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CONGRUENT TRIANGLES 466
a)
- Vertical translation - down 4.
- Rotation of
1
turn.
2
- Center of rotation : lower right vertex.
- Anti-clockwise rotation of
1
turn.
4
N
as
b)
- Center of rotation : lower right vertex.
- Vertical translation - down 10.
- Reflection in the vertical axis.
c)
- Rotation of
1
turn.
2
- Center of rotation : vertex of the concave angle of the figure.
- Vertical translation - up 6.
- Translation of 10 to the left.
d)
- Counter-clockwise rotation of
1
turn.
4
- Center of rotation : vertex of the concave angle of the figure.
- Reflection in a horizontal axis.
e)
s
- Translation of 8 to the right.
s.
1
- The figures are not congruent.
D
3
B
4
C
5
The centre C of rotation is located at the point of intersection of the right bisectors of the line segments joining
corresponding vertices.
N
as
s
2
Right bisector
1
C
s.
Right bisector
2
Work : (example)
X
r
T
B
Y
N
as
A
C
s
r
1. Rotate the cupboard -90 using centre X to lower the cupboard.
2. Perform translation T along the floor.
3. Rotate the cupboard 90 using centre Y to place the cupboard upright.
s.
6
Work : (example)
Statements
1
2
DEF  ABC
A composite of isometries is an isometry.
m ABC = 180  (90 + 15)
= 180  105
The sum of the interior angles of a triangle is 180.
N
as
= 75
3
Justifications
s
7
m DEF = m ABC = 75
In congruent triangles, the corresponding angles are
congruent.
Other justifications may be accepted, provided the meaning has not been altered.
Result
D
s.
8
The measure of angle DEF is 75.
9
Work : (example)
s
C
N
as
t
s
A
B
Result
Figure A is mapped onto figure B by a composite of
- a reflexion and a translation, t  s.
or
- a translation and a reflexion s  t.
s.
Also accept "glide reflection".
10
B
11
Work : (example)
B
A
s
C
N
as
D
STATEMENTS
JUSTIFICATIONS
1.
Segments AD and BC are congruent.
The opposite sides of a rectangle are congruent.
2.
Segments AB and DC are congruent.
The opposite sides of a rectangle are congruent.
3.
Diagonal AC is common to both triangles
ABC and CDA.
By construction
4.
Triangles ABC and CDA are congruent.
If three sides of one triangle are congruent to
three sides of another triangle, then the
triangles are congruent.
or
s.
SSS
12
C
13
A
14
Work : (example)
B
s
A
N
as
C
D
Statements
E
Justifications
1.
AC  CE
2.
BC  CD
3.
ACB  DCE
Vertically opposite angles are congruent.
4.
ABC  CDE
If two sides and the contained angle of one triangle are
congruent to two sides and the contained angle of another
triangle, then the triangles are congruent.
Point C is midpoint of segment AE.
s.
Point C is midpoint of segment BD.
or
SAS
15
Work : (example)
B
A
s
Statements
AD  DC
2.
BD  BD
3.
4.
Justifications
1.
The hypothesis states that D is the midpoint of segment AC.
N
as
1.
2.
Reflexive property.
ADB  CDB
3.
The hypothesis states that DB  AC .
ABD  CBD
4.
If two sides and the contained angle of one triangle are
congruent to two sides and the contained angle of another
triangle, then the triangles are congruent.
or
S-A-S.
C
17
C
s.
16
C
D
B
19
a)
C
G
L
D
A
N
as
F
J
H
B
E
I
K
b) Symmetric and reflexive properties
The measures of corresponding sides of similar triangles are proportional.
s.
20
21
s
18
C
23
D
24
B
Step 3
 ABC ~  EDC
because
Step 4
if the lengths of two sides of one triangle are proportional to the lengths
of the two corresponding sides of another triangle and the contained
angles are congruent, then the triangles are similar.
 ABC   EDC
because
the corresponding angles of similar figures are congruent.
s.
25
s
A
N
as
22
Example of an appropriate method
Measure of angle JLK
s
Two vertically opposite angles are congruent.
Hence
m  JLK  m  DLH
Value of x
N
as
 75
The sum of the measures of the interior angles of a triangle is 180.
