- Algebra
- Arithmetic
- Whole Numbers
- Numbers
- Types of Numbers
- Odd and Even Numbers
- Prime & Composite Numbers
- Sieve of Eratosthenes
- Number Properties
- Commutative Property
- Associative Property
- Identity Property
- Distributive Property
- Order of Operations
- Rounding Numbers
- Absolute Value
- Number Sequences
- Factors & Multiples
- Prime Factorization
- Greatest Common Factor
- Least Common Multiple
- Squares & Perfect Squares
- Square Roots
- Squares & Square Roots
- Simplifying Square Roots
- Simplifying Radicals
- Radicals that have Fractions
- Multiplying Radicals

- Integers
- Fractions
- Introducing Fractions
- Converting Fractions
- Comparing Fractions
- Ordering Fractions
- Equivalent Fractions
- Reducing Fractions
- Adding Fractions
- Subtracting Fractions
- Multiplying Fractions
- Reciprocals
- Dividing Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Multiplying Mixed Numbers
- Dividing Mixed Numbers
- Complex Fractions
- Fractions to Decimals

- Decimals
- Exponents
- Percent
- Scientific Notation
- Proportions
- Equality
- Properties of equality
- Addition property of equality
- Transitive property of equality
- Subtraction property of equality
- Multiplication property of equality
- Division property of equality
- Symmetric property of equality
- Reflexive property of equality
- Substitution property of equality
- Distributive property of equality

- Commercial Math

- Calculus
- Differential Calculus
- Limits calculus
- Mean value theorem
- L’Hôpital’s rule
- Newton’s method
- Derivative calculus
- Power rule
- Sum rule
- Difference rule
- Product rule
- Quotient rule
- Chain rule
- Derivative rules
- Trigonometric derivatives
- Inverse trig derivatives
- Trigonometric substitution
- Derivative of arctan
- Derivative of secx
- Derivative of csc
- Derivative of cotx
- Exponential derivative
- Derivative of ln
- Implicit differentiation
- Critical numbers
- Derivative test
- Concavity calculus
- Related rates
- Curve sketching
- Asymptote
- Hyperbolic functions
- Absolute maximum
- Absolute minimum

- Integral Calculus
- Fundamental theorem of calculus
- Approximating integrals
- Riemann sum
- Integral properties
- Antiderivative
- Integral calculus
- Improper integrals
- Integration by parts
- Partial fractions
- Area under the curve
- Area between two curves
- Center of mass
- Work calculus
- Integrating exponential functions
- Integration of hyperbolic functions
- Integrals of inverse trig functions
- Disk method
- Washer method
- Shell method

- Sequences, Series & Tests
- Parametric Curves & Polar Coordinates
- Multivariable Calculus
- 3d coordinate system
- Vector calculus
- Vectors equation of a line
- Equation of a plane
- Intersection of line and plane
- Quadric surfaces
- Spherical coordinates
- Cylindrical coordinates
- Vector function
- Derivatives of vectors
- Length of a vector
- Partial derivatives
- Tangent plane
- Directional derivative
- Lagrange multipliers
- Double integrals
- Iterated integral
- Double integrals in polar coordinates
- Triple integral
- Change of variables in multiple integrals
- Vector fields
- Line integral
- Fundamental theorem for line integrals
- Green’s theorem
- Curl vector field
- Surface integral
- Divergence of a vector field
- Differential equations
- Exact equations
- Integrating factor
- First order linear differential equation
- Second order homogeneous differential equation
- Non homogeneous differential equation
- Homogeneous differential equation
- Characteristic equations
- Laplace transform
- Inverse laplace transform
- Dirac delta function

- Differential Calculus
- Matrices
- Pre-Calculus
- Lines & Planes
- Functions
- Domain of a function
- Transformation Of Graph
- Polynomials
- Graphs of rational functions
- Limits of a function
- Complex Numbers
- Exponential Function
- Logarithmic Function
- Sequences
- Conic Sections
- Series
- Mathematical induction
- Probability
- Advanced Trigonometry
- Vectors
- Polar coordinates

- Probability
- Geometry
- Angles
- Triangles
- Types of Triangles
- Special Right Triangles
- 3 4 5 Triangle
- 45 45 90 Triangle
- 30 60 90 Triangle
- Area of Triangle
- Pythagorean Theorem
- Pythagorean Triples
- Congruent Triangles
- Hypotenuse Leg (HL)
- Similar Triangles
- Triangle Inequality
- Triangle Sum Theorem
- Exterior Angle Theorem
- Angles of a Triangle
- Law of Sines or Sine Rule
- Law of Cosines or Cosine Rule

