Download Geometry – AP Book 6.1

page 1
Geometry – AP Book 6.1
AP Book G6-1
page 145
1.
2.
3.
2.
Obtuse; 120º
c)
Acute; 30º
d)
Obtuse; 150º
a)
Acute; 60º
5; 5
b)
Acute; 34º
e)
8; 8
c)
Obtuse; 115º
f)
10; 10
d)
Acute; 30º
a)
3
e)
Acute; 40º
b)
4
f)
Obtuse; 137º
5; 5
b)
6; 6
c)
7; 7
d)
3.
c)
5
g)
Obtuse; 125º
d)
6
h)
Acute; 20º
a)
30º
Shapes
Letters
Triangles
C
b)
130º
Quadrilaterals
B, D, F,
G, H
c)
45º
d)
45º
Pentagons
A
e)
105º
Hexagons
E, I
f)
90º
Teacher to check.
5.
37 sides
AP Book G6-2
page 146
2.
a)
Less than
b)
Greater than
c)
Right angle
d)
Less than
a)
Teacher to check.
3.
a)
Teacher to check.
b)
In each triangle,
two sides are the
same length.
c)
(i)
(ii)
Acute-angled
(or Equilateral
– see G6-8)
(iii) Right-angled
(or Isosceles –
see G6-8)
4.
Teacher to check.
a)
5.
Two congruent
right-angled triangles.
S - 135 º , A - 45 º
b)
3.
Teacher to check.
6.
Teacher to check.
Teacher to check.
d)
1 right angle
1.
NOTE:
Side lengths and angles are
recorded clockwise, from left
to right.
1.
Obtuse-angled
(or Isosceles –
see G6-8)
Teacher to check.
2 right angles
AP Book G6-7
page 153
1.
a)
ABC: 48º
A
52º, 90º, 38º
with 3 cm, 4 cm,
5 cm side lengths
B
110º, 35º, 35º
with 4 cm, 6.5 cm,
4 cm side lengths
C
60º angles with
3 cm side lengths
D
22º, 79º, 79º
with 5.2 cm, 5.2
cm, 2.1 cm side
lengths
E
120º, 35º, 25º
with 3.1 cm, 6 cm,
4 cm side lengths
Property
Triangles
with Property
Acuteangled
C, D
Obtuseangled
B, E
Rightangled
A
Equilateral
C
Isosceles
B, D
Scalene
2.
a)
A, E
Scalene: A, E
a)
Acute-angled
b)
ABC: 42º
Right-angled: A
b)
Right-angled
c)
ABC: 90º
Both: A
Teacher to check.
c)
Obtuse-angled
a)
XYZ: 110º
Teacher to check.
d)
Acute-angled
b)
XYZ: 110º
e)
Obtuse-angled
c)
XYZ: 122º
a)
110º, 30º, 40º;
Obtuse-angled
b)
90º, 45º, 45º;
Right-angled
c)
60º, 50º, 70º; Acuteangled
2 right angles
AP Book G6-3
page 147
1.
Teacher to check.
2.
2.
c)
e)
4.
1.
Teacher to check.
B- 90 º , G - 45 º
AP Book G6-8
page 154
AP Book G6-6
page 152
1.
2 right angles
No right angles
70º, 115º, 70º, 115º;
VD – “at least one
obtuse angle” and
”at least one acute
angle”
The pentagon (A) has all
obtuse angles.
AP Book G6-4
page 150
AP Book G6-5
page 151
b)
3.
4.
c)
Acute; 60º
b)
a)
4.
1.
a)
a)
2.
Acute
b)
Obtuse
c)
Acute
d)
Acute
e)
Obtuse
f)
Obtuse
g)
Obtuse
h)
Actute
i)
Obtuse
3.
a)
b)
About 110º, 110º,
110º, 110º, 110º;
VD – “at least one
obtuse angle”
60º, 60º, 60º;
VD – “at least one
acute angle”
2.
3.
Neither: B, C, D
b)
Isosceles: D, B
AEB = 45º
Both: B
BEC = 21º
CED = 59º
From least to greatest:
BEC, AEB, CED
4.
acute angles:
AED, BEC
obtuse angles:
AEB, CED
Obtuse: E, B
Neither: A, C
3.
Teacher to check.
AP Book G6-9
page 155
1.
Teacher to check.
2.
a)
Right-angled;
Isosceles
b)
Obtuse-angled;
Scalene
Answer Key for AP Book 6.1
page 2
Geometry – AP Book 6.1 (continued)
3.
The angles of every
triangle must add up to
180 º. We know the top
angle is 120 º and the
base angles are always
the same on an isosceles
triangle, therefore the two
base angles must be 30º
120 + 30 +30 = 180.
H
2.
