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

\(180^{\circ}\) \(=x+20+110\ \ \text{(cointerior angles)} \)  
\(x^{\circ}\) \(=180-130\)  
  \(=50^{\circ}\)  
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\(\angle DBE = 180-160=20^{\circ}\ \ \text{(180° in a straight line)}\)

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

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 016

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

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\(46^{\circ}\)

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

\(180^{\circ}\) \(=a+60+74\)  
\(a^{\circ}\) \(=180-134\)  
  \(=46^{\circ}\)  

Solving Problems, SM-Bank 015

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

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\(45^{\circ}\)

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

\(180^{\circ}\) \(=x+70+65\)  
\(x^{\circ}\) \(=180-135\)  
  \(=45^{\circ}\)  

Solving Problems, SM-Bank 011

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

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

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\(\angle UTQ = 125^{\circ}\ \ \text{(corresponding angles)} \)

\(\angle STX=125^{\circ}\ \ \text{(vertically opposite angles)}\)

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

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\(\angle UTQ = 125^{\circ}\ \ \text{(corresponding angles)} \)

\(\angle STX=125^{\circ}\ \ \text{(vertically opposite angles)}\)

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

Solving Problems, SM-Bank 007

Determine if two lines in the diagram below are parallel, giving reasons for your answer.   (2 marks)
 

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\(\angle \text{unknown} = 360-310=50^{\circ}\ \ \text{(360° about a point)}\)

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

\(140 + 50= 190^{\circ} \neq 180^{\circ}\)

\(\therefore \ \text{Lines are not parallel.}\)

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\(\angle \text{unknown} = 360-310=50^{\circ}\ \ \text{(360° about a point)}\)

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

\(140 + 50 = 190^{\circ} \neq 180^{\circ}\)

\(\therefore \ \text{Lines are not parallel.}\)

Solving Problems, SM-Bank 008

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

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\(40^{\circ}\)

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

\(180\) \(=x+65+75\)  
\(180\) \(=x+140\)  
\(x^{\circ}\) \(=180-40\)  
  \(=40^{\circ}\)  

Solving Problems, SM-Bank 028

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

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\(14°\)

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\(\angle RQT + \angle UTQ\) \(=180\ \ \text{(cointerior angles)}\)  
\(110+5x\) \(=180\)  
\(5x\) \(=180-110\)  
\(x^{\circ}\) \(=\dfrac{70}{5}\)  
  \(=14^{\circ}\)  

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 025

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

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\(11°\)

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\(\angle ADE + \angle DAC = 180^{\circ}\ \ \text{(cointerior angles)}\)

\(\angle ADE = 180-92=88^{\circ}\)

\(44+4x\) \(=88\)  
\(4x\) \(=44\)  
\(x^{\circ}\) \(=\dfrac{44}{4} \)  
  \(=11^{\circ}\)  

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

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

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

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