Inscribed Angles: Definition, Examples & Formula - StudySmarter

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Inscribed Angles

A circle is unique because it does not have any corners or angles, which makes it different from other figures such as triangles, rectangles, and triangles. But specific properties can be explored in detail by introducing angles inside a circle. For instance, the simplest way to create an angle inside a circle is by drawing two chords such that they start at the same point. This might seem unnecessary at first, but by doing so, we can employ many rules of trigonometry and geometry, thus exploring circle properties in more detail.

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• 7 minutes reading time
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• Fact Checked Content
• Last Updated: 02.02.2023
• Published at: 20.07.2022
• 7 min reading time

• Applied Mathematics
• Calculus
• Decision Maths
• Discrete Mathematics
• Geometry
• 2 Dimensional Figures
• 3 Dimensional Vectors
• 3-Dimensional Figures
• Affine geometry
• Altitude
• Analytic geometry
• Angle Sum Property
• Angles in Circles
• Arc Measures
• Area and Volume
• Area of Circles
• Area of Circular Sector
• Area of Parallelograms
• Area of Plane Figures
• Area of Rectangles
• Area of Regular Polygons
• Area of Rhombus
• Area of Trapezoid
• Area of a Kite
• Birational geometry
• Central Angle
• Complex geometry
• Composition
• Computational geometry
• Cones
• Conformal geometry
• Congruence Transformations
• Congruent Triangles
• Convex geometry
• Convexity in Polygons
• Coordinate Systems
• Cube Properties
• Cylinders
• Differential geometry
• Differential topology
• Dilations
• Discrete geometry
• Distance and Midpoints
• Equation Of Circles
• Equilateral Triangles
• Ergodic theory
• Euclidean geometry
• Exterior Angle Theorem
• Figures
• Finsler geometry
• Fractal geometry
• Fundamentals of Geometry
• Galois geometry
• Geodesy
• Geometric Inequalities
• Geometric Mean
• Geometric Probability
• Geometric Transformations
• Geometric analysis
• Geometric function theory
• Geometric topology
• Glide Reflections
• HL ASA and AAS
• Harmonic analysis
• Hodge theory
• Hyperbolic geometry
• Identity Map
• Inscribed Angles
• Integral geometry
• Interior Angles
• Inversive geometry
• Isometry
• Isosceles Triangles
• Kähler manifolds
• Law of Cosines
• Law of Sines
• Linear Measure and Precision
• Manifolds
• Mathematical Physics
• Median
• Metric geometry
• Minimal surfaces
• Mirror symmetry
• Moduli spaces
• Morse theory
• Möbius transformations
• Non-Euclidean geometry
• Parallel Lines Theorem
• Parallelograms
• Perpendicular Bisector
• Plane Geometry
• Poisson geometry
• Polygon Properties
• Polygons
• Projections
• Projective geometry
• Properties of Chords
• Proportionality Theorems
• Pyramids
• Pythagoras Theorem
• Quantum geometry
• Quaternionic analysis
• Rectangle
• Rectangular Prism
• Reflection in Geometry
• Regular Polygon
• Rhombus Properties
• Rhombuses
• Ricci flow
• Right Triangles
• Rotations
• SSS and SAS
• Sasaki geometry
• Scalene Triangles
• Segment Length
• Similarity
• Similarity Transformations
• Simplicial complexes
• Special Right Triangles
• Special quadrilaterals
• Spherical geometry
• Square Properties
• Squares
• Sub-Riemannian geometry
• Surface Area of Cone
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Sphere
• Surface Area of a Solid
• Surface of Pyramids
• Symmetry
• Symplectic geometry
• Toric geometry
• Transformation groups
• Translations
• Transversal Lines
• Trapezoid Properties
• Trapezoids
• Triangle Inequalities
• Triangles
• Twistor theory
• Using Similar Polygons
• Vector Addition
• Vector Product
• Vector bundles
• Volume of Cone
• Volume of Cylinder
• Volume of Pyramid
• Volume of Solid
• Volume of Sphere
• Volume of prisms
• Vortex dynamics
• Weyl geometry
• What is Point Slope Form
• Logic and Functions
• Mechanics Maths
• Probability and Statistics
• Pure Maths
• Statistics
• Theoretical and Mathematical Physics