- Polygons
- Circles
- Circle Theorems
- Solid Geometry
- Volume of Cubes
- Volume of Rectangular Prisms
- Volume of Prisms
- Volume of Cylinders
- Volume of Spheres
- Volume of Cones
- Volume of Pyramids
- Volume of Solids
- Surface Area of a Cube
- Surface Area of a Cuboid
- Surface Area of a Prism
- Surface Area of a Cylinder
- Surface Area of a Cone
- Surface Area of a Sphere
- Surface Area of a Pyramid
- Geometric Nets
- Surface Area of Solids

- Coordinate Geometry and Graphs
- Coordinate Geometry
- Coordinate Plane
- Slope of a Line
- Equation of a Line
- Forms of Linear Equations
- Slopes of Parallel and Perpendicular Lines
- Graphing Linear Equations
- Midpoint Formula
- Distance Formula
- Graphing Inequalities
- Linear Programming
- Graphing Quadratic Functions
- Graphing Cubic Functions
- Graphing Exponential Functions
- Graphing Reciprocal Functions

- Geometric Constructions
- Geometric Construction
- Construct a Line Segment
- Construct Perpendicular Bisector
- Construct a Perpendicular Line
- Construct Parallel Lines
- Construct a 60° Angle
- Construct an Angle Bisector
- Construct a 30° Angle
- Construct a 45° Angle
- Construct a Triangle
- Construct a Parallelogram
- Construct a Square
- Construct a Rectangle
- Locus of a Moving Point

- Geometric Transformations

- Sets & Set Theory
- Statistics
- Collecting and Summarizing Data
- Common Ways to Describe Data
- Different Ways to Represent Data
- Frequency Tables
- Cumulative Frequency
- Advance Statistics
- Sample mean
- Population mean
- Sample variance
- Standard deviation
- Random variable
- Probability density function
- Binomial distribution
- Expected value
- Poisson distribution
- Normal distribution
- Bernoulli distribution
- Z-score
- Bayes theorem
- Normal probability plot
- Chi square
- Anova test
- Central limit theorem
- Sampling distribution
- Logistic equation
- Chebyshev’s theorem

- Difference
- Correlation Coefficient
- Tautology
- Relative Frequency
- Frequency Distribution
- Dot Plot
- Сonditional Statement
- Converse Statement
- Law of Syllogism
- Counterexample
- Least Squares
- Law of Detachment
- Scatter Plot
- Linear Graph
- Arithmetic Mean
- Measures of Central Tendency
- Discrete Data
- Weighted Average
- Summary Statistics
- Interquartile Range
- Categorical Data

- Trigonometry
- Vectors
- Multiplication Charts
- Time Table
- 2 times table
- 3 times table
- 4 times table
- 5 times table
- 6 times table
- 7 times table
- 8 times table
- 9 times table
- 10 times table
- 11 times table
- 12 times table
- 13 times table
- 14 times table
- 15 times table
- 16 times table
- 17 times table
- 18 times table
- 19 times table
- 20 times table
- 21 times table
- 22 times table
- 23 times table
- 24 times table

- Time Table

# The Inscribed Angle Theorem – Explanation & Examples

The circular geometry is really vast. A circle consists of many parts and angles. These parts and angles are mutually supported by certain Theorems, e.g., t**he Inscribed Angle Theorem**, Thales’ Theorem, and Alternate Segment Theorem.

**We will go through the inscribed angle theorem**, but before that, let’s have a brief overview of circles and their parts.

Circles are all around us in our world. There exists an interesting relationship among the angles of a circle. To recall, a chord of a circle is the straight line that joins two points on a circle’s circumference. Three types of angles are formed inside a circle when two chords meet at a common point known as a vertex. These angles are the central angle, intercepted arc, and the inscribed angle.

For more definitions related to circles, you need to go through the previous articles.

*In this article, you will learn:*

- The inscribed angle and inscribed angle theorem,
- we will also learn how to prove the inscribed angle theorem.

## What is the Inscribed Angle?

**An inscribed angle is an angle whose vertex lies on a circle, and its two sides are chords of the same circle.**

On the other hand, a central angle is an angle whose vertex lies at the center of a circle, and its two radii are the sides of the angle.

The intercepted arc is an angle formed by the ends of two chords on a circle’s circumference.

Let’s take a look.

In the above illustration,

**α** = The central angle

**θ** = The inscribed angle

**β** = the intercepted arc.

## What is the Inscribed Angle Theorem?

*The inscribed angle theorem, which is also known as the arrow theorem or the central angle theorem, states that:*

**The size of the central angle is equal to twice the size of the inscribed angle. The inscribed angle theorem can also be stated as:**

**α = 2****θ**

The size of an inscribed angle is equal to half the size of the central angle.

