Inscribed Angle
In short: An inscribed angle is pinned on the rim of a circle, with two chords as arms, and it measures exactly half of the rim stretch trapped between those arms.
GeometryGeometryG-C.A.2
What is Inscribed Angle?
Stand on the edge of a circular stage and look at two lamps that also sit on the edge: the opening between your two lines of sight is an inscribed angle. Its corner lies on the circle, and its arms are chords ending on the circle.
The Inscribed Angle Theorem says its measure is half the intercepted arc: \(m\angle ABC = \frac{1}{2}\,(\text{arc } AC)\). Two consequences follow. Corners that intercept the same arc are equal, and a corner intercepting a semicircle is a right angle.
How to spot it
- A figure shows a vertex on the circle, two chords from it, and a highlighted arc between their far ends.
- The wording mentions an intercepted arc, or says an angle is inscribed in a circle.
- A triangle with all three corners on a circle has one side that is a diameter.
- An expression for the angle and one for the arc are given, and the two are linked by a factor of one half.
The method
- Identify the arc that lies inside the angle, between the far endpoints of its two chords.
- Write the relation: angle \(=\frac{1}{2}\) arc, or arc \(= 2 \times\) angle.
- Substitute the given expressions and solve the equation for the variable.
- Check special cases: a diameter as a side means a right angle at the vertex.
Inscribed Angle: solved example
Problem
In a circle, \(m\angle ABC = (5x + 1)^{\circ}\) is inscribed and its intercepted arc \(AC\) measures \((12x – 10)^{\circ}\). Find \(x\), the angle and the arc.
Solution
The angle is half the arc: \(5x + 1 = \frac{12x – 10}{2} = 6x – 5\). Subtract \(5x\) and add 5 to get \(x = 6\).
Then \(m\angle ABC = 5(6) + 1 = 31\) and the arc is \(12(6) – 10 = 62\). Check: \(62 \div 2 = 31\). The results are \(31^{\circ}\) and \(62^{\circ}\).
Try it
Question 1
An inscribed angle measures \(38^{\circ}\). What is its intercepted arc?
Show the answer
Double it: \(2 \times 38 = 76\), so the arc is \(76^{\circ}\).
Question 2
A triangle is inscribed in a circle and one of its sides is a diameter. How large is the angle opposite that side?
Show the answer
That angle intercepts a semicircle of \(180^{\circ}\), so it measures \(90^{\circ}\).
Inscribed Angle vs. Central Angle
Where the corner sits decides everything. A central angle sits at the center and matches its arc degree for degree. An inscribed angle sits on the rim and measures only half of the same arc. One arc of 132 degrees gives two openings: 132 and 66.
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