Hence
m  KJL  m  LKJ  m  JLK  180
3x   2 x   75  180
5 x   105
x  21
Measure of angle FKB
s.
26
Corresponding angles formed by two parallel lines and a transversal are congruent.
Hence
m  FKB  m  KJL
 3 x 
 321
 63
The numerical measure of angle FKB is 63.
Note:
Students who use an appropriate method in order to determine the value of x have shown that
they have a partial understanding of the problem.
s.
N
as
s
Answer:
Step 1
m  ABC + m  CBD = 180
because adjacent angles whose external sides are in a
straight line are supplementary.
m  ABC + 120 = 180
Step 2
m  BCA = m  QEP = 40
Step 3
m  ABC + m  BCA + m  BAC = 180
s
m  ABC = 60
because alternate exterior angles are congruent when
formed by a transversal intersecting two parallel lines.
N
as
60 + 40 + m  BAC = 180
m  BAC = 80
s.
27
because the sum of the measures of the
interior angles of a triangle is 180.
Example of an appropriate method
Volume of the pyramid and volume of the prism
Volume of the pyramid = Volume of the prism = 3380  2
Volume of the pyramid
N
as
1690 cm3
Volume of the prism
1690 cm3
Height of the prism
s
The volume of the pyramid is equal to that of the prism because they are equivalent.
Volume = 1690 cm3
Height  Area of the base = 1690
Height  13  13 = 1690
Height = 10 cm
s.
28
Height of the pyramid
Volume = 1690 cm3
Height  Area of the base
 1690
3
Height  13  13
 1690
3
Height = 30 cm
Total height of the trophy
Height of the trophy = Height of the prism + Height of the pyramid
The total height of this trophy is 40 cm.
N
as
Answer:
s
10 + 30 = 40 cm
Note: Students who use an appropriate method in order to determine the height of the prism or the height
of the pyramid have shown that they have a partial understanding of the problem.
B
s.
29
Example of an appropriate method
Measure of angle BPC
m BPC = m ADP = 65
s
because they are corresponding angles and AD // PB since ABPD is a parallelogram.
N
as
Value of x
The sum of the measures of the interior angles of triangle BPC is equal to 180. Hence
3x + 2x + 65 = 180
5x = 115
x = 23
Measure of angle PCB
m PCB = 2x = 2  23 = 46
Measure of angle BAP
s.
30
m BAP = m PCB = 46 because the opposite angles of a parallelogram are isometric.
Answer
The measure of angle BAP is 46.
N
as
s
Students who correctly or incorrectly determine the value of x have shown that they have a partial
understanding of the problem.
s.
Note
Example of an appropriate solution
Volume of the sphere
V 
s
4r 3
V 
3
412 3
3
N
as
V = 2304π cm3
Since the cone and the sphere are equivalent, their volumes are equal.
Height of the cone
Volume of the cone =
r 2 h
3
  212  h
2304π =
3
Therefore h = 48 cm
s.
31
Answer The height of the cone is 48 cm.
s.
s
N
as
Name : _________________________________
Group : _________________________________
568436 - Mathematics
s.
N
as
Question Booklet
s
Date : _________________________________
Identify the geometric transformations by which one figure can be applied exactly onto the other.
a) figure A onto figure G.
b) figure C onto figure L.
c) figure K onto figure E.
s
d) figure B onto figure J.
N
as
e) figure B onto figure L.
A
B
C
D
E
F
G
H
I
J
K
L
s.
1
N
as
s
When manufacturing clothing by the assembly line method, a tailor has to cut out the same piece
numerous times. The tailor, who is to cut out a piece like the one below on the left, has to cut out as
many of these pieces as possible from a square piece of cloth 100 centimetres on each side.
Which of the formulas below calculates exactly the maximum number of pieces that he can cut from this piece
of cloth?
100 100  1

20
11
A)
N
100  100
110
C)
N
B)
N
100  100
220
D)
N  2
s.
2
100 100  1

20
11
In Peru, clothing is often decorated with designs of isometric figures.
N
as
Figure 1
s
The following is one such model to be reproduced.
Lucia wants to include the above design on her clothing but drawn in the following way:
Figure 2
How can she transpose figure 1 to correspond to figure 2?
s.
3
A)
Symmetry about the vertical axis followed by a rotation of 90 in the clockwise direction.