Two Pairs: A, B, E, H
3.
a)
Property
Shapes with
Property
A, E, F
1 pair of
parallel
sides
G, H
2 pairs of
parallel
sides
B, C, D, J
3 pairs of
parallel
sides
I
Triangles
A, E
1 right
angle
C
a) and c) should be
marked as parallel
2 right
angles
A, B, G, K
B, C, D
BONUS:
4 right
angles
Quadrilaterals
F, H
Pentagons
F, H
Hexagons
I, J
If the pair of lines
in b) are extended,
they will intersect
so aren’t parallel.
2.
Teacher to check.
3.
a)
1
b)
2
c)
3
d)
1
e)
2
f)
2
g)
0
h)
0
4.
Teacher to check.
5.
Teacher to check.
E has three lines that are
parallel;
H and M each have one
pair of parallel lines;
W and K have no parallel
lines.
b)
lines
4.
5.
a)
b)
AP Book G6-11
page 158
A
2 pairs
B
2 pairs
No parallel
c)
e)
No
c)
No
d)
Some
a)
Rhombus
b)
Rectangle
c)
Trapezoid
Rectangle, parallelogram,
rhombus
10. Square, rectangle
12. a)
Both have two pairs
of parallel sides.
A rhombus has
4 equal sides.
b)
Both have two pairs
of parallel sides and
all sides are equal.
A rhombus may not
have all 90º angles.
A square has 4 right
angles.
c)
A parallelogram has
two pairs of parallel
lines, each line of a
pair are equal in
length.
A trapezoid has one
pair of parallel lines,
each line in the pair
may differ in length.
C, I, J
A, B, D, G, K
AP Book G6-12
page 160
2 pairs
E, F, H
1.
4 cm, 2.5 cm, 4 cm,
2.5 cm;
(not equilateral)
b)
2.7 cm, 2.7 cm, 2.7
cm, 2.7 cm;
(equilateral)
c)
1.7 cm, 1.7 cm,
1.7 cm, 1.7 cm,
1.7 cm;
(equilateral)
d)
2 cm, 2 cm, 2 cm,
2 cm;
(equilateral)
2.
Equilateral
A, B, D, F, I
Not
Equilateral
C, E, G, H, J
No right 
A, D, F, G, I
1 right 
E
2 right 
H
3 right 
J
4 right 
B, C
No
obtuse 
A, B, C, E, G
a)
3 cm, 2 cm, 3 cm,
2 cm; parallelogram
b)
2 cm, 2 cm, 2 cm,
2 cm; square
Square – A parallelogram
with 4 right angles and 4
equal sides.
Rectangle – A
parallelogram with 4 right
angles.
Rhombus – A parallelogram
with 4 equal sides.
3.
Shapes with
Property
Property
4.
5.
a)
13. a)
Rectangle
b)
Parallelogram
c)
Square
d)
Rhombus
a)
Parallelogram
A square is a
parallelogram with
4 right angles (so it is
also a rectangle).
It has one extra
property:
equal sides.
b)
A rectangle does
not have all equal
sides.
c)
In a trapezoid, only
1 pair of sides is
parallel.
b)
Square
c)
Rhombus
d)
Rectangle
a)
2 pairs;
rectangle
b)
2 pairs;
parallelogram
AP Book G6-13
page 162
c)
2 pairs;
square
1.
d)
1 pair;
trapezoid
C
1 pair
D
1 pair
E
2 pairs
F
No pairs
6.
Teacher to check.
G
1 pair
7.
a)
1 or more
obtuse 
b)
11. Teacher to check.
1 pair
a)
8.
9.
D, E, I, J
By definition, the
lines in d) cannot
be parallel as they
are curved.
1.
No Pairs: F
No parallel
sides
No right
angles
e)
6.
d)
One Pair: C, D, G
AP Book G6-10
page 156
1.
2 pairs
D, F, H, I, J
2.
a)
yes
b)
no
c)
no
d)
no
Teacher to check.
All
Answer Key for AP Book 6.1
page 3
Geometry – AP Book 6.1 (continued)
3.
The intersection of the
diagonals is perpendicular.
4.
a)
B, D
b)
The diagonals
bisect at 90º.
One of the two
diagonals is the line
of symmetry.
5.
6.
7.
a)
b)
Diagonals meet at right
angles (only): none
∆KLM:
25º, 50º, 105º
8.
a)
2.
3.
∆ABC and ∆KLM;
∆DEF and ∆HIJ
Outside: B, D, C
b)
Answers will vary.
a)
square
c)
Answers will vary.
b)
rhombus
c)
square
d)
rhombus
9.
a)
Not congruent
b)
Congruent
c)
Congruent
a)
No; size is different.
b)
Yes; same shape
and size.
a)
Teacher to check drawings.
a)
Sides: 3cm, 4cm,
5cm
Angles: 90º, 36º,
125º
AP Book G6-14
page 163
1.
∆ABC:
25º, 50º, 105º
∆HIJ:
30º, 60º, 90º
Inside: A, E, F
7.
5.
