Question

find the measure of ∠ilj.
write your answer as a whole number or a decimal.
m∠ilj =
submit

Explanation:

Step1: Recall the sum of angles in a circle

The sum of angles around a point (in a circle) is \(360^{\circ}\).

Step2: Calculate the measure of \(\angle ILJ\)

We know that in a circle, the central angles add up to \(360^{\circ}\). Let \(m\angle ILJ = x\).
We have two angles given: \(67^{\circ}\) and \(89^{\circ}\). Since the figure has four central angles (formed by two intersecting diameters), and the opposite angles are equal.
The sum of all central angles \(= 360^{\circ}\).
\(2\times67^{\circ}+2\times89^{\circ}+ 2x=360^{\circ}\) (but a simpler way is to note that \(67^{\circ}+89^{\circ}+x + y=360^{\circ}\) and since the opposite angles are equal \(x = y\)).
Another approach: The sum of non - \(\angle ILJ\) angles (considering the circle's total angle \(360^{\circ}\)):
The sum of the known non - \(\angle ILJ\) angles (since there are two pairs of equal angles) is \(2\times67^{\circ}+2\times89^{\circ}\). But if we consider the four central angles \(a,b,c,d\) (\(a = 67^{\circ}\), \(b = 89^{\circ}\), \(c=x\), \(d=x\) as they are vertical angles).
\(67^{\circ}+89^{\circ}+x + x=360^{\circ}\)
\(156^{\circ}+2x=360^{\circ}\)
Subtract \(156^{\circ}\) from both sides: \(2x=360^{\circ}- 156^{\circ}=204^{\circ}\)
Divide by \(2\): \(x = 102^{\circ}\)

Answer:

\(102\)