Question

please help me with this question thank youufind the measure of the arc or angle indicated

Answer 1

We have to find the angle at vertex J.

We know the shortest arc JK and the shortest arc JL.

As a property of the arcs, the inscribed angle $m\angle L=\frac{1}{2}\text{arc JK}=\frac{1}{2}\cdot126\degree=63\degree$We can apply the same property to angle K:

$m\angle K=\frac{1}{2}\text{arc JL}=\frac{1}{2}\cdot94\degree=47\degree$

Now we have two angles of the triangle. We know that the sum of the measures of the interior angles of a triangle is equal to 180°, so we can write:

$\begin{gathered} m\angle J+m\angle L+m\angle K=180\degree \\ m\angle J+63+47=180 \\ m\angle J=180-63-47 \\ m\angle J=70\degree \end{gathered}$

Answer: The angle at vertex J is 70 degrees.

NOTE: we could have solve it applying the relation between inscribed angles and arcs with angle J as:

$m\angle J=\frac{1}{2}\text{arc LK}=\frac{1}{2}(360-94-126)=\frac{1}{2}\cdot140=70\degree$

The arc LK is the full circle, 360°, less the other arcs, 94° and 126°.