  YOR 
 45).
2
P. 32
12.5 More about the Construction of
Geometric Figures
C. Special Angles
(b) Angles 60 and 30
Each interior angle of an equilateral triangle is
60. We can obtain an angle of 60 by drawing
any 2 sides of an equilateral triangle.
All interior angles and
all sides of an equilateral
triangle are equal.
Step 1: Draw a horizontal line AB. Draw an arc
with A as the centre and an arbitrary
radius on the line. Mark the point of
intersection as ‘M’.
Step 2: Draw an arc with M as the centre and
the same radius as in Step 1. Mark the
As AM, MC and CA are
point of intersection as ‘C’
the radii, they are equal
(AM  MC  CA).
in length.
Step 3: Use a straightedge to join CA. CA is a side of the
equilateral triangle AMC with CAM  60.
P. 33
12.5 More about the Construction of
Geometric Figures
C. Special Angles
An angle of 30 can be obtained by bisecting an angle of 60.
Step 4: Draw 2 arcs with A as the centre and an arbitrary radius
on lines AC and AM. Mark the points of intersection as
‘P’ and ‘Q’ (AP  AQ) respectively.
Step 5: Draw 2 arcs with P and Q as centres
and an arbitrary radius. Mark the point
of intersection as ‘R’ (PR  QR).
Step 6: Use a straightedge to join AR.
AR is the angle bisector of CAM
60
(CAM  RAM 
 30).
2
P. 34
With the help of special
angles, we can construct
other angles such as
75 ( 30  45) or
105 ( 45  60) .
Chapter Summary
12.1 Introduction to Congruence
Congruent figures have the same shape and size. They will remain
congruent after translation, reflection and rotation.
For 2 congruent triangles, the size of their corresponding angles and
the length of their corresponding sides are equal.
P. 35
Chapter Summary
12.2 Conditions for Congruent Triangles
1. SSS
2. ASA
3. AAS
4. SAS
5. RHS
P. 36
Chapter Summary
12.3 Introduction to Similarity
Similar figures have the same shape but can be different in size.
They will overlap each other after suitable enlargement or reduction.
For 2 similar triangles, the size of their corresponding angles and
the corresponding sides are proportional.
P. 37
Chapter Summary
12.4 Conditions for Similar Triangles
1. AAA
2. 3 sides proportional
3. ratio of 2 sides, inc. ∠
P. 38
Chapter Summary
12.5 More about the Construction of Geometric Figures
Using a pair of compasses and a straightedge, we can construct
angle bisectors, perpendicular bisectors and special angles such as
90, 60, 45 and 30.
P. 39