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Properties of Geometric Figures, SM-Bank 041

A pentagon is pictured below.
 

  1. By drawing triangles from one vertex, or otherwise, calculate the sum of the internal angles of a pentagon.   (1 mark)

  2. Determine the value of \(x^{\circ}\) in the pentagon.     (2 marks)
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i.    \(540^{\circ}\)

ii.   \(110^{\circ}\)

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i.    \(\text{Pentagon can be divided into 3 triangles (from one chosen vertex).}\)

\(\text{Sum of internal angles}\ = 3 \times 180 = 540^{\circ}\)
 

ii.    \(540\) \(=x + 2 \times 90 + 135+115 \)  
\(540\) \(=x+430\)  
\(x^{\circ}\) \(=540-430\)  
  \(=110^{\circ}\)  

Properties of Geometric Figures, SM-Bank 040

 

Determine the value of the two unknown angles in the quadrilateral above, giving reasons for your answer.     (3 marks)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\) \(=5x+3x+79+105 \)  
\(8x\) \(=360-184\)  
\(x^{\circ}\) \(=\dfrac{176}{8}\)  
  \(=22^{\circ}\)  

 
\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Properties of Geometric Figures, SM-Bank 038

\(ABCD\) is a trapezium.
 

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

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\(67^{\circ}\)

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\(DA \parallel CB \ \ (ABCD\ \text{is a trapezium}) \)

\(x+113\) \(=180\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-113\)  
  \(=67^{\circ}\)  

Properties of Geometric Figures, SM-Bank 037

\(ABCD\) is a trapezium.
 

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

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\(83^{\circ}\)

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\(AD \parallel BC \ \ (ABCD\ \text{is a trapezium}) \)

\(x+97\) \(=180\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-97\)  
  \(=83^{\circ}\)  

Properties of Geometric Figures, SM-Bank 036

\(ABCD\) is a parallelogram.
 

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

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\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\) \(=180\ \ \text{(cointerior angles)} \)  
\(a^{\circ}\) \(=180-125\)  
  \(=55^{\circ}\)  
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\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\) \(=180\ \ \text{(cointerior angles)} \)  
\(a^{\circ}\) \(=180-125\)  
  \(=55^{\circ}\)  

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

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

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

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

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

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

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

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

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