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Chapter 3 Extra Practice Answer Key
Get Ready
1. A: equilateral triangle, 3-gon; B: square, 4-gon; C: regular pentagon, 5-gon; D: regular
hexagon, 6-gon; E: regular octagon, 8-gon
2. a)
A
B
C
D
E
b) A: 3, B: 4, C: 5, D: 6, E: 8
c) A regular polygon has the same number of lines of symmetry and the same order of
symmetry as it has sides.
3. a), d) congruent; b), c) not congruent
4. to 6.
10
F'
8
D'
E'
6
F
F''
4
E
D
0
-10
-8
-6
-4
E''
2
-2
-2
D''
0
0
2
4
H
6
8
10
G
-4
-6
I
-8
-10
4. Congruent: DE and D'E', EF and E'F', FD and F'D';  D and  D',  E and  E',  F
and  F'
5. Congruent: DE and D"E", EF and E"F", FD and F"D";  D and  D",  E and  E",
 F and  F"
6. a 180° rotation about the origin
3.1 Unique Triangles
1. Jane. Knowing three side lengths of a triangle produces a unique triangle so Mike must
have draw a triangle that was identical to Jane's.
2. a) No, two side measures and one angle that is not contained could result in different
triangles.
b) No, three angles could result in different triangles.
c) Yes, two angles and one side measure describes one unique triangle.
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3. No. With three given angles but no given side lengths, other triangles could be drawn
that are similar but different sizes.
4. No. Two angles and one side measure describe a unique triangle.
5. Adding the measure of KM, since three side measures describe a unique triangle;
adding  M, since two sides measures and a contained angle describes a unique triangle.
6. a) Unique; two angles and one side measure describe a unique triangle.
b) Unique; three side measures describe a unique triangle.
c) Not unique; three angles are not enough information to describe a unique triangle.
7. Yes. Two angles and one side measure describe a unique triangle.
8. a) Not unique. Two side measures and one angle that is not contained by those sides
are not enough to describe a unique triangle.
b) Unique. Two angles and one side measure describe a unique triangle.
c) Not unique. Two side measures and one angle that is not contained by those sides are
not enough to describe a unique triangle.
d) Not unique. Three angles are not enough information to describe a unique triangle.
3.2 Prove Triangles are Congruent
1.The sum of the angles in a triangle is always 180° so if two angles measure 60° the
third angle must also measure 60°.
2.  D   G,  E   H,  F   I, DE  GH, EF  HI, FD  IG
3. a) Congruent. Two side measures and the contained angle describe a unique triangle.
b) Not congruent. Many similar triangles can have the same three angles.
4. a) to c) Measure two sides and the contained angle, two angles and any one side, or
three sides. Could use a ruler and/or a protractor.
d) The triangle are congruent because the chosen measurements were sufficient to
describe a unique triangle. e) three measurements
5. You could measure all three sides. Could not use just a protractor since all three angles
could be the same, and the triangles could still not be congruent, just similar.
6. You could measure the base of the triangle and the angles at both ends of it.
7. You know two sides of the triangles are congruent. If EG bisects the quadrilateral, then
DH  FH, so the three sides are also congruent. If two triangles have three congruent
sides, the triangles are congruent.
8. Since DEH  FEH, you know that the three corresponding side measures will
also be congruent. You know that DE  EF, and EG is a side shared by both triangles, so
DG  FG.
3.3 Properties of Transformations
1. a)
Measure the distance of each point from mirror line, draw image points an equal distance
from mirror line, then join the points.
b) The points at either end of the mirror line. They do not change position because they
are exactly on the mirror line, so their reflection is in the same position as they are.
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This page may be reproduced for classroom use by the purchaser of this book without the written permission of the publisher.
c) regular hexagon; yes, one side was constructed as a reflected copy of the other side, so
it has symmetry about the mirror line.
2. a), b)
B
72º C
A
c) No. A is the centre of rotation, so it did not change position.
B
d)
72º C
A
e) Yes, the original triangle was copied as a rotation, so the resulting figure has rotational
symmetry.
3. Yes. The image is congruent to the original. Line segments joining corresponding
points on the original and the image are congruent and parallel.
4. Rotation. The image is congruent to the pre-image, and the orientation is the same, so
it is not a reflection. Line segments joining corresponding points on the pre-image and
the image are not parallel, so it is not a translation.
5. i) It is congruent to the pre-image, and the lines joining the corners could be parallel
and congruent, depending on which corner on the image corresponded to which on the
pre-image
ii) It is congruent to the pre-image, and the corresponding corners could be equidistant
from a mirror line between the two rectangles.
iii) It is congruent to the pre-image and the corresponding corners could be rotated 180°
about a point between the two rectangles.
3.4 Regular Polygons
1. a) it is a pentagon. b) it is an octagon. c) it is a hexagon. d) it is a pentagon.
