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

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

\((z+y)^{\circ}\) \(=110^{\circ}\ \)  
\((x+y)^{\circ}\) \(=110^{\circ}\ \)  
\(x^{\circ}\) \(=110-70\)  
  \(=40^{\circ}\)  
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\(\text{Extend middle parallel line:}\)
 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solving Problems, SM-Bank 010

In the diagram below, \(BC\) is parallel to \(DE\) and \(\angle ACB\) is a right-angle.
 

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

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\(\text{Extend line}\ BC: \)
 

\(\angle GCF=180-120=60^{\circ}\ \ \text{(180° in a straight line)}\)

\(x^{\circ} = 60^{\circ} \ \ \text{(corresponding angles)}\)

Show Worked Solution

\(\text{Extend line}\ BC: \)
 

\(\angle GCF=180-120=60^{\circ}\ \ \text{(180° in a straight line)}\)

\(x^{\circ} = 60^{\circ} \ \ \text{(corresponding angles)}\)

Solving Problems, SM-Bank 026

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

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\(15°\)

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\(\text{Extend the parallel line on the left:}\)
 

\(\text{Angle opposite}\ \angle ABC = 3x^{\circ}\ \ \text{(vertically opposite)}\)

\(\angle DEB = 360-(90+135) = 135^{\circ}\ \ \text{(360° about a point)} \)

\(3x+135\) \(=180\ \ \text{(cointerior angles)} \)  
\(3x\) \(=180-135\)  
\(x^{\circ}\) \(=\dfrac{45}{3}\)  
  \(=15^{\circ}\)  

Solving Problems, SM-Bank 023

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

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\(50°\)

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

\(\text{Alternate angles are equal}\ (x^{\circ}) \).

\(\text{Cointerior angles sum to 180° (110° and 70°)}\)

\(x^{\circ} = 120-70=50^{\circ} \)