Download Geometry and measure

3
Geometry and measure
You will be expected to:
// know
the properties of angles, polygons, symmetrical 2-D and 3-D shapes and circles
// construct geometrical shapes and loci
// use three-figure bearings and Pythagoras
// calculate perimeters, surface areas and volumes of 2-D and 3-D shapes
// use units of mass, length, area, volume and capacity and convert between different units
// use similarity to solve problems
// use vectors
// identify and describe a range of transformations.
3.1 Lines and angles
Angles and triangles
An acute angle is between 0° and 90°.
A right angle is exactly 90°.
An obtuse angle is between 90° and 180°.
A reflex angle is between 180° and 360°.
The angles on a straight line add up to 180°.
The angles at a point add up to 360°.
The angles in a triangle add up to 180°.
An equilateral triangle has all three angles the same size and all three sides the same length.
An isosceles triangle has two angles the same size and two sides the same length.
A scalene triangle has all three angles different sizes and all three sides different lengths.
A right-angled triangle has one right angle.
The exterior angle in a triangle makes a straight line with the side of the triangle.
In the triangle, a + b + c = d + c = 180°. d = a + b
a
exterior angle
d
c
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Test yourself
1.
Explain whether a right-angled triangle is always a scalene triangle.
2.
Find the sizes of the angles shown.
c
38°
68°
e
128°
32°
f
d
a
b
h
i
135°
41°
k
j
g
Lines and angles
A transversal is a line that crosses two or more parallel lines.
Angles d and f are alternate angles – alternate angles are equal.
a
d
Angles a and e are corresponding angles – corresponding angles are equal.
Angles c and f are interior angles – interior angles are on the same side
of the transversal and inside the parallel lines. They add up to 180°.
e
h
g
c
b
f
Angles a and c are opposite angles – opposite angles are equal.
Worked example
In the diagram above, angle a = 142°.
Find the sizes of angles b to h.
TIP
Note that there are two,
and sometimes three or
four, different ways of
finding each angle.
Answer
b = 180° − 142° = 38° (angles on a straight line add up to 180°)
c = a = 142° (opposite angles); or b + c = 180° (angles on a straight line)
d = b = 38° (opposite angles); or a + d = 180° (angles on a straight line)
e = a = 142° (corresponding angles); or e = c (alternate angles) or d + e = 180° (interior angles)
f = d = 38° (alternate angles); or f = b (corresponding angles); or c + f = 180° (interior angles); or e + f = 180°
(angles on a straight line)
g = c = 142° (corresponding angles); or g = e (opposite angles); or f + g = 180° (angles on a straight line)
h = f = 38° (opposite angles); or h = d (corresponding angles); or e + h = g + h = 180° (angles on a straight line)
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Test yourself
1.
Find the sizes of the marked angles in the diagram.
a
b
122° c
d
e
f
i
g
h
2.
Find the sizes of the marked angles in the diagram.
54°
g
d
f
a
b
c
e
h
i
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3.2 Polygons
Quadrilaterals
The properties of a square, rectangle, parallelogram, rhombus, trapezium and kite are shown in the diagrams.
square
rectangle
parallelogram
rhombus
trapezium
kite
The sum of the interior angles in a quadrilateral is 360°.
Worked example
Find the sizes of the marked angles in the diagram.
x
2x + 8
4x
4x – 25
3x + 10
Answer
The angles 4x and x are interior angles so 4x + x = 180°
5x= 180°
x= 36°
So the two angles are 36° and 144°.
2x + 8 = 2 × 36 + 8 = 80°
4x − 25 = 4 × 36 − 25 = 119°
3x + 10 = 3 × 36 + 10 = 118°
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Test yourself
1.
Find the sizes of the marked angles in the diagrams.
(a)
a
b
(b)
c
k
j
51°
f
123°
e d
a
c
h
i
41°
b
f
e
103°
g
d
2.
Find the sizes of the marked angles in the diagrams.
(a)
(b)
5x +12
6x – 4
6x – 8
y
5y + 7
6x – 8
2x – 12
2y + 5
x
8x
3.
The four angles of a quadrilateral are (x + 15)°, (x + 20)°, (2x − 10)° and (3x − 15)°. Find the value of x.
4.
Explain whether all squares are rectangles.
5.
Explain whether all rhombuses are parallelograms.
6.
Explain whether all squares are rhombuses.
7.
Explain whether all kites are rhombuses.
Polygons
A polygon is a shape with three or more sides.
A regular polygon has all sides the same length and all interior angles the same size.
The angle sum of the interior angles of an n-sided polygon is s = 180(n − 2)° or (2n − 4) right angles.
360°
The exterior angle of a regular n-sided polygon is
.
n
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The sum of the exterior and interior angles of a polygon is 180°.
Polygons which have the same shape and size are congruent.
If you cannot remember the formula for the sum of the interior
angles of a polygon, split the polygon into triangles by drawing
lines across the polygon from one vertex to each of the other
vertices that is not already connected to it. The sum of the
interior angles is the number of triangles multiplied by 180°.
A hexagon splits into four triangles so, sum of interior
angles = 4 × 180° = 720°.
TIP
Worked example
1.
The interior angle of a regular polygon is 160°. How many sides does the polygon have?
2.
The sum of the interior angles of a polygon is 900°. How many sides does the polygon have?
3.
Find the value of x in this shape.
2x + 3
3x – 10
52°
3x – 7
Answer
1.
Exterior angle + interior angle = 180°
Exterior angle = 180° − 160° = 20°
Exterior angle = 360° , so n =
n
360°
= 18
20°
The polygon has 18 sides.
2.
Sum of interior angles = 180(n − 2)°
So
900° = 180(n − 2)°
Solving for n:
900 = 180n − 360
180n = 1260
n = 7
The polygon has 7 sides.
3.
Use the sum of the interior angles = 360° and exterior angle + interior angle = 180°.
180 − (2x + 3) + 180 − (3x − 10) + 3x − 7 + 52 = 360
Solve for x.
180 − 2x − 3 + 180 − 3x + 10 + 3x − 7 + 52 = 360
− 2x − 3 − 3x + 10 + 3x − 7 + 52 = 0
− 2x + 52 = 0
2x = 52
x = 26
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