B)
Symmetry about the vertical axis followed by a rotation of 90 counterclockwise.
Symmetry about the horizontal axis followed by a rotation of 180 clockwise.
D)
Symmetry about the horizontal axis followed by a rotation of 180 counterclockwise.
s.
N
as
s
C)
The figure below shows triangle ABC and triangle ADE. The data given on the figure can be used to prove that
these triangles are congruent.
C
B
D
65
25
s
4.5 cm
E
4.5 cm
N
as
A
Below is the reasoning which shows that triangle ABC is congruent to triangle ADE.
STATEMENT
1.
m C = m E = 90
JUSTIFICATION
1.
These data are given in the problem.
m AC  m AE  4.5 cm
m BAC = 25
m ADE = 65
2.
m DAE = 25
2.
The acute angles in a right triangle are
complementary.
3.
m BAC = m DAE
3.
4.
ABC  ADE
4.
s.
4
By transitivity
?
Which of the following is the justification for statement 4?
B)
Two triangles are congruent if they have three corresponding congruent sides.
C)
Two triangles are congruent if they have one congruent side between two corresponding
congruent angles.
D)
Two triangles are congruent if they have two corresponding congruent angles.
N
as
s
Two triangles are congruent if they have one congruent angle bounded by two
corresponding congruent sides.
The two figures below are congruent because a rotation can make figure 1 coincide with figure 2.
In the answer booklet, indicate the centre C of this rotation.
Show all your construction lines.
1
s.
5
A)
2
A cupboard has to be moved from position A to position B, passing under obstacle C during the move.
A
s
C
B
Show all your work
N
as
What geometric transformations can be used to move the cupboard from A to B?
s.
6
Triangle DEF is the image of triangle ABC under the translation t followed by a rotation r of 70 around point D.
The measure of angle BAC is 15.
E
s
F
B
N
as
15
A
D
C
t
Find the measure of angle DEF and state the properties that justify each step of your work.
Show your work.
s.
7
8
y
In the adjacent Cartesian plane, figure
ABCD and A'B'C'D' are congruent.
C
A
A’
B’
1
x
C’
D’
N
as
Which isometric transformations would
make figure ABCD coincide with figure
A'B'C'D'?
1
s
D
B
A counterclockwise rotation of 90 centred at (0, 0) followed by the translation (9, 0).
B)
A clockwise rotation of 90 centred at (0, 0) followed by the translation (-9, 0).
C)
The translation (-1, -6) followed by a clockwise rotation of 90 centred at (0, 0).
D)
A clockwise rotation of 90 centred at (0, 0) followed by the translation (-1, -6).
s.
A)
While admiring a mural, Mary notices that figures A and B are congruent.
N
as
s
A
B
Name and illustrate the composite of isometric transformations that will map figure A onto figure B.
Show your work.
s.
9
Right triangles ABC and DEF are isometric (congruent).
If
E
A
m AC = 20 cm
m EF = 10 cm
m EA = 7 cm
What is the length of AD to the nearest tenth?
A)
8.4 cm
B)
10.3 cm
D
N
as
B
F
s
C
s.
10
C)
15.4 cm
D)
17.3 cm
Given rectangle ABCD to the right, prove that
triangles ABC and CDA are congruent.
B
A
N
as
Show your work.
s
C
D
s.
11
Given triangles ABC and DEF.
A
D
13.7 cm
12.6 cm
8 cm
80
B
80
8 cm
s
65
65
F
E
N
as
C
Which of the following reasons can be used to explain why triangles ABC and DEF are congruent?
A)
If two angles of one triangle are congruent to two angles of another triangle, then the
triangles are congruent. (AA)
B)
If two sides and the contained angle of one triangle are congruent to two sides and the
contained angle of another triangle, then the triangles are congruent. (SAS)
C)
If two angles and the contained side of one triangle are congruent to two angles and the
contained side of another triangle, then the triangles are congruent. (ASA)
D)
If three sides of one triangle are congruent to three sides of another triangle, then the
triangles are congruent. (SSS)
s.
12
In the figure below
AB  BC  FE  ED
B
s
A  BCF  CFE  D
N
as
C
A
D
F
E
Using this information, which two triangles can be proven congruent?
