From the diagram, we can see that ∠ J intercepts arc MK.
This arc consists of two smaller arcs — ML and LK. Hence, adding their measures, we can calculate the measure of MK.
mMK&=mML+mLK
& ⇓
mMK&= 130^(∘)+ 110^(∘)=240^(∘)
According to the stated theorem, the measure of ∠ J is half the measure of its intercepted arc MK.
m∠ J=mMK/2=240^(∘)/2=120^(∘)
Similarly, we can find the measure of ∠ K. This angle intercepts arc JL, which consists of JM and ML.
Thus, by adding the given measures of JM and ML, we can find the measure of JL.
mJL&=mJM+mML
& ⇓
mJL&= 54^(∘)+ 130^(∘)=184^(∘)
Finally, dividing the measure of mJL by 2, we can calculate the measure of m∠ K.
m∠ K=mJL/2=184^(∘)/2=92^(∘)
Now, since JKLM is an inscribed quadrilateral, we can apply the Inscribed Quadrilateral Theorem. Let's recall what it states!
Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
According to this theorem, opposite angles ∠ J and ∠ L, as well as ∠ K and ∠ M, are supplementary. This means that the sum of their measures is equal to 180^(∘).
m∠ J+m∠ L=180^(∘)
m∠ K+m∠ M=180^(∘)
Let's substitute the measures of these angles and solve each equation for the unknown variable. We will start with the first equation!
