Download Higher Unit 5: Angles and trigonometry

Higher Unit 5: Angles and trigonometry
Numerical fluency
1 Work out
a 32 + 42 b (3 + 4)2 Work out the expression in brackets first.
9 + 16 = 25 = c
32  4 2 =
d
52 122 e
52  42 10  2 5
2 Work out these.
Give your answers
i in surd form
a
c
ii as a decimal correct to 2 decimal places.
3  22
12  72
b
i i ii ii 52  22
A surd is a square root that cannot be
simplified any further. It is written with the
root symbol, for example, 5 .
All the numbers under the square
root symbol must be prime numbers.
For example, 10 = 2 5
22  62
d
i i ii ii Geometrical fluency
3 Reasoning Work out the angles marked with letters.
Give reasons for your answers.
a = b = c = d = e = f = g = h = i = j = k = You don’t have to
work out the angles in
the order of the letters
but it may help.
The properties of a shape are facts about its sides, angles, diagonals and symmetry.
Here are some of the properties of some well-known quadrilaterals.
Square




all sides are equal in length
opposite sides are parallel
all angles are 90º
diagonals bisect each other at 90º
Rectangle




opposite sides are equal in length
opposite sides are parallel
all angles are 90º
diagonals bisect each other
Rhombus




all sides are equal in length
opposite sides are parallel
opposite angles are equal
diagonals bisect each other at 90º
Parallelogram




opposite sides are equal in length
opposite sides are parallel
opposite angles are equal
diagonals bisect each other
Trapezium
 1 pair of parallel sides
Isosceles trapezium
 2 sides are equal in length
Kite
 2 pairs of sides are equal in length
 no parallel lines
 1 pair of equal angles
 1 pair of parallel sides
 diagonals bisect each other at 90º
 2 pairs of equal angles
4 Write down which quadrilaterals
a have four equal sides
b have opposite equal angles
c have one pair of equal angles
e have opposite parallel sides
5
d have diagonals bisecting at 90º
f can have 4 different sized sides.
a Write the name of each shape.
b Tick the regular polygons.
Are all of the sides
and angles equal?
6
a Find the missing lengths and angles in these triangles.
i
ii
iii
The dashed lines are
lines of symmetry.
One half fits exactly on
top of the other half.
b Name each triangle. Give a reason for your answer.
i Isosceles
Reason:Two sides are equal and two angles are equal.
ii Reason: iii
Reason: 7 Work out the angles and lengths marked with letters.
rectangle
rhombus
kite
isosceles trapezium
8 Work out angle a in the quadrilateral.
Give your reason.
Angles in a
quadrilateral
sum to 360°.
Algebraic fluency
9 Make x the subject of each formula.
a 3x = y c x = 4y 5
Divide both sides
by 3.
b x = y 3
d 7 = y x
The subject of a formula is the
variable on its own on one side
of the equals sign. A is the
subject of A = l × b
First multiply both sides by x.
10 a a = 5 and b = 9. Work out
i x = a2 + b2
ii x = b2 – a2
x = 52 + 2
x = 92 − = 25 + = 81 − = = Substitute the numbers
that you know into the
formula.
b p = 11 and q = 6. Work out
i r = p2 + q2 ii r = p2 – q2 11 Give your answers as fractions in their simplest terms.
a x = 6 and y = 18. Work out x .
y
You can divide by the HCF of the
numerator and denominator to write
a fraction in its simplest form.
x = 6 = 1
y 18 . . . . .
b a = 5 and b = 15. Work out a . b
c m = 12 and n = 30. Work out m . n
12 a Write an equation for these six angles.
b Solve your equation to find the value of t.
What do the angles
on a straight line
add up to?
13 Modelling Work out the sizes of the angles.
a
Work out the value of
the letter first.
Use the angle facts
you know about angles
on a straight line.
b
14 Problem-solving Work out the size of each angle in this quadrilateral.
What do you know
about the grey angle?
Write an equation using
what you know about
the sum of the angles
in a quadrilateral.