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Special Properties, SMB-025

A quadrilateral is pictured below.
 

What is the value of `x`?   (3 marks)

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`126^@`

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`text{Sum of exterior angles = 360°}`

`y^{\circ}` `=360-(127+114+65)`  
  `=360-306`  
  `=54^{\circ}`  

 
`:.x^{\circ}=180-54 = 126^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-024

A quadrilateral is drawn below.
 

What is the value of `x`?   (3 marks)

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`117^@`

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`y^{\circ}=180-75 = 105^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`z^{\circ}` `=360-(130+105+62)`  
  `=360-297`  
  `=63^{\circ}`  

 
`:.x^{\circ}=180-63 = 117^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-024

A quadrilateral is drawn below.

What is the value of `x`?   (3 marks)

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`117^@`

Show Worked Solution

`y^{\circ}=180-75 = 105^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`z^{\circ}` `=360-(130+105+62)`  
  `=360-297`  
  `=63^{\circ}`  

 
`:.x^{\circ}=180-63 = 117^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-013 MC

Which statement is always true?

  1. Scalene triangles have two angles that are equal.
  2. All angles in a parallelogram are equal.
  3. The opposite sides of a trapezium are equal in length.
  4. The diagonals of a rhombus are perpendicular to each other.
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`D`

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`text{Consider each option:}`

`A:\ \text{Isosceles (not scalene) have two equal angles.}`

`B:\ \text{Only opposite angles in a parallelogram are equal.}`

`C:\ \text{At least one pair of opposite sides of a trapezium are not equal.}`

`D:\ \text{Rhombuses have perpendicular diagonals.}`

`=>D`

Special Properties, SMB-010 MC

Which of these are always equal in length?

  1. the diagonals of a rhombus
  2. the diagonals of a parallelogram
  3. the opposite sides of a parallelogram
  4. the opposite sides of a trapezium
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`C`

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`text{Consider each option:}`

`A:\ \text{rhombus diagonals are perpendicular but not always equal}`

`B:\ \text{parallelogram diagonals not always equal (see below)}`

`C:\ \text{always true (see above)}`

`D:\ \text{at least 1 pair of opposite sides of a trapezium are not equal}`
\(\Rightarrow C\)

Special Properties, SMB-009 MC

`PQRS` is a parallelogram.

Which of these must be a property of `PQRS`?

  1. Line `PS` is perpendicular to line `PQ`.
  2. Line `PQ` is parallel to line `PS`.
  3. Diagonals `PR` and `SQ` are perpendicular.
  4. Line `PS` is parallel to line `QR`.
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`D`

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`text{By elimination:}`

`A\ \text{and}\ B\ \text{clearly incorrect.}`

`C\ \text{true if all sides are equal (rhombus) but not true for all parallelograms.}`

`text(Line)\ PS\ text(must be parallel to line)\ QR.`

`=>D`

Special Properties, SMB-008

The sum of the internal angles of a polygon can be calculated by drawing triangles from any given vertex as shown below. 
 

What is the size of the angle marked `x` in the diagram below?   (2 marks)

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`107°`

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`text{Since the quadrilateral was divided into two triangles}`

`=>\ \text{Sum of internal angles}\ = 2 xx 180 = 360^{\circ}`

`:. x` `= 360-(103 + 88 + 62)`
  `= 360-253`
  `= 107°`

Plane Geometry, 2UA 2005 HSC 5b

The diagram shows a parallelogram `ABCD` with `∠DAB = 120^@`. The side `DC` is produced to `E` so that `AD = BE`.

Prove that `ΔBCE` is equilateral.  (3 marks)

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`text(See Worked Solutions)`

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`BC` `= AD\ text{(opposite sides of parallelogram}\ ABCD)`
`∠BCD` `= 120^@\ text{(opposite angles of parallelogram}\ ABCD)`
`∠BCE` `= 60^@\ (∠DCE\ text{is a straight angle)}`
`∠CEB` `= 60^@\ text{(base angles of isosceles}\ \Delta BCE)`
`∠CBE` `= 60^@\ text{(angle sum of}\ ΔBCE)`

 
`:.ΔBCE\ text(is equilateral)`