A)
BCF  CFE
C)
CFE  CED
B)
BCF  ABF
D)
ABF  CED
s.
13
In the adjacent figure, C is the midpoint of segments AE
and BD.
A
Prove that triangles ABC and CDE are congruent.
Justify your statements.
C
D
E
N
as
Show your work.
To draw the plans for a roof, we first draw horizontal segment AC. Then, from the midpoint of
segment AC, we draw perpendicular DB. Finally, we draw segments AB and BC.
B
A
Show that triangles ABD and CBD are congruent.
s.
15
B
s
14
Justify your statements.
D
C
B)
C)
D)
m A = 70
m B = 35
m C = 75
m D = 70
m E = 35
m F = 75
m A = 70
m B = 35
m AB = 5 cm
m D = 70
m E = 35
m EF = 5 cm
N
as
A)
s
In which of the following situations is there sufficient information to conclude that triangles ABC and DEF are
congruent?
m A = 70
m C = 75
m AC = 10 cm
m D = 70
m F = 75
m DF = 10 cm
m A = 70
m AB = 5 cm
m AC = 10 cm
m D = 70
m DE = 5 cm
m EF = 10 cm
s.
16
In the figure on the right,
H
G
BF  DH
A
F
B
O
CG  AE
E
AOB  COD
C
s
D
N
as
O is the midpoint of segments BF, DH, CG and AE.
Which property can be used to prove that the 4 triangles are congruent?
A)
A.A.
B)
S.S.S.
s.
17
C)
S.A.S.
D)
A.S.A.
B)
C)
D)
m B = 45
m AB = 12 cm m AC = 6 cm
m E = 45
m EF = 12 cm m DF = 6 cm
m C = 60
m AC = 4 cm
m BC = 9 cm
m F = 60
m DF = 4 cm
m EF = 9 cm
N
as
A)
s
In which of the following situations is there enough information to conclude that triangles ABC and DEF are
congruent?
m A = 75
m B = 45
m AB = 8 cm
m D = 75
m E = 45
m DF = 8 cm
m A = 75
m C = 60
m AB = 10 cm
m D = 75
m F = 60
m EF = 10 cm
s.
18
Compare the congruence of the different figures given below. Represent the comparison by drawing
arrows on the Venn Diagram.
B
C
E
F
G
I
b)
D
s
A
H
N
as
a)
J
K
L
What properties are common to the subsets of the congruent figures thus formed?
s.
19
In the following figure, PQR and MNR are similar triangles.
Q
b
N
h
R
M
N
as
P
h’
s
a
Measures a and b are known.
What theorem allows you to deduce that heights h and h' are in the following proportion?
s.
20
h ab

h'
a
21
D
In the adjacent figure, AB is parallel to DE .
9
B
C
The following steps can be used to determine the
3
A
Justifications
N
as
Statements
E
s
6
measure of BC .
1
m ACB = m DCE
1
Vertically opposite angles are congruent.
2
m BAC = m CDE
2
If a secant cuts two parallel lines, then the alternate
interior angles are congruent.
3
ABC ~ CDE
3
Two triangles are similar if they have respectively at least
two congruent angles (AA).
4
m AC
4
m CD

m BC
m CE
s.
3 m BC

9
6
Which justification completes the reasoning in step 4?
?
Triangles are similar if their corresponding angles are congruent.
B)
Triangles are similar if their corresponding sides are congruent.
C)
Triangles are similar if the measures of their corresponding sides are proportional.
D)
Triangles are similar if the measures of their corresponding angles are proportional.
s.
N
as
s
A)
Polygons ABCDE and A'B'C'D'E' are similar. Segment
AC and A'C' are diagonals.
B
B’
C
A
C’
A’
D’
STATEMENT
m AB
2.
m A' B'

JUSTIFICATION
In similar polygons, corresponding angles are congruent.
m BC
In similar polygons, corresponding sides are proportional.
m B' C'
ABC ~ A'B'C'
3.
E’
N
as
B  B'
1.
D
s
The following proof can be used to show that
triangles ABC and A'B'C'D' are similar.
?
Which of the following can be used to justify the third statement of this proof?
A)
Two triangles are similar if an angle of one is congruent to an angle of the other and the
sides about these angles are proportional.
s.
