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Properties of Geometric Figures, SM-Bank 021

 

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  

Properties of Geometric Figures, SM-Bank 011

In the diagram, \(AB\) is parallel to \(DE\).
 

  1. On the diagram, label the alternate angles to \(a^{\circ}\) and \(b^{\circ}\).   (1 mark)

  2. Using part (a), show that the sum of internal angles of a triangle equals 180°.   (2 marks)
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a.    
     

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Show Worked Solution

a.    
       

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Volume, SM-Bank 091

An ancient building has the shape of a trapezoidal prism.

The shaded side is a trapezium.
 

 
What is the volume of the building in m³ ?  (2 marks)

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\(1344\ \text{m}^3\)

Show Worked Solution

\(\text{Area of trapezium}\)

\(A\) \(=\dfrac{h}{2}\times (a+b)\)
  \(=\dfrac{6}{2}\times (12+16)\)
  \(=3\times 28\)
  \(=84\ \text{m}^2\)

 

\(\therefore\ V\) \(=Ah\)
  \(=84\times 16\)
  \(=1344\ \text{m}^3\)

Volume, SM-Bank 090

A rectangular trough in a paddock provides water for horses.

Its measurements can be seen below:
  

  1. Calculate the volume of the trough in cubic metres.  (2 marks)
  2. Given that one cubic metre holds 1000 litres of water, what is the capacity of the trough in litres?  (1 mark)

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a.    \(3\ \text{m}^3\)

b.    \(\text{3000 litres}\)

Show Worked Solution
a.    \(\text{Volume}\) \(=Ah\)
    \(=(0.5\times 0.75)\times 8\)
    \(=0.375\times 8\)
    \(=3\ \text{m}^3\)

 

b.    \(\text{Capacity}\) \(=1000\times 3\)
    \(=3000\ \text{litres}\)

Properties of Geometrical Figures, SM-Bank 018

The diagram below shows a right-angled triangle.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(138^{\circ}\)

Show Worked Solution

\(\text{Right angle}\ = 90^{\circ} \)

\(a^{\circ}\) \(=48+90\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=138^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 017

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(40^{\circ}\)

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\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 70)\ \ \text{(180° in triangle)} \)  
  \(=180-140\)  
  \(=40^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 016

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(110^{\circ}\)

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\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 35)\ \ \text{(180° in triangle)} \)  
  \(=110^{\circ}\)  

Volume, SM-Bank 089

A large sculpture is made in the shape of a cube.

The total length of all of its edges is 60 metres.

What is the volume of the cube in cubic metres?  (2 marks)

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\(125\ \text{m}^3\)

Show Worked Solution

\(\text{A cube has 12 edges.}\)

\(\text{Length of 1 edge} =\dfrac{60}{12} = 5\ \text{m}\)
 

\(\therefore\ \text{Volume of cube}\) \(=5\times 5\times 5\)
  \(=125\ \text{m}^3\)

Properties of Geometrical Figures, SM-Bank 015

The diagram below shows an isosceles triangle.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(71^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=180-38\ \ \text{(180° in triangle)} \)  
\(a^{\circ}\) \(=\dfrac{142}{2}\)  
  \(=71^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 014

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(59^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2x^{\circ}\) \(=180-62\ \ \text{(180° in triangle)} \)  
\(x^{\circ}\) \(=\dfrac{118}{2}\)  
  \(=59^{\circ}\)  

Volume, SM Bank 088 MC

Alan is moving house and is packing his belongings in rectangular cardboard boxes.

The height of each box is 0.5 metres.

Which box has a volume of 0.12 cubic metres?
 

 

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\(D\)

Show Worked Solution

\(\text{Volume}\)

\(=0.4\times 0.6\times 0.5\)

\(= 0.12\ \text{m}^3\)

\(\Rightarrow D\)

Volume, SM-Bank 087 MC

Two bricks can be joined to make three different rectangular prisms. Two of them are shown here.
 

 
What would be the measurements of the third prism?

