1 Answer

Sajjan Singh · Answer 1 · 2021-11-25T21:42:43+0000

By the inscribed angle theorem, we have

$m\angle C=\frac12m\angle AMB=\frac12m\widehat{AB}$

(where
$\widehat{AB}$ denotes the minor arc AB)

$\implies m\angle C=\frac{86^\circ}2=43^\circ$

The interior angles of any triangle sum to 180º in measure, so

$m\angle A+m\angle B+m\angle C=180^\circ\implies m\angle B=86^\circ$

Use the inscribed angle theorem again to find the measure of arc BC:

$m\angle A=\frac12m\angle BMC=\frac12m\widehat{BC}$

$\implies m\widehat{BC}=2\cdot51^\circ=102^\circ$

Similarly find the measure of arc AC:

$m\angle B=\frac12m\widehat{AC}$

$\implies m\widehat{AC}=2\cdot86^\circ=172^\circ$

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