22
B)
Two triangles are similar if their three corresponding sides are proportional.
C)
Two triangles are similar if a side of one is proportional to a side of the other and the angles
about these sides are congruent.
D)
Two triangles are similar if two angles of one triangle are congruent to two angles of the
other triangle.
Given the triangles illustrated below.
B
D
E
I
G
C
N
as
A
F
s
H
Which of the following statements are TRUE?
1.
Triangle ABC is similar to triangle DFE.
2.
Triangle ABC is equivalent to triangle HGI.
3.
Triangles ABC and DFE are isometric.
A)
1 and 2 only
C)
2 and 3 only
B)
1 and 3 only
D)
1, 2 and 3
s.
23
Given the following diagram, which isometry maps points A, B, C, and D onto points A', B', C' and D'
respectively?
y
C
A'
A
D'
D
s
B
x
C'
N
as
B'
A)
A reflection about the x-axis, followed by a reflection about the y-axis
B)
A translation that maps point A onto A', followed by a reflection about the x-axis
C)
A reflection about line CD, followed by a clockwise rotation of 90 about point D
D)
A rotation of 180 about point D
s.
24
In the diagram below, line segments AE and BD intersect at C.
In addition:
D
A
m AC  50 cm
m DC  100 cm
B
m EC  125 cm
N
as
E
The following is part of a procedure used to show that  ABC   EDC.
Step 1
 BCA   DCE
because vertically opposite angles are congruent.
Step 2
Step 3
m AC
m EC

m BC
m DC
because
50 cm
40 cm

125 cm 100 cm
 ABC ~  EDC
because ...
Step 4
s
C
m BC  40 cm
 ABC   EDC
because ...
s.
25
In your Answer Booklet, complete steps 3 and 4 of this procedure.
The following diagram shows lines AB, CD, EF and GH
In addition:
G
F
AB // CD
m  DLH = 75
A
K
B
?
m  LKJ = (2x)
(2x)
C
J
(3x)
L
s
m  KJL = (3x)
D
N
as
75
E
What is the numerical measure of angle FKB?
Show all your work.
s.
26
H
In the following figure:
Lines RC and DQ are parallel.
Lines BD and CP intersect at point A.
m  CBD = 120
s
m  QEP = 40
A
N
as
?
R
B
C
120
D
Q
E
40
P
The following is part of a procedure used to determine the measure of angle BAC.
Step 1
m  ABC + m  CBD = 180
s.
27
m  ABC + 120 = 180
m  ABC = 60
because ...
Step 2
m  BCA = m  QEP = 40
because ...
Step 3
m  ABC + m  BCA + m  BAC = 180
because the sum of the measures of the
interior angles of a triangle is 180.
60 + 40 + m  BAC = 180
s.
N
as
In your answer booklet, complete steps 1 and 2 of this procedure.
s
m  BAC = 80
The trophy illustrated on the right is composed of a
right prism topped by a right pyramid.
The prism and the pyramid are equivalent.
s
The base of the pyramid and the bases of the prism
are square and congruent.
13 cm
13 cm
N
as
The edge of the base of the prism measures 13 cm.
The total volume of the trophy is 3380 cm3.
What is the total height of this trophy?
Show all your work.
s.
28
Congruent quadrilaterals ABCD and A'B'C'D' are drawn in the Cartesian plane below.
y
A
B
C
1
A'
1
s
D
x
N
as
D'
B'
C'
Which one of the following isometries must be used if quadrilateral A'B'C'D' is to be the image of quadrilateral
ABCD?
A)
A reflection in the line x = 0 followed by a translation of (0, 6)
B)
A translation of (4, -6) followed by a reflection in the line x = 2
C)
A reflection in the line y = 0 followed by a reflection in the line x = 0
D)
A rotation of 90 about centre (0, 0) followed by a translation of (8, 2)
s.
29
30
In the figure below, quadrilaterals ABCP and ABPD are parallelograms.
A
B
?
2x
65
D
Show all your work.
A right circular cone and a sphere are equivalent. The radius of the sphere and the radius of the base of
the cone are both 12 cm.
What is the height of the cone in cm?
Show all your work.
s.
31
C
N
as
What is the measure of angle BAP?
P
s
3x