L6 Reasoning angles
33 minutes
33 marks
Page 1 of 21
Q1.
Completing shapes
(a)
The diagram shows a rhombus.
One angle is 46°
Calculate the size of the angle marked m
Show your working.
.............................°
2 marks
(b)
Complete the accurate drawing below to make a rhombus of side length 6 cm.
Show your method, either by showing working or by leaving in your construction lines.
2 marks
Page 2 of 21
Q2.
Parallel lines
In this diagram the two parallel lines are marked with arrows.
Not drawn accurately
Work out the value of y
y = ...................
2 marks
Page 3 of 21
Q3.
Computer
Kay is drawing shapes on her computer.
(a)
She wants to draw this triangle. She needs to know angles a , b and c.
Calculate angles a , b and c.
a = .............................. °
b = .............................. °
c = .............................. °
2 marks
(b)
Kay draws a rhombus:
Calculate angles d and e .
d = .............................. °
1 mark
e = .............................. °
1 mark
Page 4 of 21
(c)
Kay types the instructions to draw a regular pentagon:
repeat 5 [forward 10, left turn 72]
Complete the instructions to draw a regular hexagon.
repeat 6 [forward 10, left turn ............................]
1 mark
Q4.
Star
The shape below has 3 identical white tiles and 3 identical grey tiles.
The sides of each tile are all the same length.
Opposite sides of each tile are parallel.
One of the angles is 70°
NOT TO SCALE
(a)
Calculate the size of angle k .
angle k = ...............................°
1 mark
Page 5 of 21
(b)
Calculate the size of angle m.
Show your working.
angle m = ...............................°
2 marks
Q5.
Finding Angles
The diagram shows two isosceles triangles inside a parallelogram.
(a)
On the diagram, mark another angle that is 75°
Label it 75°
1 mark
Page 6 of 21
(b)
Calculate the size of the angle marked k
Show your working.
........................ °C
2 marks
Now look at the triangle drawn on the straight line PQ.
(c)
Write x in terms of y
1 mark
(d)
Now write x in terms of t and w
1 mark
(e)
Use your answers to parts (c) and (d) to show that y = t + w
1 mark
Page 7 of 21
Q6.
Finding Angles
The diagram shows two isosceles triangles inside a parallelogram.
(a)
On the diagram, mark another angle that is 75°
Label it 75°
1 mark
(b)
Calculate the size of the angle marked k
Show your working.
........................ °C
2 marks
Page 8 of 21
Q7.
Shapes
The drawing shows how shapes A and B fit together to make a right-angled triangle.
Work out the size of each of the angles in shape B.
Write them in the correct place in shape B below.
3 marks
Page 9 of 21
Q8.
Angle p
This shape has been made from two congruent isosceles triangles.
Not drawn accurately
What is the size of angle p ?
p = .........................°
2 marks
Page 10 of 21
Q9.
Three straight lines
The diagram shows three straight lines.
Work out the sizes of angles a , b and c
Give reasons for your answers.
a = ................° because ..........................................................................................
..........................................................................................
..........................................................................................
1 mark
b = ................° because ..........................................................................................
..........................................................................................
..........................................................................................
1 mark
c = ................° because ..........................................................................................
..........................................................................................
..........................................................................................
1 mark
Page 11 of 21
Q10.
Angle k
Look at the diagram.
Not drawn accurately
AB is a straight line.
Work out the size of angle k
k = ...................... °
2 marks
Page 12 of 21
M1.
(a)
134
2
or
Shows a correct method with not more than one computational error, eg
•
180 – 46
•
360 – (2 × 46), then ÷ 2
•
360 – 92 = 270 (error), 270 ÷ 2 = 135
•
2 × 46 = 94 (error),
= 33
1
(b)
Rhombus completed within the accuracy as specified on the overlay
2
or
122 shown and correctly placed at either open end of the 6 cm lines,
even if the angle is incorrectly drawn, eg
•
or
Two arcs of equal radius (even if not 6 cm) are shown, centred at the
open ends of the 6 cm lines, indicating compasses have been used
1
[4]
Page 13 of 21
M2.
36
2
or
Forms or implies a correct equation
eg
•
5y = 180
•
The 3y and 2y together are 180
•
180 – 3y = 2y
•
2y = 72
•
10 lots is 360°
•
180 ÷ 5
•
72 ÷ 2
1
[2]
M3.
(a)
Indicates a = 100 and b = 140
1
Indicates c= 120
Allow follow through , ie 360 – (angles a and b), provided angle c
is given as greater than 90°.
1
(b)
Indicates d = 50
1
Indicates e = 130
Allow follow through, ie 180 – angle d, provided angle e is given
as greater than 90°.
1
(c)
Indicates a correct value, eg:
•
60
•
300
Correct values include 360n ± 60, for any integer n.
1
[5]
M4.
