In geometry, an inscribed angle is formed when two secant lines of a circle (or, in a degenerate case, when one secant line and one tangent line of that circle) intersect on the circle. Table of Geometry, from the 1728 Cyclopaedia. ... A secant line of a curve is a line that intersects two or more points on the curve. ... In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ... A secant line of a curve is a line that intersects two or more points on the curve. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...

Typically, it is easiest to think of an inscribed angle as being defined by two chords of the circle sharing an endpoint. A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... An endpoint or end point is a mark of termination or completion. ...

Property

An inscribed angle is said to intercept an arc on the circle. The intercepted arc is the portion of the circle that is in the interior of the angle. The measure of the intercepted arc (equal to its central angle) is exactly twice the measure of the inscribed angle. In Euclidean geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of a circle. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... A central angle is an angle whose vertex is the center of a circle, and whose sides pass through a pair of points on the circle, thereby subtending an arc between those two points whose angle is (by definition) equal to the central angle itself. ...

This single property has a number of consequences within the circle. For example, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal. It also allows one to prove that the opposite angles of a cyclic quadrilateral are supplementary. In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. ... A pair of angles are supplementary if their respective measures sum to 180 degrees. ...

Proof

To understand this proof, it is useful to draw a diagram.

1. Inscribed angles where one chord is a diameter

Let O be the center of a circle. Choose two points on the circle, and call them V and A. Draw line VO and extended past O so that it intersects the circle at point B which is diametrically opposite the point V. Draw an angle whose vertex is point V and whose sides pass through points A and B. Image File history File links No higher resolution available. ... Antipodal points on the surface of a sphere are diametrically opposite; on the other side of a globe. ...

Angle BOA is a central angle; call it θ. Draw line OA. Lines OV and OA are both radii of the circle, so they have equal lengths. Therefore triangle VOA is isosceles, so angle BVA (the inscribed angle) and angle VAO are equal; let each of them be denoted as ψ. Remote Authentication Dial In User Service (RADIUS) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Angles BOA and AOV are supplementary. They add up to 180°, since line VB passing through O is a straight line. Therefore angle AOV measures 180° − θ. A pair of angles are supplementary if their respective measures sum to 180 degrees. ...

It is known that the three angles of a triangle add up to 180°, and the three angles of triangle VOA are: For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

180° − θ

ψ

ψ.

Therefore

Subtract 180° from both sides,

where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB.

2. Inscribed angles with the center of the circle in their interior

Given a circle whose center is point O, choose three points V, C, and D on the circle. Draw lines VC and VD: angle DVC is an inscribed angle. Now draw line VO and extend it past point O so that it intersects the circle at point E. Angle DVC subtends arc DC on the circle. Image File history File links No higher resolution available. ...

Suppose this arc includes point E within it. Point E is diametrically opposite to point V. Angles DVE and EVC are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

Therefore

then let

so that

Draw lines OC and OD. Angle DOC is a central angle, but so are angles DOE and EOC, and

Let

so that

From Part One we know that θ₁ = 2ψ₁ and that θ₂ = 2ψ₂. Combining these results with equation (2) yields

therefore, by equation (1),

3. Inscribed angles with the center of the circle in their exterior

[The previous case can be extended to cover the case where the measure of the inscribed angle is the difference between two inscribed angles as discussed in the first part of this proof.] Image File history File links No higher resolution available. ...

Given a circle whose center is point O, choose three points V, C, and D on the circle. Draw lines VC and VD: angle DVC is an inscribed angle. Now draw line VO and extend it past point O so that it intersects the circle at point E. Angle DVC subtends arc DC on the circle.

Suppose this arc does not include point E within it. Point E is diametrically opposite to point V. Angles DVE and EVC are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

Therefore

.

then let

so that

Draw lines OC and OD. Angle DOC is a central angle, but so are angles DOE and EOC, and

Let

so that

From Part One we know that θ₁ = 2ψ₁ and that θ₂ = 2ψ₂. Combining these results with equation (4) yields

θ₀ = 2ψ₂ − 2ψ₁

therefore, by equation (3),

θ₀ = 2ψ₀.

4. Conclusion

The measure of any inscribed angle is half the measure of its intercepted arc.

External links

• Munching on Inscribed Angles at cut-the-knot
• Arc Central Angle With interactive animation
• Arc Peripheral (inscribed) Angle With interactive animation
• Arc Central Angle Theorem With interactive animation