Unit8 Geometry
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Unit 8 : Geometry Friendly Notes Points, Lines, Line Segments, Rays, and Angles A point is a location in space. A point is often represented by a dot and usually named with a capital letter.
X
Y
All geometric figures are made up of points.
A line goes on forever in both directions.
A line is often drawn with arrows at both ends.
125 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 125
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
A line segment is part of a line between two points. Line segment PQ is shorter than the Line segment RS.
P
Q
R
S
A ray is a line with one end point. E
C
D Ray DE is not the same as ray ED.
An angle is formed when two rays have the same endpoint. The point where the two end points meet is called the vertex.
Vertex
126 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 126
© 2014 Marshall Cavendish Education Pte Ltd
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Right Angles
90
o
180o
360o
270o 1
A -turn 4 forms a right angle.
1
3
A -turn 2 forms two right angles.
A -turn 4 forms 3 right angles.
A complete turn forms 4 right angles.
1 right angle = 90˚ 2 right angles = 180˚ 3 right angles = 270˚ 4 right angles = 360˚
This angle is less than 90˚.
It is called an acute angle.
This angle is more than 90˚.
It is called an obtuse angle.
An acute angle is < 90˚. An obtuse angle is > 90˚ but < 180˚.
127 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 127
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
Measuring Angles We use a protractor to measure angles. We measure angles in degrees.
170 180 10 0
30
160 20
40
0 14
15
0
100 110 120 80 7 0 13 60 0 50
170 180 10 0
m a = 55˚
140˚ 80 90 70 100 60 110 120
160 20
a
40˚
0 10 20 180 170 160 30 1 50 40 14 0
30
0
15
40
0 10 20 180 170 160 30 150 40 14 0
b
0
110 120 70 13 60 0 50
13
80
50
100
0 14
80 90 70 00 60 10 1 1 120 50 0 13
m b = 180˚ + 40˚ = 220˚ OR m b = 360˚ − 140˚ = 220˚
Perpendicular Lines Perpendicular lines meet at right angles. A C
90
o
D
AB and CD are perpendicular lines. B We say AB is perpendicular to CD. We write AB CD.
128 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 128
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
Parallel Lines Parallel lines never meet. A B AB and CD are parallel lines. We draw arrowheads on the C D lines to show they are parallel.
We say AB is parallel to CD. We write AB // CD.
Quadrilaterals A polygon is a closed figure with straight sides. A quadrilateral is a 4-sided polygon. These are quadrilaterals. A
D
B
C Trapezoid
ABCD is a trapezoid. It has at least one pair of parallel lines.
129 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 129
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
E
F
H
G Parallelogram
EFGH is a parallelogram. It has two pairs of parallel lines which are equal on opposite sides. So, a parallelogram is a trapezaid. A square has 4 equal sides. It also has 4 equal angles. Each angle = 90˚ A rectangle has equal opposite sides. It also has 4 equal angles. Each angle = 90˚ So, a square and a rectangle are both parallelograms.
J M
K L Rhombus
JKLM is a rhombus. It has two pairs of parallel lines. It has 4 equal sides. It is a parallelogram with equal sides.
We use arrowheads ( ) to show parallel lines. We use strokes ( / ) to show equal sides.
130 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 130
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
Triangles A triangle is a 3-sided polygon. An equilateral triangle has 3 equal sides. An isosceles triangle has 2 equal sides. A scalene triangle has no equal sides. An equilateral triangle is also an isosceles triangle.
60
60
o
o
60
o
Equilateral triangle
40
o
40
o
Isosceles triangle
Scalene triangle
Line Symmetry A symmetric figure has one or more lines of symmetry. A line of symmetry divides a figure into two equal parts. These equal parts fit exactly. Each of the following figures has one line of symmetry.
131 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 131
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
The figure has more than one line of symmetry.
132 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 132
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
X
Y
All geometric figures are made up of points.
A line goes on forever in both directions.