(a)
Indicates 110
1
Page 14 of 21
(b)
For 2m indicates 50
For only 1m shows a complete, correct method, eg:
•
360 – 3 × 70 = 150, 150 ÷ 3
•
70 + m = 120
•
120 – 70
•
70 + 70 + 70 + m + m + m = 360
•
3 × 70 = 210, 150 + 210 = 360, 150 ÷ 3 = 60
•
360 – 70 – 70 – 70 = 160, ÷ 3
1
[3]
M5.
(a)
A correct angle of 75 indicated.
Do not accept extra lines added to the diagram to create an angle
of 75
1
(b)
50
2
or
Shows a correct method
eg
•
(180 – 80) ÷ 2
•
100 ÷ 2
Do not accept follow through from an incorrectly marked 75 in the
lower triangle
1
(c)
Correct expression or equation with x as the subject
eg
•
180 – y
•
x = 180 – y
1
Page 15 of 21
(d)
Correct expression or equation with x as the subject
eg
•
180 – t –w
•
x = 180 –(t + w)
!
Units inserted
Ignore
eg, accept
• x = 180° – y
!
Correct equations in (c) and (d) but with x not the subject
eg
• x + y = 180 and x + t + w = 180
Mark as 0, 1
1
(e)
Correct explanation
eg
•
180 – y = 180 – (t + w), so y = t + w
•
x = 180 – y, x = 180 – (t + w), so y = t = w
•
x + t + w = 180 and x + y = 180,
so y = t + w
Do not accept spurious explanation
eg
• y = 180 – x, x = 180 – t – w,
so y = t + w
1
[6]
M6.
(a)
A correct angle of 75 indicated.
Do not accept extra lines added to the diagram to create an angle
of 75
1
Page 16 of 21
(b)
50
2
or
Shows a correct method
eg
•
(180 – 80) ÷ 2
•
100 ÷ 2
Do not accept follow through from an incorrectly marked 75 in
the lower triangle
1
[3]
M7.
All four angles correct and correctly positioned, ie
!
Accept units omitted
units incorrect, eg
• 50%
Withhold one mark only for the first occurrence
3
or
At least three angles correct and correctly positioned
or
All four correct angles shown but identification of which angle
is which size is not clear
2
or
At least two angles correct and correctly positioned
Accept follow through
For 2m or 1m, follow through for 47°
as 360 – sum of their other three angles
or
97 – their 50
1
[3]
Page 17 of 21
M8.
140
2
or
Shows the value 110 or 220
or
Shows or implies a complete correct method with not more than one
computational error
eg
•
360 – 2 × (180 – 35 × 2)
•
360 – (360 – 4 × 35)
•
70 × 2
•
•
35 + 35 = 80 (error), 180 – 80 = 100
360 – 100 × 2 = 160
1
[2]
Page 18 of 21
M9.
Gives a = 50 and gives a correct reason
eg
•
Angle a is on a straight line with 130, so a = 180 – 130
•
a is supplementary with 130, so a + 130 = 180
•
The angle vertically opposite 130 is 130, 360 – (130 + 130) = 100,
(angles at a point) a is
= 50 (also vertically opposite)
Accept: minimally acceptable reason
eg
• On a straight line
• Supplementary
• Opposite angles and angles at a point
Do not accept: informal reason without the correct
geometrical property identified
eg
• 180 – 130
• 360 – 260
2
Do not accept: incomplete reason
eg
• It is adjacent to the 130° angle
U1
Gives b = 60 and gives a correct reason
eg
•
Angle b is vertically opposite the 60° angle, so it is also 60°
•
The angle on a straight line with b is 120, so b is 360 – 120 – 120 – 60
(angles at a point)
Accept: minimally acceptable reason
eg
• Opposite
• Angles on a straight line and angles at a point
Do not accept: informal reason without the correct
geometrical property identified
eg
• b is equal to the 60° angle next to it
Do not accept: incomplete reason
eg
• It is the same as the 60° angle
U1
Page 19 of 21
Gives c = 70 and gives a correct reason
eg
•
There are 180° in a triangle, so c = 180 – 50 – 60
•
The exterior angle of a triangle is equal to the sum of the two opposite
interior angles, so c = 130 – 60
Accept: minimally acceptable reason
eg
• Angles in a triangle
• Exterior angle = sum of two opposite interior angles
• We’ve already got 50 and 60 in the triangle
!
Follow through
Accept as 180 – (their a + b), alongside a correct reason referring
to angles in a triangle, or as 130 – their b alongside a correct
reason referring to an exterior angle of a triangle
Do not accept: informal reason without the correct
geometrical property identified
eg
• 180 – (a + b)
• 130 – b
Do not accept: incomplete reason
eg
• It is in a triangle
• All the inside angles add up to 180°
U1
[3]
M10.
35
2
or
Shows the values 50 and 95 or the value 145
or
Shows a complete correct method with not more than one computational error
eg
•
180 – 130 = 50,
180 – 85 = 105 (error),
180 – 50 – 105 = 25
•
(130 + 85) – 180
1
[2]
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Page 21 of 21