2. The interior and exterior angles will add to 360°. Subtract the interior angle from 360°
to find the exterior angle. Examples may vary.
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3. The order of rotational symmetry and the number of lines of symmetry of a regular
polygon are the same as the number of its sides,
4. interior: 177°, exterior: 183°
5. a) to d) Drawing not to scale.
A
e) regular hexagon; each side is the same length, since the triangles are all congruent.
6. a) Drawing not to scale.
b) interior angle: 30°, side length: 2.7 cm
c) Answers may vary. Draw a circle with a diameter of 10 cm, then construct 30° angles
around the midpoint of the circle and mark where they intersect the circumference. Then
connect the intersection points to form the dodecagon.
7. No, The exterior angle is approximately 213°. The interior angle is approximately
147°. 213 is not a multiple of 147.
Chapter 3 Review
1. a) Unique. Two side measures and the angle contained by them describe a unique
triangle.
b) Unique. Two angles and any one side measure describe a unique triangle.
c) Unique. Three side measures describe a unique triangle.
2. ABC and DEF are congruent, since two angles and one side measure are the
same. Since the triangles are congruent and side AC corresponds to side DF, then AC =
DF.
3. a) Draw a line segment between corresponding points on the two trapezoids and mark
the midpoint. Use the Bullseye compass to construct the perpendicular bisector of this
line segment through the midpoint. The bisector is the mirror line.
b) Draw a line segment between a pair of corresponding vertices on the trapezoids. Use
the Bullseye compass to construct the perpendicular bisector of this line segment. Repeat
with another pair of corresponding vertices. The point where the two perpendicular
bisectors meet is the centre of rotation.
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4. a) Methods may vary. Method 1: The central angle for each triangle in the octagon will
be 360° ÷ 8 = 45°. The two other angles in each triangle will be (180° – 45°)  2 = 67.5°.
Two of these angles comprise each interior angle, so each interior angle measures 135°.
Method 2: Divide the octagon into triangles. Multiply the number of triangles by 180°
and divide the result by the number of sides (8). 6  180 ÷ 8 = 135°.
b) The sides are all of equal length. c) rotational symmetry of 8, 8 lines of symmetry
5. a) Congruent. Three side measures describe a unique triangle.
b) Congruent. Three side measures describe a unique triangle.
c) Congruent. Two angle measures and one side measure describe a unique triangle.
6. Rotation about the point shared by both quadrilaterals; The image is congruent to the
pre-image and could be rotated 180° about the common point of the two quadrilaterals.
A translation right and up; the image is congruent to the pre-image, and the lines joining
the corners could be parallel and congruent, depending on which corner on the image
corresponded to which on the pre-image
7., 8.
N'
M
O
O''
O'
N
N''
M''
7. MN = M'N', NO = N'O', MO = M'O',  M =  M',  N =  N',  O =  O'
8. MN = M'N', NO = N'O', MO = M'O',  M =  M',  N =  N',  O =  O'
9. a) Drawing not to scale.
¥P
b) Methods may vary. Draw a circle with centre P and radius 6 cm. Calculate the measure
of each central angle (360  6 = 60°), draw one radius, and mark other radii at central
angles of 60° all around the circle. Connect the points where the radii met the circle to
complete the hexagon.
Chapter 3 Practice Test
1. B 2. C 3. D
4. Divide 360° by 40, and subtract the result from 180°. 180 – (360 ÷ 40) = 171. Each
interior angle will be 171°/
5. Suzie. Two parts of a triangle will not describe a unique triangle. Need three sides, two
angles and any side, or two sides and the contained angle. Explanations may vary.
Copyright 2008 McGraw-Hill Ryerson Limited, a subsidiary of the McGraw-Hill Companies.
This page may be reproduced for classroom use by the purchaser of this book without the written permission of the publisher.
6. a) i) Congruent. Three side measures describe a unique triangle.
ii) Congruent. Two angles and any one side describe a unique triangle.
iii) Congruent. Two side measures and the contained angle describe a unique triangle.
iv) Unknown. Two angles are not enough to describe a unique triangle.
b) Corresponding vertices of should be labelled with the same letter and a prime symbols.
c) i)  A =  A',  B =  B',  C =  C' ii)  H =  H', ,GH = G'H', GI = G'I'
iii)  N =  N',  O =  O', NO = N'O' iv) none
d) i) reflection ii) rotation iii) translation
7. The regular pentagon will have all interior angles equal and all sides equal, and will
have rotational symmetry. The not regular pentagon will not have all interior angles equal
or all sides equal, and will not have rotational symmetry.
Copyright 2008 McGraw-Hill Ryerson Limited, a subsidiary of the McGraw-Hill Companies.
This page may be reproduced for classroom use by the purchaser of this book without the written permission of the publisher.