What is an Inscribed Angle?
Contents
Definition: An inscribed angle is an angle whose vertex lies on the circumference of the circle. The vertex is the common endpoint of the two sides of the angle.
An inscribed angle can be defined as the angle subtended at a point on the circle by two given points on the circle. An inscribed angle is an angle formed in the interior of a circle by two chords that have a common endpoint on the circle.
This common endpoint forms the vertex of the inscribed angle.
Inscribed Angle of a Circle
Inscribed Angle Theorems & Characteristics
- Angle at the center is double the angle at the circumference.
The angle formed at the center of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. - An angle inscribed in a semi-circle is a right angle.
Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle always. - In a circle, inscribed angles that intercept the same arc are congruent.
Angles in the same segment are equal. Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Uses
The inscribed angle is used in many proofs of elementary Euclidean geometry of the plane. Inscribed angle is the basis for several other theorems related to the power of a point with respect to a circle.
Importance
Inscribed angle is a very important part of the circle theorems. Four circle theorems are directly based on the inscribed angle.
Angle at the center is double the angle at the circumference. An angle inscribed in a semi-circle is a right angle.
In a circle, inscribed angles that intercept the same arc are congruent.
In a cyclic quadrilateral, there are four inscribed angles. Opposite angles in a cyclic quadrilateral are supplementary.
Inscribed Angle Formula
The angle is inscribed in a circle if an angle has its vertex on that circle and has sides containing two chords of the same circle.
An inscribed angle is half in measure of its intercepted arc or can say angle at the center is double the angle at the circumference (inscribed angle).
An angle inscribed in a semi-circle is a right angle.
In a circle, inscribed angles that intercept the same arc are congruent.
Opposite angles in a cyclic quadrilateral adds to 180.
Inscribed Angle Examples
Inscribed Angle in Semi-Circles
An angle inscribed in a semi-circle is a right angle
A diameter subtends an inscribed angle of measure 90° always on the circumference of a circle.
Example
In the above circle, O is the center point, AC is the diameter. From the ends of the diameter, two lines are drawn AB and CB that are meeting on the circumference of the circle at point B.
Angle B is an inscribed angle drawn from the ends of the diameter of a circle to its circumference, angle B is a right angle (90°).
In the above circle, O is the center point, AB is the diameter. From the ends of the diameter, two lines are drawn AC and BC that are meeting on the circumference of the circle at point C.
Angle C is an inscribed angle drawn from the ends of the diameter of a circle to its circumference, angle C is a right angle (90°), and angle AOB is 180°, double of 90°. Angle at the center (∠AOB) is double the angle at the circumference (∠ACB).
Inscribed Angle of a Circle
In a circle, inscribed angles that intercept the same arc are congruent.
Any two inscribed angles in a circle with the same intercepted arcs are always congruent (equal in measure).
Example
In the above circle, O is the center point; angle A and angle D are inscribed angles that share the same arc BC. The measure of angle A is equal to the measure of angle D. Angle A is congruent to angle D.
∠A ≅ ∠D
Inscribed Angle in Cyclic Quadrilaterals
The Opposite angles in a cyclic quadrilateral are supplementary.
A cyclic quadrilateral has all its vertices on the circumference of the circle. In a cyclic quadrilateral, there are four inscribed angles, opposite angles in a cyclic quadrilateral are supplementary. Sum of opposite angles of a cyclic quadrilateral is equal to 180° always.
Example
In the figure above, a quadrilateral is inscribed in a circle. It’s a cyclic quadrilateral as all its vertices are on the circumference of the circle. The four inscribed angles are ∠a, ∠b, ∠c and ∠d. Opposite angles in a cyclic quadrilateral are supplementary.
∠a + ∠c = 180°
∠b + ∠d = 180°
Example
In the figure above, ABCD is a cyclic quadrilateral. ∠A is opposite to ∠B and ∠C is opposite to ∠D.
∠A + ∠B = 97° + 83° = 180°
∠C + ∠D = 98° + 82° = 180°
Sum of all of the angles of a cyclic quadrilateral is a full angle 360°.
97° + 83° + 98° + 82° = 360°