Contents

• Applied Mathematics
• Calculus
• Decision Maths
• Discrete Mathematics
• Geometry
• 2 Dimensional Figures
• 3 Dimensional Vectors
• 3-Dimensional Figures
• Affine geometry
• Altitude
• Analytic geometry
• Angle Sum Property
• Angles in Circles
• Arc Measures
• Area and Volume
• Area of Circles
• Area of Circular Sector
• Area of Parallelograms
• Area of Plane Figures
• Area of Rectangles
• Area of Regular Polygons
• Area of Rhombus
• Area of Trapezoid
• Area of a Kite
• Birational geometry
• Central Angle
• Complex geometry
• Composition
• Computational geometry
• Cones
• Conformal geometry
• Congruence Transformations
• Congruent Triangles
• Convex geometry
• Convexity in Polygons
• Coordinate Systems
• Cube Properties
• Cylinders
• Differential geometry
• Differential topology
• Dilations
• Discrete geometry
• Distance and Midpoints
• Equation Of Circles
• Equilateral Triangles
• Ergodic theory
• Euclidean geometry
• Exterior Angle Theorem
• Figures
• Finsler geometry
• Fractal geometry
• Fundamentals of Geometry
• Galois geometry
• Geodesy
• Geometric Inequalities
• Geometric Mean
• Geometric Probability
• Geometric Transformations
• Geometric analysis
• Geometric function theory
• Geometric topology
• Glide Reflections
• HL ASA and AAS
• Harmonic analysis
• Hodge theory
• Hyperbolic geometry
• Identity Map
• Inscribed Angles
• Integral geometry
• Interior Angles
• Inversive geometry
• Isometry
• Isosceles Triangles
• Kähler manifolds
• Law of Cosines
• Law of Sines
• Linear Measure and Precision
• Manifolds
• Mathematical Physics
• Median
• Metric geometry
• Minimal surfaces
• Mirror symmetry
• Moduli spaces
• Morse theory
• Möbius transformations
• Non-Euclidean geometry
• Parallel Lines Theorem
• Parallelograms
• Perpendicular Bisector
• Plane Geometry
• Poisson geometry
• Polygon Properties
• Polygons
• Projections
• Projective geometry
• Properties of Chords
• Proportionality Theorems
• Pyramids
• Pythagoras Theorem
• Quantum geometry
• Quaternionic analysis
• Rectangle
• Rectangular Prism
• Reflection in Geometry
• Regular Polygon
• Rhombus Properties
• Rhombuses
• Ricci flow
• Right Triangles
• Rotations
• SSS and SAS
• Sasaki geometry
• Scalene Triangles
• Segment Length
• Similarity
• Similarity Transformations
• Simplicial complexes
• Special Right Triangles
• Special quadrilaterals
• Spherical geometry
• Square Properties
• Squares
• Sub-Riemannian geometry
• Surface Area of Cone
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Sphere
• Surface Area of a Solid
• Surface of Pyramids
• Symmetry
• Symplectic geometry
• Toric geometry
• Transformation groups
• Translations
• Transversal Lines
• Trapezoid Properties
• Trapezoids
• Triangle Inequalities
• Triangles
• Twistor theory
• Using Similar Polygons
• Vector Addition
• Vector Product
• Vector bundles
• Volume of Cone
• Volume of Cylinder
• Volume of Pyramid
• Volume of Solid
• Volume of Sphere
• Volume of prisms
• Vortex dynamics
• Weyl geometry
• What is Point Slope Form
• Logic and Functions
• Mechanics Maths
• Probability and Statistics
• Pure Maths
• Statistics
• Theoretical and Mathematical Physics

Contents

• Fact Checked Content
• Last Updated: 02.02.2023
• 7 min reading time

• Content creation process designed by Lily Hulatt
• Content sources cross-checked by Gabriel Freitas
• Content quality checked by Gabriel Freitas

Sign up for free to save, edit & create flashcards. Save Article Save Article

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What is an Inscribed Angle of a Circle?

Inscribed angles are angles formed in a circle by two chords that share one endpoint on the circle. The common endpoint is also known as the vertex of the angle. This is shown in figure 1, where two chords AB¯ and BC¯ form an inscribed angle m<ABC, where the symbol ‘m<' is used to describe an inscribed angle.

Inscribed Angles, StudySmarter Originals

The other endpoints of the two chords form an arc on the circle, which is the arc AC shown below. There are two kinds of arcs that are formed by an inscribed angle.

• When the measure of the arc is less than a semicircle or 180°, then the arc is defined as a minor arc which is shown in figure 2a.

• When the measure of the arc is greater than a semicircle or 180°, then the arc is defined as a major arc which is shown in figure 2b.

But how do we create such an arc? By drawing two cords, as we discussed above. But what exactly is a chord? Take any two points on a circle and join them to make a line segment:

A chord is a line segment that joins two points on a circle.

