Mathematics
1
Grade 9
Geometry
Straight Lines
1. Study the diagrams and fill in the correct statement or appropriate reason.
a)
Statement: 𝐶𝐶̂ 1 + 𝐶𝐶̂ 2 = 180°
Reason:
____________________________
____________________________
b)
Statement: ________________________
Reason: exterior angle of a triangle equals
the sum of the two opposite interior angles
c)
In right angled triangle ABC:
Statement:
𝐴𝐴𝐴𝐴 2 = ____________________________
𝐵𝐵𝐵𝐵 2 = ____________________________
Reason: Pythagoras
2. Name the type of triangle or quadrilateral according to the information given.
Diagram
a)
Name
Mathematics
b)
2
Grade 9
c)
d)
e)
𝐴𝐴̂ > 90°
3. Determine the value of 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧, in alphabetical order, for each diagram. All reasons must be given.
a)
50°
b)
𝑦𝑦
Reason
Statement
Reason
𝑥𝑥
𝑦𝑦
𝑥𝑥
Statement
38°
Mathematics
3
c) The quadrilateral below is a
parallelogram.
𝑦𝑦
Grade 9
Statement
Reason
Statement
Reason
𝑥𝑥
d) The quadrilateral below is a
square.
𝑦𝑦
𝑥𝑥
e)
𝑦𝑦
𝑧𝑧
50°
𝑧𝑧
𝑦𝑦
Reason
Statement
Reason
𝑥𝑥
f) The quadrilateral below is a
rhombus
𝑥𝑥
Statement
Mathematics
4
Grade 9
4. Determine the value of 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 and 𝑒𝑒, in alphabetical order, for each diagram. All reasons must be
given.
a)
Q
80°
U
a
S
b
120°
R
c
V
T
Statement
Reason
b)
P
X
Q
b
35 °
U
R
V
Statement
c
a
e
S
d
W
Reason
T
Y
Mathematics
5
Grade 9
�1 with reasons.
5. In the rhombus below, determine 𝑁𝑁
L
K
50 °
N
Reason
�2 .
6. Determine the value of 𝐷𝐷
D
F
1
2
80°
A
Statement
2
1
Statement
M
100°
E
20°
C
B
Reason
Mathematics
6
Grade 9
7. You are given the following information regarding the diagram below:
𝑃𝑃𝑃𝑃 ∥ 𝑆𝑆𝑆𝑆 ;
𝑄𝑄𝑃𝑃�𝑅𝑅 = 30°
𝑅𝑅𝑆𝑆̂𝑇𝑇 = 50°
;
P
Q
R
T
S
Calculate, giving reasons, the size of 𝑃𝑃𝑅𝑅� 𝑆𝑆. (Hint: Construct a line through 𝑅𝑅 parallel to 𝑃𝑃𝑃𝑃)
Statement
Reason
Mathematics
MEMO:
7
Grade 9
1. Study the diagrams and fill in the correct statement or appropriate reason.
a)
Statement: 𝐶𝐶̂ 1 + 𝐶𝐶̂ 2 = 180°
Reason: ∠’s on a str. line
b)
Statement: 𝐶𝐶̂ 2 = 𝐵𝐵𝐴𝐴̂𝐶𝐶 + 𝐴𝐴𝐵𝐵� 𝐶𝐶
Reason: exterior angle of a triangle equals
the sum of the two opposite interior angles
c)
In right angled triangle ABC:
Statement:
𝐴𝐴𝐴𝐴 2 = 𝐴𝐴𝐵𝐵 2 + 𝐵𝐵𝐶𝐶 2
𝐵𝐵𝐵𝐵 2 = 𝐴𝐴𝐶𝐶 2 − 𝐴𝐴𝐵𝐵 2
Reason: Pythagoras
2. Name the type of triangle or quadrilateral according to the information given.
Diagram
Name
a)
Equilateral Triangle
b)
Obtuse angled triangle
c)
Scalene Triangle
d)
Kite
e)
Rhombus
3. Determine the value of 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧, in alphabetical order, for each diagram. All reasons must be given.
a)
50°
𝑦𝑦
𝑥𝑥
Statement
Reason
𝑥𝑥 = 50°
Vert. opp. ∠’s
𝑦𝑦 = 130°
∠’s on a str. line
Mathematics
b)
8
𝑦𝑦
38°
𝑥𝑥
c) The quadrilateral below is a
parallelogram.