  1. 18 cm by 16 cm by 7 cm
  2. 36 cm by 8 cm by 7 cm
  3. 32 cm by 18 cm by 7 cm
  4. 36 cm by 14 cm by 8 cm
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\(B\)

Show Worked Solution

\(\text{36 cm by 8 cm by 7 cm}\)
 

\(\Rightarrow B\)

Volume, SM-Bank 080

Wes made a small model staircase by stacking blocks.

There are no gaps between blocks.
 

 If each block is 1 cubic centimetre, what is the volume of the model staircase?  (2 marks)

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\(80\ \text{cm}^2\)

Show Worked Solution

\(\text{Area of cross-section = 20 cm}^2\)

\(V\) \(=Ah\)
  \(=20\times 4\)
  \(=80\ \text{cm}^3\)

 
\(\therefore\ \text{The volume of the model staircase = 80 cm}^3\)

Volume, SM-Bank 079 MC

A sculpture is pictured from 3 different angles below:
 

 
The sculpture is in the shape of

  1. a trapezium and a circle.
  2. a trapezium and a cylinder.
  3. a sphere and a prism.
  4. a cylinder and a prism.
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\(D\)

Show Worked Solution

\(\text{The bottom shape is a trapezoidal prism (not a trapezium).}\)

\(\therefore\ \text{The sculpture is in the shape of a cylinder and a prism.}\)

\(\Rightarrow D\)

Properties of Geometric Figures, SM-Bank 007

In the diagram \(AB\) is a straight line.

Calculate the size of the angle marked \(x^{\circ}\), giving reasons for your answer.    (3 marks)

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\(\text{Equilateral triangle}\ \ \Rightarrow\ \ \text{all angles}\ = 60^{\circ}\)

\(\text{Vertically opposite angles of 60° are equal (see diagram)}\)

\(x^{\circ} = 180-(90+60) = 30^{\circ}\ \ \text{(180° in straight line)}\)

Show Worked Solution

\(\text{Equilateral triangle}\ \ \Rightarrow\ \ \text{all angles}\ = 60^{\circ}\)

\(\text{Vertically opposite angles of 60° are equal (see diagram)}\)

\(x^{\circ} = 180-(90+60) = 30^{\circ}\ \ \text{(180° in straight line)}\)

Properties of Geometric Figures, SM-Bank 006

Pablo creates a design that is made up of 3 rectangles and 2 straight lines, as shown below.
 

What is the size of angle \(x^{\circ}\)?   (3 marks)

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\(\text{135 degrees}\)

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\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal} \)

\(\text{Since there is 180° in a  straight line:}\)

\(x + 45\) \(= 180\)
\(x^{\circ}\) \(= 135^{\circ}\)

Solving Problems, SM-Bank 029

What is the value of \(x^{\circ}\) in this diagram?   (2 marks)

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\(54^{\circ}\)

Show Worked Solution

\(\text{Adjacent angle to 144°}\ = 180-144=36^{\circ}\ \ \text{(180° in straight line)}\)

\(x^{\circ}= 180-(90 + 36)=54^{\circ}\ \ \text{(180° in straight line)}\)

Properties of Geometrical Figures, SM-Bank 004 MC

A triangle is divided into 2 parts by a straight line.

The angles are then labelled.
 

Which statement is true about the sum of angles?

  1. `b + c + d = 180^@`
  2. `c + d + e = 360^@`
  3. `a + b + f + g = 360^@`
  4. `d + e + f + g = 180^@`
Show Answers Only

`C`

Show Worked Solution

`text(Consider each option:)`

`text(Option A:)\ \ b + c + d != 180\ \ => \ b+c = 180^@`

`text(Option B:)\ \ c + d + e != 360^@\ \ => \ c + d + e = 180^@\ \ text{(angle sum of triangle)}`

`text(Option C:)\ \ a + b + f + g = 360^@`

  `=>\ text(Correct since the angle sum of a quadrilateral = 360°)`

`text(Option D:)\ \ d + e + f + g != 180\ \ => \ e + f  = 180^@`

`=> C`

Properties of Geometrical Figures, SM-Bank 003 MC

Tom drew this shape on grid paper.

Which one of the shapes below when joined to Tom's shape without an overlap, will not make isosceles triangle?

