SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Properties of Geometric Figures, SM-Bank 026

Determine the value of \(a^{\circ}\), \(b^{\circ}\), and \(c^{\circ}\), giving reasons for your answer.   (3 marks)

Show Answers Only

\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  
Show Worked Solution

\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  

Properties of Geometric Figures, SM-Bank 025

  1. Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

  2. Determine the value of \(b^{\circ}\), giving reasons for your answer.   (2 marks)
Show Answers Only

a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 024

An isosceles triangle is pictured below.
 

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

Show Answers Only

\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 023

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

Show Answers Only
\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  
Show Worked Solution
\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  

Properties of Geometric Figures, SM-Bank 022

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)  

Properties of Geometric Figures, SM-Bank 021

 

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

Show Answers Only

\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  

Properties of Geometric Figures, SM-Bank 011

In the diagram, \(AB\) is parallel to \(DE\).
 

  1. On the diagram, label the alternate angles to \(a^{\circ}\) and \(b^{\circ}\).   (1 mark)

  2. Using part (a), show that the sum of internal angles of a triangle equals 180°.   (2 marks)
Show Answers Only

a.    
     

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Show Worked Solution

a.    
       

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Properties of Geometrical Figures, SM-Bank 018

The diagram below shows a right-angled triangle.
 

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

Show Answers Only

\(138^{\circ}\)

Show Worked Solution

\(\text{Right angle}\ = 90^{\circ} \)

\(a^{\circ}\) \(=48+90\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=138^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 017

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(40^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 70)\ \ \text{(180° in triangle)} \)  
  \(=180-140\)  
  \(=40^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 016

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(110^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 35)\ \ \text{(180° in triangle)} \)  
  \(=110^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 015

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(71^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=180-38\ \ \text{(180° in triangle)} \)  
\(a^{\circ}\) \(=\dfrac{142}{2}\)  
  \(=71^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 014

The diagram below shows an isosceles triangle.
 

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

Show Answers Only

\(59^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2x^{\circ}\) \(=180-62\ \ \text{(180° in triangle)} \)  
\(x^{\circ}\) \(=\dfrac{118}{2}\)  
  \(=59^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 004 MC

A triangle is divided into 2 parts by a straight line.

The angles are then labelled.
 

Which statement is true about the sum of angles?

  1. `b + c + d = 180^@`
  2. `c + d + e = 360^@`
  3. `a + b + f + g = 360^@`
  4. `d + e + f + g = 180^@`
Show Answers Only

`C`

Show Worked Solution

`text(Consider each option:)`

`text(Option A:)\ \ b + c + d != 180\ \ => \ b+c = 180^@`

`text(Option B:)\ \ c + d + e != 360^@\ \ => \ c + d + e = 180^@\ \ text{(angle sum of triangle)}`

`text(Option C:)\ \ a + b + f + g = 360^@`

  `=>\ text(Correct since the angle sum of a quadrilateral = 360°)`

`text(Option D:)\ \ d + e + f + g != 180\ \ => \ e + f  = 180^@`

`=> C`

Properties of Geometrical Figures, SM-Bank 003 MC

Tom drew this shape on grid paper.

Which one of the shapes below when joined to Tom's shape without an overlap, will not make isosceles triangle?

A.  
 B.  
 C.  
 D.  
Show Answers Only

\(C\)

Show Worked Solution

\(\text{An isosceles triangle has two sides of the same length.}\)

\(\text{Option C will form a scalene triangle (all sides different lengths).}\)

\(\Rightarrow C\)

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
Show Answers Only

`D`

Show Worked Solution

`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-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
Show Answers Only

`D`

Show Worked Solution

`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`