𝑦𝑦
𝑥𝑥
d) The quadrilateral below is a
square.
y
Grade 9
Statement
Reason
𝑥𝑥 = 52°
Int. ∠’s of a ∆
Alt. ∠’s ; parallel lines
𝑦𝑦 = 52°
Statement
Reason
𝑥𝑥 = 70°
Opp. ∠’s of a parm. is =
Statement
Reason
𝑥𝑥 = 90°
Diag. of a sqr. bisects perpendicularly
Corr. ∠’s ; opp. sides of parm. is parallel
(opp sides of parm. is ||)
𝑦𝑦 = 70°
𝑦𝑦 = 45°
x
Diag. of a sqr. bisects the corners
Statement
e)
𝑥𝑥 =
180°−48°
2
Reason
= 66°
Int. ∠’s of a ∆ ; ∠’s opp. = sides
𝑦𝑦 = 180° − 66° − 90° = 24° Co-int. ∠’s ; parallel lines
𝑧𝑧 2 = 122 + 52 = √169 = 13 Pythag.
f)
𝑥𝑥
50°
𝑧𝑧
𝑦𝑦
Statement
Reason
𝑥𝑥 = 50°
Diag. of rhom. bisect corners
𝑦𝑦 = 50°
𝑧𝑧 = 90°
Opp. ∠’s of rhom are =
Diag. of rhom. bisect each other
perpendicularly
4. Determine the value of 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 and 𝑒𝑒, in alphabetical order, for each diagram. All reasons must
be given.
Mathematics
a)
9
Q
Grade 9
U
80°
a
b
S
120°
V
R
c
T
Statement
Reason
𝑎𝑎 = 80°
Vert. opp. ∠’s
𝑐𝑐 = 60°
Alt. ∠’s ; QR // ST
∠’s on a str. line
𝑏𝑏 = 60°
P
b)
X
Q
b
35°
U
R
c
a
T
e
Y
S
d
W
V
Statement
Reason
𝑎𝑎 = 35°
∠’s opp. = sides
𝑐𝑐 = 70°
Co-int. ∠’s ; QR // TS
Int. ∠’s of a ∆
𝑏𝑏 = 110°
Corr. ∠’s ; QT // RS
𝑑𝑑 = 110°
Atl. ∠’s ; QT // RS
𝑒𝑒 = 35°
�1 with reasons.
5. In the rhombus below, determine 𝑁𝑁
L
K
50 °
1
M
2
N
Mathematics
10
Statement
�1 = 𝐾𝐾𝐿𝐿�𝑁𝑁
𝑁𝑁
�1 =
∴ 𝑁𝑁
Grade 9
Reason
180°−50°
2
KL = KN, all sides of rhom. are equal ∴ ∠’s opp. =
sides are =.
Int. ∠’s of a ∆
= 65°
D
�2 .
6. Determine the value of 𝐷𝐷
F
2
1
80°
A
100°
E
20°
C
B
Statement
Reason
𝐷𝐷𝐵𝐵� 𝐸𝐸 = 60°
∠’s on a str. line
�2 = 180° − 40° − 80° = 60°
𝐷𝐷
Co-int. ∠’s ; DF // BC
�1 = 40°
𝐷𝐷
Int. ∠’s of a ∆
7. You are given the following information regarding the diagram below:
𝑃𝑃𝑃𝑃 ∥ 𝑆𝑆𝑆𝑆 ;
𝑄𝑄𝑃𝑃�𝑅𝑅 = 30°
𝑅𝑅𝑆𝑆̂𝑇𝑇 = 50°
;
Calculate, giving reasons, the size of 𝑃𝑃𝑅𝑅� 𝑆𝑆. (Hint: Construct a line through 𝑅𝑅 parallel to 𝑃𝑃𝑃𝑃)
P
Q
A
R
B
T
S
Statement
Reason
𝑃𝑃𝑅𝑅� 𝐴𝐴 = 𝑄𝑄𝑃𝑃� 𝑅𝑅 = 30°
Atl. ∠’s ; PQ // AB
𝑆𝑆𝑅𝑅� 𝐴𝐴 = 𝑅𝑅𝑆𝑆̂𝑇𝑇 = 50°
∴ 𝑃𝑃𝑅𝑅� 𝑆𝑆 = 80°
Atl. ∠’s ; AB // ST