An angle whose vertex lies on the circle and whose sides are chords of the circle. Its measure is exactly half the measure of the intercepted arc.
Imagine sitting on the edge of a circular stadium and looking at two players on the field. The angle your eyes make is an inscribed angle. No matter where you sit on the same arc, that viewing angle stays the same—and it's always half of what you'd see from the center. It's like the circle is 'halving' your perspective compared to the center's view.
Showing a random 20 of 50 problems.
Example 1
medium
In circle O, inscribed angle ∠ABC intercepts arc AC. If arc AC=134°, and arc CD=70°, find inscribed angle ∠ADC that intercepts arc AC from the same side.Both ∠ABC and ∠ADC intercept arc AC = 134°; find ∠ADC.
Example 2
medium
An inscribed angle is (2x)° and intercepts an arc of (3x+20)°. Find x.
Example 3
medium
A tangent-chord angle intercepts an arc of 80°. Find the tangent-chord angle.
Example 4
medium
In a circle, an inscribed angle intercepts an arc, and the rest of the circle is 260°. Find the inscribed angle.Remaining arc = 260°; find inscribed angle x intercepting the other arc.
Example 5
medium
Triangle ABC is inscribed in a circle with BC as a diameter. Angle A is what?BC is a diameter; find inscribed angle x at vertex A.
Example 6
medium
A triangle inscribed in a circle has arcs of 100°, 140°, 120° opposite its vertices. Find all three angles.
Example 7
easy
In a circle, a central angle measures 140°. An inscribed angle intercepts the same arc. Find the inscribed angle.Central angle = 140°; find inscribed angle x on the same arc.
Example 8
hard
Quadrilateral ABCD is inscribed in a circle. If ∠A=82°, find ∠C. Then, if arc AB=96° and arc BC=110°, find ∠ADC.
Example 9
easy
An inscribed angle intercepts an arc of 80°. What is the measure of the inscribed angle?Find inscribed angle x given arc = 80°.
Example 10
easy
A central angle is 72°. Find the inscribed angle on the same arc.Central angle = 72°; find inscribed angle x.
Example 11
easy
Arc AB in circle O measures 96°. Inscribed angle ∠ACB has its vertex on the major arc. Find ∠ACB.Arc AB = 96°; find inscribed angle ∠ACB = x.
Example 12
hard
Cyclic quadrilateral ABCD has ∠A=(2x+10)° and ∠C=(3x−5)°. Find x and ∠A.
Example 13
easy
Why does the inscribed angle stay the same as you move its vertex along the same arc?
Example 14
easy
An angle inscribed in a semicircle intercepts the diameter's arc (180°). Find the angle.Angle inscribed in a semicircle: find x.
Example 15
medium
Points A, B, C, D lie on a circle in order. Inscribed angle BAC = 30° and inscribed angle BDC = ? (both subtend chord BC from the same side).
Example 16
easy
An inscribed angle measures 25°. What is the measure of its intercepted arc?Inscribed angle = 25°; find intercepted arc x.
Example 17
medium
Two inscribed angles in different circles each intercept an arc of 90°. Compare their measures.
Example 18
hard
In circle O, two chords AB and CD intersect inside at P. Arc AC=84° and arc BD=36°. Find ∠APC.
Example 19
medium
Why are opposite angles of a cyclic quadrilateral supplementary?
Example 20
hard
A tangent and a chord meet at point T on a circle. The chord cuts an arc of 110° on the near side. Find the tangent-chord angle.