Major arc and Minor arc of a circle, StudySmarter Originals

Now that a chord has been defined, what can one build around a chord? Let‘s start with an arc, and as obvious as it sounds, it is a simple part of the circle defined below:

An arc of a circle is a curve formed by two points in a circle. The length of the arc is the distance between those two points.

• An arc of a circle that has two endpoints on the diameter, then the arc is equal to a semicircle.
• The measure of the arc in degrees is the same as the central angle that intercepts that arc.

The length of an arc can be measured using the central angle in both degrees or radians and the radius as shown in the formula below, where θ is the central angle, and π is the mathematical constant. At the same time, r is the radius of the circle.

Arc length (degrees)= θ 360 · 2π·r Arc length ( radians) = θ·r

Inscribed Angles Formula

Several types of inscribed angles are modeled by various formulas based on the number of angles and their shape. Thus a generic formula cannot be created, but such angles can be classified into certain groups.

Inscribed Angle Theorems

Let's look at the various Inscribed Angle Theorems.

Inscribed angle

The inscribed angle theorem relates the measure of the inscribed angle and its intercepted arc.

It states that the measure of the inscribed angle in degrees is equal to half the measure of the intercepted arc, where the measure of the arc is also the measure of the central angle.

m<ABC = 12·m<AOC

Inscribed Angle Theorem, StudySmarter Originals

Inscribed angles in the same arc

When two inscribed angles intercept the same arc, then the angles are congruent. Congruent angles have the same degree measure. An example is shown in figure 4, where m<ADC and m<ABC and m<ABC are equal as they intercept the same arc AC:

m<ABC=m<ADC

Congruent Inscribed Angles, StudySmarter Originals

Inscribed angle in a Semicircle

When an inscribed angle intercepts an arc that is a semicircle, the inscribed angle is a right angle equal to 90°. This is shown below in the figure, where arc AB is a semicircle with a measure of 180° and its inscribed angle m<ACB is a right angle with a measure of 90°.

Inscribed Angle in a Semicircle, StudySmarter Originals

Inscribed Quadrilateral

If a quadrilateral is inscribed in a circle, which means that the quadrilateral is formed in a circle by chords, then its opposite angles are supplementary. For example, the following diagram shows an inscribed quadrilateral, where m<A is supplementary to m<C and m<B is supplementary to m<D:

m<B+m<D=180°

m<A+m<C=180°

Inscribed Quadrilateral, StudySmarter Originals

Inscribed Angles Examples

Find angles m<ABC and m<ACD if the central angle m<AQD shown below is 75°.

Inscribed angles example, StudySmarter Originals

Solution:

Since angles m<ACD and m<ABD intercept the same arc AD, then they are congruent.

m<ABD= m <ACD

Using the inscribed angle theorem, we know that the central angle is twice the inscribed angle that intercepts the same arc.

m<AQD = 2·m<ACD 75° = 2·m<ACD M<ACD = 37.5°

Hence the angle is 37.5°.

What is the measure of angle m<ABD in the circle shown below if m<ACD is 30°?

Congruent Inscribed Angles, StudySmarter Originals

Solution:

As angles m<ABD and m<ACD intercept the same arc , then they are equal . Hence, if m<ACD is 30° then m<ABD must also be 30°.

Method for Solving Inscribed Angle Problems

To solve any example of inscribed angles, write down all the angles given. Recognize the angles given by drawing a diagram if not given. Let’s look at some examples.

Find m<ABC if its intercepted arc has a measure of 80°.

Solution:

Using the inscribed angle theorem, we derive that the inscribed angle equals half of the central angle.

m<ABC = 12·m<AOC m<ABC = 802=40 °

Find m<C and m<D in the inscribed quadrilateral shown below.

Inscribed quadrilateral Example, StudySmarter Originals

Solution:

As the quadrilateral shown is inscribed in a circle, its opposite angles are complementary.

<A + <C = 180° <B + <D = 180 °

Then we substitute the given angles into the equations, and we re-arrange the equations to make the unknown angle the subject.

98°+<C = 180° <C= 180°-98° = 82° 85° +<D = 180° <D = 180°- 85°=95°

Find m<b, m<d, and m<c in the diagram below.

An inscribed quadrilateral, StudySmarter Originals

Solution:

Inscribed angles m<BAC and m<BDC intercept the same arc BC. Hence they are equal, therefore

<d = 50°

Angle m<BCD is inscribed in a semicircle. Hence <c must be a right angle.

<c = 90°

As quadrilateral ABCD is inscribed in a circle, its opposite angles must be supplementary.

