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

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

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

\(x^{\circ}\) \(=105-70\)  
  \(=35^{\circ}\)  
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\(\text{Add middle parallel line:}\)
 

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

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

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

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

Solving Problems, SM-Bank 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

\(\text{Add parallel line:}\)
 

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

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

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

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

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

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

Solving Problems, SM-Bank 014

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

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\(\angle y^{\circ} = 40^{\circ}\ \ \text{(alternate angles)} \)

\(\angle x^{\circ}=360=40 = 320^{\circ}\ \ \text{(360° about a point)}\)

Show Worked Solution

\(\angle y^{\circ} = 40^{\circ}\ \ \text{(alternate angles)} \)

\(\angle x^{\circ}=360=40 = 320^{\circ}\ \ \text{(360° about a point)}\)

Solving Problems, SM-Bank 013

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

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\(\angle b^{\circ} = 360-325 = 35^{\circ}\ \ \text{(360° about a point)} \)

\(\angle a^{\circ}=35^{\circ}\ \ \text{(alternate angles)}\)

Show Worked Solution

\(\angle b^{\circ} = 360-325 = 35^{\circ}\ \ \text{(360° about a point)} \)

\(\angle a^{\circ}=35^{\circ}\ \ \text{(alternate angles)}\)

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

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

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

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

Show Worked Solution

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

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

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

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

Solving Problems, SM-Bank 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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\(\text{Extend the parallel line on the left:}\)
 

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

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

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