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°

Find: All angles of KLIJ

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=\frac12m\widehat{KJ}`

`m\angle ILJ=\frac12m\widehat{IJ}`

and since

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

we have

`m\angle L=\frac{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}`