**θ = ½****α**

Where α and θ are the central angle and inscribed angle, respectively.

## How do you Prove the Inscribed Angle Theorem?

*The inscribed angle theorem can be proved by considering three cases, namely:*

- When the inscribed angle is between a chord and the diameter of a circle.
- The diameter is between the rays of the inscribed angle.
- The diameter is outside the rays of the inscribed angle.

**Case 1: When the inscribed angle is between a chord and the diameter of a circle:**

**To prove α = 2θ:**

- △
*CBD*is an isosceles triangle whereby*CD = CB*= the radius of the circle. - Therefore, ∠ CDB = ∠ DBC = inscribed angle = θ
- The diameter AD is a straight line, so ∠
*BCD*= (180**–**α) ° - By triangle sum theorem, ∠
*CDB*+ ∠DBC + ∠BCD = 180°

θ + θ + (180 **–** α) = 180°

Simplify.

⟹ θ + θ + 180 **–** α = 180°

⟹ 2θ + 180 – α = 180°

Subtract 180 on both sides.

⟹ 2θ + 180 – α = 180°

⟹ 2θ – α = 0

**⟹**** 2θ = α. Hence proved.**

**Case 2: when the diameter is between the rays of the inscribed angle.**

To prove 2θ = α:

- First, draw the diameter (in dotted line) of the circle.

- Let the diameter bisects θ into θ
_{1}and θ Similarly, the diameter bisects α into α_{1 }and α_{2}.

⟹ θ_{1} + θ_{2} = θ

⟹ α_{1 }+ α_{2} = α

- From the first case above, we already know that,

⟹ 2θ_{1 }= α_{1}

⟹ 2θ_{2} = α_{2}

- Add the angles.

⟹ α_{1 }+ α_{2} = 2θ_{1 }+ 2θ_{2}

⟹ α_{1 }+ α_{2} = 2 (θ_{1 }+ 2θ_{2})

**Hence, ****2θ = α:**

**Case 3: When the diameter is outside the rays of the inscribed angle.**

To prove 2θ = α:

- Draw the diameter (in dotted line) of the circle.

- Since 2θ
_{1}= α_{1}

⟹ 2 (θ_{1 }+ θ) = α + α_{1}

⟹ But, 2θ_{1 }= α_{1 }and 2θ_{2} = α_{2}

⟹ By substitution, we get,

2θ = α:

**Solved examples about inscribed angle theorem**

*Example 1*

Find the missing angle x in the diagram below.

__Solution__

By inscribed angle theorem,

The size of the central angle = 2 x the size of the inscribed angle.

Given, 60° = inscribed angle.

Substitute.

The size of the central angle = 2 x 60°

= 120°

*Example 2*

Give, that ∠*QRP* = (2x + 20) ° and ∠*PSQ *= 30°. Find the value of x.

__Solution__

By inscribed angle theorem,

Central angle = 2 x inscribed angle.

∠*QRP =2*∠*PSQ*

∠*QRP *= 2 x 30°.

= 60°.

Now, solve for x.

⟹ (2x + 20) ° = 60°.

Simplify.

⟹ 2x + 20° = 60°

Subtract 20° on both sides.

⟹ 2x = 40°

Divide both sides by 2.

⟹ x = 20°

So, the value of x is 20°.

*Example 3*

Solve for angle x in the diagram below.

__Solution__

Given the central angle = 56°

2∠*ADB =*∠*ACB*

2x = 56°

Divide both sides by 2.

x = 28°

*Example 4*

If ∠ *YMZ* = 150°, find the measure of ∠*MZY* and ∠ *XMY.*

__Solution__

Triangle MZY is an isosceles triangle, Therefore,

∠*MZY =*∠*ZYM*

Sum of interior angles of a triangle = 180°

∠*MZY = *∠*ZYM = *(180° – 150°)/2

= 30° /2 = 15°

Hence, ∠*MZY = *15°

And by inscribed angle theorem,

2∠*MZY = *∠ *XMY*

∠ *XMY *= 2 x 15°

= 30°

*Practice Questions*

1. What is the vertex of a central angle?

A. Ends of a chord.

B.Center of a circle.

C. Any point on the circle.

D. None of these.

2. The degree measure of a central angle is equal to the degree measure of its _________.

A. Chord

B. Inscribed angle

C. Intercepted arc

D. Vertex

3. According to the Inscribed angle theorem, the measure of an inscribed angle is ____ the measure of its intercepted arc.

A. Half

B. Twice

C. Four times

D. None of these

4.

For the circle above, *XY* is the diameter, and *O *is the circle. The vertex of the angle is at its center.

Calculate the value of *n*.

__Answers__

- B
- C
- A
- 45