Answer:
Part 1) The measure of arc EHL is [tex]108\°[/tex]
Part 2) The measure of angle LVE is [tex]54\°[/tex]
Step-by-step explanation:
step 1
Let
x-----> the measure of arc EHL
y----> the measure of arc EVL
we know that
The measurement of the outer angle is the semi-difference of the arcs it encompasses.
so
[tex]m<EYL=\frac{1}{2}(y-x)[/tex]
we have
[tex]m<EYL=72\°[/tex]
substitute
[tex]72\°=\frac{1}{2}(y-x)[/tex]
[tex]144\°=(y-x)[/tex]
[tex]y=144\°+x[/tex] ------> equation A
Remember that
[tex]x+y=360\°[/tex] -----> equation B ( complete circle)
substitute equation A in equation B and solve for x
[tex]x+(144\°+x)=360\°[/tex]
[tex]2x=360\°-144\°[/tex]
[tex]x=216\°/2=108\°[/tex]
Find the value of y
[tex]y=144\°+x[/tex]
[tex]y=144\°+108\°=252\°[/tex]
therefore
The measure of arc EHL is [tex]108\°[/tex]
The measure of arc EVL is [tex]252\°[/tex]
step 2
Find the measure of angle LVE
we know that
The inscribed angle measures half that of the arc comprising
Let
x-----> the measure of arc EHL
[tex]m<LVE=\frac{1}{2}(x)[/tex]
we have
[tex]x=108\°[/tex]
substitute
[tex]m<LVE=\frac{1}{2}(108\°)=54\°[/tex]
