Question

Find the value of the unknowns $x$ and $y$ in the following diagrams:

Answer 1

(i) $y+{120}^{\circ }={180}^{\circ }$ (Linear pair)

$y={180}^{\circ }-{120}^{\circ }={60}^{\circ }$

$x+y+{50}^{\circ }={180}^{\circ }$ (Angle sum property)

$x+{60}^{\circ }+{50}^{\circ }={180}^{\circ }$

$x+{110}^{\circ }={180}^{\circ }$

$x={180}^{\circ }-{110}^{\circ }={70}^{\circ }$

(ii) $y={80}^{\circ }$ (Vertically opposite angles)

$y+x+{50}^{\circ }={180}^{\circ }$ (Angle sum property)

${80}^{\circ }+x+{50}^{\circ }={180}^{\circ }$

$x+{130}^{\circ }={180}^{\circ }$

$x={180}^{\circ }-{130}^{\circ }={50}^{\circ }$

(iii) $y+{50}^{\circ }+{60}^{\circ }={180}^{\circ }$(Angle sum property)

$y={180}^{\circ }-{60}^{\circ }-{50}^{\circ }={70}^{\circ }$

$x+y={180}^{\circ }$ (Linear pair)

$x={180}^{\circ }-y={180}^{\circ }-{70}^{\circ }={110}^{\circ }$

(iv) $x={60}^{\circ }$ (Vertically opposite angles)

${30}^{\circ }+x+y={180}^{\circ }$

${30}^{\circ }+{60}^{\circ }+y={180}^{\circ }$

$y={180}^{\circ }-{30}^{\circ }-{60}^{\circ }={90}^{\circ }$

(v) $y={90}^{\circ }$ (Vertically opposite angles)

$x+x+y={180}^{\circ }$ (Angle sum property)

$2x+y={180}^{\circ }$

$2x+{90}^{\circ }={180}^{\circ }$

$2x={180}^{\circ }-{90}^{\circ }={90}^{\circ }$

$x=\frac{{90}^{\circ }}{2}={45}^{\circ }$

(vi)

$y=x$ (Vertically opposite angles)

$a=x$ (Vertically opposite angles)

$b=x$ (Vertically opposite angles)

$a+b+y={180}^{\circ }$ (Angle sum property)

$x+x+x={180}^{\circ }$

$3x={180}^{\circ }$

$x=\frac{{180}^{\circ }}{3}={60}^{\circ }$

$y=x={60}^{\circ }$