A line is often drawn with arrows at both ends.
125 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 125
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
A line segment is part of a line between two points. Line segment PQ is shorter than the Line segment RS.
P
Q
R
S
A ray is a line with one end point. E
C
D Ray DE is not the same as ray ED.
An angle is formed when two rays have the same endpoint. The point where the two end points meet is called the vertex.
Vertex
126 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 126
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
Right Angles
90
o
180o
360o
270o 1
A -turn 4 forms a right angle.
1
3
A -turn 2 forms two right angles.
A -turn 4 forms 3 right angles.
A complete turn forms 4 right angles.
1 right angle = 90˚ 2 right angles = 180˚ 3 right angles = 270˚ 4 right angles = 360˚
This angle is less than 90˚.
It is called an acute angle.
This angle is more than 90˚.
It is called an obtuse angle.
An acute angle is < 90˚. An obtuse angle is > 90˚ but < 180˚.
127 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 127
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
Measuring Angles We use a protractor to measure angles. We measure angles in degrees.
170 180 10 0
30
160 20
40
0 14
15
0
100 110 120 80 7 0 13 60 0 50
170 180 10 0
m a = 55˚
140˚ 80 90 70 100 60 110 120
160 20
a
40˚
0 10 20 180 170 160 30 1 50 40 14 0
30
0
15
40
0 10 20 180 170 160 30 150 40 14 0
b
0
110 120 70 13 60 0 50
13
80
50
100
0 14
80 90 70 00 60 10 1 1 120 50 0 13
m b = 180˚ + 40˚ = 220˚ OR m b = 360˚ − 140˚ = 220˚
Perpendicular Lines Perpendicular lines meet at right angles. A C
90
o
D
AB and CD are perpendicular lines. B We say AB is perpendicular to CD. We write AB CD.
128 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 128
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
Parallel Lines Parallel lines never meet. A B AB and CD are parallel lines. We draw arrowheads on the C D lines to show they are parallel.
We say AB is parallel to CD. We write AB // CD.
Quadrilaterals A polygon is a closed figure with straight sides. A quadrilateral is a 4-sided polygon. These are quadrilaterals. A
D
B
C Trapezoid
ABCD is a trapezoid. It has at least one pair of parallel lines.
129 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 129
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
E
F
H
G Parallelogram
EFGH is a parallelogram. It has two pairs of parallel lines which are equal on opposite sides. So, a parallelogram is a trapezaid. A square has 4 equal sides. It also has 4 equal angles. Each angle = 90˚ A rectangle has equal opposite sides. It also has 4 equal angles. Each angle = 90˚ So, a square and a rectangle are both parallelograms.
J M
K L Rhombus
JKLM is a rhombus. It has two pairs of parallel lines. It has 4 equal sides. It is a parallelogram with equal sides.
We use arrowheads ( ) to show parallel lines. We use strokes ( / ) to show equal sides.
130 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 130
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
Triangles A triangle is a 3-sided polygon. An equilateral triangle has 3 equal sides. An isosceles triangle has 2 equal sides. A scalene triangle has no equal sides. An equilateral triangle is also an isosceles triangle.
60
60
o
o
60
o
Equilateral triangle
40
o
40
o
Isosceles triangle
Scalene triangle
Line Symmetry A symmetric figure has one or more lines of symmetry. A line of symmetry divides a figure into two equal parts. These equal parts fit exactly. Each of the following figures has one line of symmetry.
131 © 2014 Marshall Cavendish Education Pte Ltd
10_(M)PMCCEdn_EP4_08.indd 131
Primary Mathematics (Common Core Edition) Extra Practice 4
5/21/14 4:28 PM
The figure has more than one line of symmetry.
132 Primary Mathematics (Common Core Edition) Extra Practice 4
10_(M)PMCCEdn_EP4_08.indd 132
© 2014 Marshall Cavendish Education Pte Ltd
5/21/14 4:28 PM
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