A.  
 B.  
 C.  
 D.  
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\(C\)

Show Worked Solution

\(\text{An isosceles triangle has two sides of the same length.}\)

\(\text{Option C will form a scalene triangle (all sides different lengths).}\)

\(\Rightarrow C\)

Unit Conversion, SM-Bank 023

  1. By considering the dimensions of a square with an area of 1 square kilometre, complete the unit conversion equation below:

1 km² = ________ m  ×  ________ m = ______________ km²   (1 mark)

  1. The Barangaroo precinct in Sydney has a total floor space of 681 000 square metres.
  2. Find the area of this floor space in square kilometres.  (1 mark)
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a.    \(1\ \text{km}^{2} = 1000\ \text{m}\ \times\ 1000\ \text{m}\ = 1\ 000\ 000\ \text{km}^{2} \)

b.    \(0.681\ \text{km}^{2}\)

Show Worked Solution

a.    \(1\ \text{km}^{2} = 1000\ \text{m}\ \times\ 1000\ \text{m}\ = 1\ 000\ 000\ \text{km}^{2} \)
 

b.    \(\text{Area}\) \(= \dfrac{681\ 000}{1\ 000\ 000}\)  
  \(=0.681\ \text{km}^{2}\)  

Unit Conversion, SM-Bank 022

  1. By considering the dimensions of a square with an area of 1 square kilometre, complete the unit conversion equation below:

1 km² = _______ m  ×  _______ m = ____________ km²   (1 mark)

  1. A rural property is in the shape of a rectangle with sides of length 1.20 kilometres and 2.36 kilometres.
  2. Find the area of the property in square kilometres and using the conversion equation, express the area in square metres.  (2 marks)
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a.    \(1\ \text{km}^{2} = 1000\ \text{m}\ \times\ 1000\ \text{m}\ = 1\ 000\ 000\ \text{km}^{2} \)

b.    \(2\ 832\ 000\ \text{m}^{2}\)

Show Worked Solution

a.    \(1\ \text{km}^{2} = 1000\ \text{m}\ \times\ 1000\ \text{m}\ = 1\ 000\ 000\ \text{km}^{2} \)
 

b.    \(\text{Area}\) \(= 1.20 \times 2.36\)  
  \(=2.832\ \text{km}^{2}\)  
  \(=2.832 \times 1\ 000\ 000\  \text{m}^{2}\)  
  \(=2\ 832\ 000\ \text{m}^{2}\)  

Unit Conversion, SM-Bank 021

The square below has an area of 1 square metre.
 

  1. Complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = __________ cm²   (1 mark)

  1. A circular poster has an area of 0.09\(\pi\) square metres.
  2. Express this area in square centimetres, giving your answer in exact form.  (1 mark)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(900\pi\ \text{cm}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(\text{0.09}\pi\ \text{m}^{2}\) \(= 0.09\pi \times 10\ 000 \)  
  \(=900\pi\ \text{cm}^2 \)

Unit Conversion, SM-Bank 020

The square below has an area of 1 square metre.
 

  1. Complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = __________ cm²   (1 mark)

  1. A circular stage has an area of 265 000 square centimetres.
  2. Express this area in square metres.  (1 mark)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(26\ \text{m}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(\text{265 000 cm}^{2}\) \(= \dfrac{26\ 000}{10\ 000}\)  
  \(=26\ \text{m}^{2}\)  

Unit Conversion, SM-Bank 019

The square below has an area of 1 square metre.
 

  1. Complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = __________ cm²   (1 mark)

  1. A rectangular serviette has an area of 0.075 square metres.
  2. Express this area in square centimetres.  (1 mark)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(750\ \text{cm}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(\text{0.075 m}^{2}\) \(=0.075 \times 10\ 000\)  
  \(=750\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 018

The square below has an area of 1 square metre.
 