<B + <D = 180 ° B + (d+35) = 180° B= 180-50-35 <b= 95 °

Inscribed Angles - Key takeaways

• An inscribed angle is an angle formed in a circle by two chords with a common end point that lies on the circle.
• Inscribed angle theorem states that the inscribed angle is half the measure of the central angle.
• Inscribed angles that intercept the same arc are congruent.
• Inscribed angles in a semicircle are right angles.
• If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

Similar topics in Math

• Probability and Statistics
• Statistics
• Mechanics Maths
• Geometry
• Calculus
• Pure Maths
• Decision Maths
• Logic and Functions
• Discrete Mathematics
• Theoretical and Mathematical Physics
• Applied Mathematics

Related topics to Geometry

• Glide Reflections
• Symmetry
• Area of a Kite
• Polygons
• Altitude
• Triangles
• Trapezoids
• Area of Regular Polygons
• Surface Area of a Solid
• Area of Parallelograms
• Isometry
• Congruence Transformations
• Volume of Solid
• Volume of Sphere
• Regular Polygon
• Equilateral Triangles
• Rotations
• Area of Plane Figures
• Surface Area of Sphere
• Rectangle
• Distance and Midpoints
• Identity Map
• Translations
• Area of Trapezoid
• SSS and SAS
• Volume of Cylinder
• Vector Product
• Properties of Chords
• Plane Geometry
• Arc Measures
• Dilations
• Composition
• Volume of prisms
• Right Triangles
• Volume of Cone
• Congruent Triangles
• 2 Dimensional Figures
• Convexity in Polygons
• Fundamentals of Geometry
• Area and Volume
• Median
• Perpendicular Bisector
• Geometric Probability
• Area of Circular Sector
• Area of Rhombus
• Law of Sines
• Linear Measure and Precision
• Pythagoras Theorem
• Parallelograms
• Law of Cosines
• Geometric Mean
• Isosceles Triangles
• Parallel Lines Theorem
• Special quadrilaterals
• Proportionality Theorems
• Similarity Transformations
• Volume of Pyramid
• Squares
• 3-Dimensional Figures
• Rhombuses
• 3 Dimensional Vectors
• Surface Area of Prism
• Inscribed Angles
• Coordinate Systems
• Surface Area of Cone
• Segment Length
• Projections
• Similarity
• HL ASA and AAS
• Using Similar Polygons
• Equation Of Circles
• Angles in Circles
• Vector Addition
• Surface Area of Cylinder
• Reflection in Geometry
• Surface of Pyramids
• Figures
• Triangle Inequalities
• Area of Rectangles
• Area of Circles
• What is Point Slope Form
• Geometric Inequalities
• Computational geometry
• Sasaki geometry
• Kähler manifolds
• Integral geometry
• Mirror symmetry
• Quaternionic analysis
• Geodesy
• Manifolds
• Non-Euclidean geometry
• Discrete geometry
• Geometric topology
• Conformal geometry
• Symplectic geometry
• Projective geometry
• Inversive geometry
• Ricci flow
• Euclidean geometry
• Quantum geometry
• Finsler geometry
• Hyperbolic geometry
• Harmonic analysis
• Toric geometry
• Simplicial complexes
• Möbius transformations
• Minimal surfaces
• Fractal geometry
• Metric geometry
• Galois geometry
• Differential geometry
• Convex geometry
• Twistor theory
• Vortex dynamics
• Complex geometry
• Weyl geometry
• Transformation groups
• Differential topology
• Ergodic theory
• Affine geometry
• Geometric function theory
• Hodge theory
• Poisson geometry
• Mathematical Physics
• Geometric analysis
• Spherical geometry
• Vector bundles
• Birational geometry
• Analytic geometry
• Sub-Riemannian geometry
• Moduli spaces
• Morse theory
• Cones
• Scalene Triangles
• Geometric Transformations
• Trapezoid Properties
• Square Properties
• Cylinders
• Rectangular Prism
• Cube Properties
• Central Angle
• Transversal Lines
• Rhombus Properties
• Polygon Properties
• Pyramids
• Angle Sum Property
• Interior Angles
• Exterior Angle Theorem
• Special Right Triangles

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Frequently Asked Questions about Inscribed Angles

What is an inscribed angle?

An inscribed angle is an angle that is formed in a circle by two chords that have a common end point that lies on the circle.

What is the difference between inscribed and central angles?

A central angle is formed by two line segments that are equal to the radius of the circle and inscribed angles are formed by two chords, which are line segments that intersect the circle in two points.

How to solve inscribed angles?

Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

What is the formula for calculating inscribed angles?

There is not a general formula for calculating inscribed angles. Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

What is an example of an inscribed angle?

A typical example would be a quadrilateral inscribed in a circle where the angles formed at the corners are inscribed angles.

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• Checked by StudySmarter Editorial Team

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