Solving an inscribed angle from intercepted arc measure

Key takeaway: This problem uses the inscribed angle theorem: an inscribed angle measures half its intercepted arc.

Step 1: Use the inscribed angle theorem

For an inscribed angle, the measure of the angle is half the measure of its intercepted arc. Here,

$\widehat{EC}=120^\circ.$

So

$m\angle EBC=\frac{1}{2}(120^\circ)=60^\circ.$

Step 2: Set up the equation

The angle is also given as

$(5x-44)^\circ.$

Set the expressions equal:

$5x-44=60.$

Add 44 to both sides:

$5x=104.$

Divide by 5:

$x=20.8.$

Final answer

$\boxed{20.8}$

If rounding is requested, this value is already correct to the nearest tenth.

Why the theorem applies

Angle $EBC$ has its vertex on the circle and its sides cut off arc $EC$. That makes it an inscribed angle, not a central angle. The key relationship is always:

$\text{inscribed angle} = \frac{1}{2}(\text{intercepted arc}).$

Quick check

A 120-degree arc should produce a 60-degree inscribed angle. Since $5x-44$ must equal 60, the solution $x=20.8$ is consistent.

---

Pitfalls the pros know 👇 The most common error is using the arc measure as if it were the angle measure. Students sometimes write $5x-44=120$, but that would be correct only for a central angle. Another mistake is forgetting to check whether the angle vertex lies on the circle. If it does, the angle is inscribed and must be half the arc. Finally, if you solve for $x$ and get a decimal, do not round too early before substitution; keep the exact value until the end.

What if the problem changes? If the intercepted arc were $150^\circ$ instead of $120^\circ$, then the inscribed angle would be $75^\circ$. The equation would become $5x-44=75$, giving $5x=119$ and $x=23.8$. If the angle were a central angle with the same arc, then you would set the expression equal to the full arc measure instead of half, which changes the solution completely.

Tags: inscribed angle theorem, intercepted arc, central angle