Question

Given: m arc KJ = 124°, m arc IC =38° Find: m∠CQJ, m∠LIJ.

Answer 1

Part 1) The measure of angle LIJ is
`m<LIJ=118\°`

Part 2) The measure of angle CQJ is
`m<CQJ=99\°`

Explanation:

step 1

Find the measure of angle KIJ

we know that

The inscribed angle measures half that of the arc comprising

so

`m<KIJ=(1)/(2)(arc\ KJ)`

substitute

`m<KIJ=(1)/(2)(124\°)=62\°`

step 2

Find the measure of angle LIJ

we know that

`m<LIJ+m<KIJ=180\°` -----> by supplementary angles

substitute

`m<LIJ+62\°=180\°`

`m<LIJ=180\°-62\°=118\°`

step 3

Find the measure of angle IKQ

we know that

The inscribed angle measures half that of the arc comprising

so

`m<IKQ=(1)/(2)(arc\ IC)`

`m<IKQ=(1)/(2)(38\°)=19\°`

step 4

Find the measure of angle IQK

Remember that the sum of the internal angles of a triangle must be equal to 180 degrees

In the triangle IQK

`m<IKQ+m<KIJ+m<IQK=180\°`

substitute the values

`19\°+62\°+m<IQK=180\°`

`m<IQK=180\°-(19\°+62\°)=99\°`

step 5

Find the measure of angle CQJ

we know that

m<CQJ=m<IQK -----> by vertical angles

so

`m<CQJ=99\°`