  1. Complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = __________ cm²   (1 mark)

  1. A doubles tennis court has an area of 260 square metres.
  2. Express this area in square centimetres.  (1 mark)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(2\ 600\ 000\ \text{cm}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(\text{260 m}^{2}\) \(=260 \times 10\ 000\)  
  \(=2\ 600\ 000\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 017

  1. By considering the dimensions of a square with an area of 1 square metre, complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = __________ cm²   (1 mark)

  1. A circular elevator has a diameter of 1.6 metres.
  2. Find the area of the elevator's floor in square metres and using the conversion equation, express the area to the nearest square centimetre.  (2 marks)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(20\ 106\ \text{cm}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(\text{Radius}\ =\dfrac{1.6}{2} = 0.8\ \text{m} \)

\(\text{Area}\) \(=\pi \times 0.8^2\)  
  \(=2.01061…\ \text{m}^{2}\)  
  \(=2.01061… \times 10\ 000\ \text{cm}^{2}\)  
  \(=20\ 106\ \text{cm}^{2}\)  

Unit Conversion, SM-Bank 016

The square below has an area of 1 square metre.
 

  1. Complete the unit conversion equation below:

1 m² = _____ cm  ×  _____ cm = _______ cm²   (1 mark)

  1. A circular playground has an area of 2000 square metres.
  2. Using the conversion equation, or otherwise, express the area in square centimetres.  (1 mark)
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a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)

b.    \(20\ 000\ 000\ \text{mm}^{2}\)

Show Worked Solution

a.    \(1\ \text{m}^{2} = 100\ \text{cm}\ \times\ 100\ \text{cm}\ = 10\ 000\ \text{cm}^{2} \)
 

b.    \(2000\ \text{m}^{2}\) \(= 2000 \times 10\ 000\ \text{cm}^{2} \)
  \(= 20\ 000\ 000\ \text{cm}^{2} \)

Unit Conversion, SM-Bank 015

The square below has an area of 1 square centimetre.
 

  1. Complete the unit conversion equation below:

1 cm² = _____ mm  ×  _____ mm = _______ mm²   (1 mark)

  1. A rectangle has an area of 1100 square millimetres.
  2. Using the conversion equation, or otherwise, express the area of the circle in square centimetres.  (1 mark)
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a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)

b.    \(11\ \text{cm}^{2}\)

Show Worked Solution

a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)
 

b.    \(1100\ \text{mm}^{2}\) \(= \dfrac{1100}{100} \)
  \(=11\ \text{cm}^{2} \)

Unit Conversion, SM-Bank 014

  1. By considering the dimensions of a square with an area of 1 square centimetre, complete the unit conversion equation below:

1 cm² = _____ mm  ×  _____ mm = _______ mm²   (1 mark)

  1. A circle has an area of 950 square centimetres.
  2. Using the conversion equation, or otherwise, express the area of the circle in square millimetres.  (1 mark)
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a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)

b.    \(95\ 000\ \text{mm}^{2}\)

Show Worked Solution

a.    \(1\ \text{cm}^{2} = 10\ \text{mm}\ \times\ 10\ \text{mm}\ = 100\ \text{mm}^{2} \)
 

b.    \(900\ \text{cm}^{2}\) \(= 900 \times 100\ \text{mm}^{2} \)
  \(= 90\ 000\ \text{mm}^{2} \)

Solving Problems, SM-Bank 022

In the diagram below, \(DG\) is parallel to \(BC\), and \(\angle ABC = 115^{\circ} \).
  

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

Show Answers Only

\(\angle CBE = 180-115=65^{\circ}\ \ \text{(180° in a straight line)}\)

\(x^{\circ} = 65^{\circ}\ \ \text{(alternate angles)}\)

Show Worked Solution

\(\angle CBE = 180-115=65^{\circ}\ \ \text{(180° in a straight line)}\)

\(x^{\circ} = 65^{\circ}\ \ \text{(alternate angles)}\)

Solving Problems, SM-Bank 021

In the diagram below, \(BE\) is parallel to \(CD\), and \(\angle ABE = 160^{\circ} \).
  

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(\angle DBE = 180-160=20^{\circ}\ \ \text{(180° in a straight line)}\)

\(180^{\circ}\) \(=x+20+110\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-130\)  
  \(=50^{\circ}\)  
Show Worked Solution

\(\angle DBE = 180-160=20^{\circ}\ \ \text{(180° in a straight line)}\)

\(180^{\circ}\) \(=x+20+110\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-130\)  
  \(=50^{\circ}\)  

Solving Problems, SM-Bank 020

In the diagram below, \(QR\) is parallel to \(SU\).
  

