True or False

Geometry Level 2

If AOB is diameter, and A D C = 12 0 ° \angle ADC = 120 ^{°} then C A B = 3 0 ° \angle CAB = 30^{°}

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False True

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1 solution

Michael Fuller
Jan 4, 2016

Construct all the dotted red lines as shown. We are trying to find C A B = C A O \angle CAB = \angle CAO .

Consider the cyclic quadrilateral A D C F ADCF . As opposite angles in a cyclic quadrilateral equal 180 ° 180° , A F C = 60 ° \angle AFC = 60° .

As angles at the centre are equal to twice the angles at the circumference, A O C = 2 A F C = 120 ° \angle AOC = 2 \angle AFC=120° .

As A O AO and C O CO are radii, A O C \triangle AOC is isosceles, and hence the base angles are both equal to x x .

120 ° + 2 x = 180 ° x = 30 ° \Rightarrow~~ 120°+2x=180° ~~\Rightarrow~~ x= 30°

Hence the statement is True \large\color{#20A900}{\boxed{\text{True}}} .

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