∆DEF:
30º, 60º, 90º
The angles opposite
the line of symmetry
are equal.
Two pairs of equal adjacent
sides (only): none
If rounding to
the nearest cm,
all triangles will
have sides of length
3 cm, 5 cm and
6 cm.
b)
Sides: 4cm, 4cm.
4cm, 4cm
Teacher to check
(must be same size
and shape as
original).
2.
In rectangle A, the longer
side is double the shorter
side. Therefore, in triangle
B, the missing side is
2 × 2 = 4 cm.
a)
3
Teacher to check.
b)
3
5.
a)
c)
5
d)
4
a)
6
b)
15
c)
12
d)
20
Answers will vary.
A:
3 possible shapes
B:
5 possible shapes
3.
C:
1 possible shapes
6.
b)
B
a)
BAC or CAB
b)
DFE or EFD
c)
IHJ or JHI
d)
MKL or LKM
A and C
7.
No
8.
A and C are similar.
9.
A & H are congruent;
A & H are both similar
to F.
4.
b)
The number of
edges is equal to
the number of lines
of symmetry.
c)
Order of rotation
symmetry:
Triangle − 3
Square − 4
Regular Pentagon
−5
10. Answers will vary.
11. Answers will vary.
Regular Hexagon
−6
12. Answers will vary.
13. No: similar figures must
have the same angles and
– since a trapezoid can
only have one set of
parallel lines – this will
never be the case.
A shape has an order
of rotation symmetry
equal to the number
of lines of symmetry.
For example,
an equilateral triangle
has an order of
rotation symmetry of
3.
AP Book G6-16
page 167
1.
Teacher to check.
2.
Teacher to check.
3.
Teacher to check.
4.
a)
3
b)
5
c)
2
d)
6
e)
5
f)
4
g)
2
AP Book G6-15
page 165
4.
b)
6.
Angles: 60º, 120º,
60º, 120º
1.
2 units
h)
5.
6.
8.
No, the shapes are not
mirror images.
9.
Answers will vary.
AP Book G6-17
page 170
1.
a)
1
nd
th
The 2 and 5 should be
shaded.
a)
Less than two
lines of symmetry:
B, C, D, F, G
F1
F2
3
4
√
3
4
√
0
1
√
0
2
√
3
1
√
0
1
√
3
0
√
Y
N
√
More than two lines
of symmetry: A, E
7.
b)
C, B
a)
Number of edges;
number of lines of
symmetry:
S?
D?
S? D?
b)
Vertices
√
Edges
√
Pairs of
parallel
pides
√
Teacher to check
drawings.
Equilateral triangle
− 3; 3
Right 
√
a)
Square − 4; 4
Acute 
√
Pentagon − 5; 5
Obtuse 
√
Hexagon − 6; 6
Lines of
symmetry
√
Equilateral
√
Length of B will be
4 units.
b)
Length of B will be
6 units.
c)
Length of B will be
6 units.
Answer Key for AP Book 6.1
Geometry – AP Book 6.1 (continued)
2.
Answers will vary.
3.
Teacher to check.
4.
AP Book G6-18
page 171
1.
b)
Teacher to check.
Figures
AP Book G6-20
page 174
1
B, C, F, H
1.
2
B, C, F
2.
Teacher to check.
3.
a)
ABC
b)
If line is perfectly
diagonal and bisects
a 90º angle, the
angle will
be 45º.
Prop
Figures
1
A, C, D, E,
F, H
2
C, F
Q
E
2+ 90º
A
O
A
Y
N
Y
N
Y
Y
B
Y
N
N
N
Y
Y
C
Y
N
Y
Y
N
N
D
N
Y
Y
N
N
Y
N
Y
N
N
N
Y
Teacher to check.
5.
1 name:
B (parallelogram),
E (trapezoid)
2 names:
C (rectangle,
parallelogram),
D (rhombus, parallelogram)
4 names:
A (parallelogram, rectangle,
square, rhombus)
6.
Answers will vary.
a)
F; F;
F; F;
b)
F; F;
T; T;
c)
F; F;
T; T;
d)
T; T;
F; T;
e)
T; F;
T;
F;
f)
F;
T;
T; F
a)
Equilateral triangle
b)
Isosceles triangle
c)
Parallelogram
d)
Trapezoid
a)
Teacher to check.
Teacher to check.
4.
AP Book G6-19
page 173
3.
Teacher to check.
Prop
2.
2.
Teacher to check.
a)
b)
C & F have both
properties.
1.
c)
C, H
c)
3.
Teacher to check.
a)
B, C & F have both
properties.
E
b)
page 4
7.
a)
3 acute angles
b)
2 acute angles and
1 obtuse angle
c)
2 acute angles and
1 right angle
a)
Equilateral triangle
b)
Obtuse-angled
triangle
c)
Right-angled triangle
d)
Acute-angled
triangle
Answer Key for AP Book 6.1