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Congruency, SMB-016

Igor was designing a shield using 10 congruent (isosceles) triangles, as shown in the diagram below.
 

How many degrees in the angle marked `x`?   (3 marks)

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`72^@`

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`text(Angles at centre of circle)\ = 360/10 = 36^@`

`text{Since triangles are isosceles:}`

`180` `= 36 + 2x`
`2x` `= 180-36`
  `= 144 `
`:. x` `= 72^@`

Special Properties, SMB-020 MC

`AB` is the diameter of a circle, centre `O`.

There are 3 triangles drawn in the lower semi-circle and the angles at the centre are all equal to `x^@`.

The three triangles are best described as:

  1. isosceles
  2. scalene
  3. right-angled
  4. equilateral
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`D`

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`3x=180^{\circ}\ \=>\ x=60^{\circ} `

`AO=OC=OD=OB\ \ text{(radii of circle)}`

`=>\ text{Since angles opposite equal sides of a triangle are}`

`text(equal, all triangle angles can be found to equal 60°.)`

`:.\ text(The three triangles are equilateral.)`

`=>D`

Special Properties, SMB-004 MC

Which one of the following triangles is impossible to draw?

  1. a right angled triangle with two acute angles
  2. an isosceles triangle with one right angle
  3. a scalene triangle with three acute angles
  4. a right angled triangle with one obtuse angle
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`D`

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`text(A right angle = 90°.)`

`text{Since an obtuse angle is greater than 90°, it is impossible for}`

`text(a triangle, with an angle sum less than 180°, to have both.)`

`=>D`

Special Properties, SMB-003 MC

Select the statement that is true about triangle `ABC`.

  1. Triangle `ABC` is a scalene triangle.
  2. Triangle `ABC` has exactly 2 equal sides.
  3. Triangle `ABC` is an equilateral triangle.
  4. Triangle `ABC` is an obtuse triangle.
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`C`

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`angleA = 180-(60 + 60) = 60^@`

`:. text(All angles are)\ 60^@.`

`:. text(Triangle)\ ABC\ text(is an equilateral.)`

`=>C`

Special Properties, SMB-002 MC

A triangle has two acute angles.

What type of angle couldn't the third angle be?

  1. an acute angle
  2. an obtuse angle
  3. a right-angle
  4. a reflex angle
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`D`

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`text(A triangle’s angles add up to 180°, and a reflex angle is)`

`text(greater than 180°.)`

`:.\ text(The third angle cannot be reflex.)`

`=>D`

Special Properties, SMB-001 MC

Which of the following triangle types is impossible to draw?

  1. a right-angled, scalene triangle
  2. a right-angled, equilateral triangle
  3. an obtuse-angled, isosceles triangle
  4. an acute-angled, scalene triangle
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`B`

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`text(An equilateral triangle has all angles = 60°.)`

`:.\ text(A right-angled, equilateral triangle is impossible.)`

`=>B`

Plane Geometry, 2UA 2004 HSC 2b

In the diagram, `ABC`  is an isosceles triangle with  `AB = AC`  and  `/_BAC = 38^@`. The line `BC` is produced to `D`. 

Find the size of `/_ACD`. Give reasons for your answer.   (2 marks)

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`109^@`

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Plane Geometry, 2UA 2004 HSC 2b Answer

`/_ABC` `= 1/2 (180-38)\ \ \ text{(base angle of isosceles}\ Delta ABC text{)}`
  `= 71^@`

 

`:.\ /_ACD` `= 71 + 38\ \ \ text{(exterior angle of}\ Delta ABC text{)}`
  `= 109^@`

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)`

Plane Geometry, 2UA 2008 HSC 4a

In the diagram, `XR` bisects `/_PRQ` and `XY\ text(||)\ QR`.

Prove that `Delta XYR` is an isosceles triangle.   (2 marks)

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`text(Proof)\ text{(See Worked Solutions)}`

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`text{Since}\ XR\ text{bisects}\ /_PRQ`

`/_XRQ` `= /_YRX = theta`
`/_RXY` `= theta\ \ \ text{(} text(alternate angles,)\ XY\ text(||)\ QR text{)}`

 
`:.\ Delta XYR\ \ text(is isosceles)`

Plane Geometry, 2UA 2011 HSC 6a

The diagram shows a regular pentagon `ABCDE`. Sides `ED` and `BC` are produced to meet at `P`.
  

  1. Find the size of `/_CDE`.    (1 mark)

  2. Hence, show that `Delta EPC` is isosceles.    (2 marks)

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  1. `108°`
  2. `text(Proof)\ \ text{(see Worked Solutions)}`
Show Worked Solution
i.  

`text(Angle sum of pentagon)=(5-2) xx 180°=540°`

`:.\ /_CDE` `= 540/5\ \ \ text{(regular pentagon has equal angles)}`
  `= 108°`
MARKER’S COMMENT: Very few students solved part (i) efficiently. Remember the general formula for the sum of internal angles equals (# sides – 2) x 90°.

 
ii.  `text(Show)\ Delta EPC\ text(is isosceles)`

`text(S)text(ince)\ ED=CD\ \ text{(sides of a regular pentagon)}`

`Delta ECD\ text(is isosceles)`

`/_DEC=1/2 xx (180-108)= 36^{\circ}\ \ \ text{(Angle sum of}\ Delta DEC text{)}`

`/_CDP=72^@\ \ \ (\angle PDE\ \text{is a straight angle})`

`/_DCP=72^@\ \ \ (\angle PCB\ \text{is a straight angle})`

`=> /_CPD= 180-(72 + 72)=36^{\circ}\ \ \ text{(angle sum of}\ Delta CPD text{)}`

`:.\ Delta EPC\ \text(is isosceles)\ \ \ text{(2 equal angles)}`