Question

90 POINTS!!!!! Given: KLIJ inscr. in k(O),m∠K = 64°, measure of arc LI = 69°, measure of arc IJ = 59°, measure of arc KJ =97°

Answer 1

In any cyclic quadrilateral, angles opposite one another are supplementary, meaning

$m\angle K+m\angle I=m\angle L+m\angle J=180^\circ$

and given that $\boxed{m\angle K=64^\circ}$, we have $\boxed{m\angle I=116^\circ}$.

By the inscribed angle theorem,

$m\angle JLK=\dfrac12m\widehat{KJ}$

$m\angle ILJ=\dfrac12m\widehat{IJ}$

and since

$m\angle L=m\angle JLK+m\angle ILJ$

we have

$m\angle L=\dfrac{97^\circ+59^\circ}2\implies\boxed{m\angle L=78^\circ}$

and it follows that

$m\angle J=180^\circ-m\angle L\implies\boxed{m\angle J=102^\circ}$