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(\angle STP = 38^{\circ}\ \ \text{(corresponding angles)}\)

\((x+30)^{\circ}\) \(=180-38\ \ \text{(180° in straight line)} \)  
\(x^{\circ}\) \(=142-30\)  
  \(=112^{\circ}\)  
Show Worked Solution

\(\angle STP = 38^{\circ}\ \ \text{(corresponding angles)}\)

\((x+30)^{\circ}\) \(=180-38\ \ \text{(180° in straight line)} \)  
\(x^{\circ}\) \(=142-30\)  
  \(=112^{\circ}\)  

Solving Problems, SM-Bank 019

In the diagram below, \(PR\) is parallel to \(TU\) and reflex \(\angle QST = 255^{\circ}\)
  

Find the value of \(x^{\circ}\), giving reasons for your answer.   (3 marks)

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\(\angle QST = 360-255 = 105^{\circ}\ \ \text{(360° about a point)}\)

\(\angle VSQ =70^{\circ} \ \ \text{(alternate angles)} \)

\(\angle VST\ =x^{\circ} \ \ \text{(alternate angles)} \)

\(x^{\circ}\) \(=105-70\)  
  \(=35^{\circ}\)  
Show Worked Solution

\(\text{Add middle parallel line:}\)
 

\(\angle QST = 360-255 = 105^{\circ}\ \ \text{(360° about a point)}\)

\(\angle VSQ =70^{\circ} \ \ \text{(alternate angles)} \)

\(\angle VST\ =x^{\circ} \ \ \text{(alternate angles)} \)

\(x^{\circ}\) \(=105-70 \)  
  \(=35^{\circ}\)  

Solving Problems, SM-Bank 018

In the diagram below, find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)
  

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\(y^{\circ}\ =70^{\circ} \ \ \text{(alternate angles)} \)

\(x^{\circ}\ =z^{\circ} \ \ \text{(alternate angles)} \)

\((z+y)^{\circ}\) \(=110^{\circ}\ \)  
\((x+y)^{\circ}\) \(=110^{\circ}\ \)  
\(x^{\circ}\) \(=110-70\)  
  \(=40^{\circ}\)  
Show Worked Solution

\(\text{Extend middle parallel line:}\)
 

\(y^{\circ}\ =70^{\circ} \ \ \text{(alternate angles)} \)

\(x^{\circ}\ =z^{\circ} \ \ \text{(alternate angles)} \)

\((z+y)^{\circ}\) \(=110^{\circ}\ \)  
\((x+y)^{\circ}\) \(=110^{\circ}\ \)  
\(x^{\circ}\) \(=110-70\)  
  \(=40^{\circ}\)  

Solving Problems, SM-Bank 017

In the diagram below, find the value of \(x^{\circ}\), giving reasons for your answer.   (3 marks)
  

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\(\text{Full interior angle}\ = 360-275=85^{\circ} \ \ \text{(360° about a point)} \)
 

\(\text{Since cointerior angles sum to 180°,}\)

\(\Rightarrow \text{interior angle (1)}\ = 180-125=55^{\circ} \)

\(\text{Since angles about a point sum to 360°,}\)

\(\Rightarrow \text{interior angle (2)}\ = 85-55=30^{\circ} \)
 

\(x^{\circ}\) \(=180-30\ \ \text{(cointerior angles)} \)  
  \(=150^{\circ}\)  
Show Worked Solution

\(\text{Add parallel line:}\)
 

\(\text{Full interior angle}\ = 360-275=85^{\circ} \ \ \text{(360° about a point)} \)
 

\(\text{Since cointerior angles sum to 180°,}\)

\(\Rightarrow \text{interior angle (1)}\ = 180-125=55^{\circ} \)

\(\text{Since angles about a point sum to 360°,}\)

\(\Rightarrow \text{interior angle (2)}\ = 85-55=30^{\circ} \)
 

\(x^{\circ}\) \(=180-30\ \ \text{(cointerior angles)} \)  
  \(=150^{\circ}\)