Question

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

Answer 1

Answer:

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

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

Step-by-step explanation:

step 1

Find the measure of angle KIJ

we know that

The inscribed angle measures half that of the arc comprising

so

$m<KIJ=\frac{1}{2}(arc\ KJ)$

substitute

$m<KIJ=\frac{1}{2}(124\°)=62\°$

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

$m<IKQ=\frac{1}{2}(arc\ IC)$

$m<IKQ=\frac{1}{2}(38\°)=19\°$

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

$m<CQJ=99\°$