smc-5008-20-Exterior angles

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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)

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

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)

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

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)

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

Show Worked Solution

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