- 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

# Angles of a Triangle – Explanation & Examples

We know that every shape in the universe is based on angles. The square is basically four lines connected so that each line makes an angle of 90 degrees with the other line. In this way, a square has four 90 degrees angles on its four sides.

Similarly, a straight line stretched on both sides at 180 degrees. If it turns at any point, it becomes two lines separated by a certain angle. In the same manner, a triangle is basically three lines connected at certain values of angles.

These measures of angles define the type of triangle. Therefore, angles are essential in studying any geometric shape.

In this article, you’ll learn the **angles of a triangle** and **how to find the unknown angles of a triangle** when you only know some of the angles. To know the important concepts of triangles, you can consult the previous articles.

## What are the Angles of a Triangle?

**The angle of a triangle is the space formed between two side lengths of a triangle.** A triangle contains interior angles and exterior angles. **Interior angles** are three angles found inside a triangle. **Exterior angles** are formed when the sides of a triangle are extended to infinity.

Therefore, exterior angles are formed outside a triangle between one side of a triangle and the extended side. Each exterior angle is adjacent to an interior angle. Adjacent angles are angles with a common vertex and side.

The figure below shows the **angle of a triangle**. The interior angles are a, b and c, while exterior angles are d, e, and f.

## How to Find the Angles of a Triangle?

*To find the angles of a triangle, you need to recall the following three properties about triangles:*

- Triangle angle sum theorem: This states that the sum of all the three interior angles of a triangle is equal to 180 degrees.

a + b + c = 180º

- Triangle exterior angle theorem: This states that the exterior angle is equal to the sum of two opposite and non-adjacent interior angles.

f = b + a

e = c + b

d = b + c

- Straight line angles. The measure of angles on a straight line is equal to 180º

c + f = 180º

a + d = 180º

e + b = 180º

Let’s work out a few example problems.

*Example 1*

Calculate the size of the missing angle x in the triangle below.

__Solution__

By triangle angle sum, theorem, we have,

x + 84º + 43º = 180º

Simplify.

x + 127º = 180º

Subtract 127º on both sides.

x + 127º – 127º = 180º – 127º

x = 53 º

Hence, the size of the missing angle is 53º.

*Example 2*

Find the size of the interior angles of a triangle that form consecutive positive integers.

__Solution__

Since, a triangle has three interior angles, then, let the consecutive angles be:

⇒1^{ST} angle = x

⇒ 2^{ND} angle = x + 1

⇒3^{RD} angle = x + 2

But we know that, the sum of the three angles is equal to 180 degrees, therefore,

⇒ x + x + 1 + x + 2 = 180°

⇒ 3x + 3 = 180°

⇒ 3x = 177°

x = 59°

Now, substitute the value of x in the original three equations.

⇒1^{ST} angle = x = 59°

⇒ 2^{ND} angle = x + 1 =59° + 1 = 60°

⇒3^{RD} angle = x + 2 = 59°+ 2 = 61°

So, the triangle’s consecutive interior angles are; 59°, 60°, and 61°.

*Example 3*

Find the triangle’s interior angles whose angles are given as; 2y°, (3y + 15) ° and (2y + 25) °.

__Solution__

In triangle, um of interior angles = 180°

2y° + (3y + 15) ° + (2y + 25) ° = 180°

Simplify.

2y + 3y + 2y + 15° + 25° = 180°

7y + 40° = 180°

Subtract 40° on both sides.

7y + 40° – 40° = 180° – 40°

7y = 140°

Divide both sides by 7.

y = 140/7

y = 20°

Substitute,

2y°= 2(20) ° = 40°

(3y + 15) ° = (3 x 20 + 15) ° = 75°

(2y + 25) ° = (2 x 20 + 25) ° = 65°

So, the three interior angles of a triangle are 40°, 75°, and 65°.

* Example 4*

Find the value of the missing angles in the diagram below.

__Solution__

By triangle exterior angle theorem, we have;

(2x + 10) ° = 63° + 87°

Simplify

2x + 10° = 150°

Subtract 10° on both sides.

2x + 10° – 10 = 150° – 10

2x = 140°

Divide both sides by 2 to get;

x = 70°

Now, by substitution;

(2x + 10) ° = 2(70°) + 10 ° = 140 ° + 10 ° = 150 °

Hence, the exterior angle is 150 °

But, straight line angles add up to 180 °. So, we have;

y + 150 ° = 180 °

Subtract 150 ° on both sides.

y + 150 ° – 150 ° = 180 ° – 150 °

y = 30 °

Therefore, the missing angles are 30 ° and 150 °.

*Example 5*

The interior angles of a triangle are in the ratio 4: 11: 15. Find the angles.

__Solution__

Let x be the common ratio of the three angles. So, the angles are,

4x, 11x and 15x.

In a triangle, sum of the three angles = 180°

4x + 11x + 15x = 180°

Simplify.

30x = 180°

Divide 30 on both sides.

x =180°/30

x = 6°

Substitute the value of x.

4x = 4(6) ° = 24°

11x = 11(6) ° = 66°

15x = 15(6) ° = 90°

So, the angles of the triangle are 24°, 66°, and 90°.

*Example 6*

Find the size of angles x and y in the diagram below.

__Solution__

Exterior angle = sum of two non-adjacent interior angles.

60° + 76° = x

x = 136°

Similarly, sum of interior angles = 180°. Therefore,

60° + 76° + y = 180°

136° + y = 180°

Subtract 136° on both sides.

136° – 136° + y = 180° – 136

y = 44°

Hence, the size of angle x and y is 136° and 44°, respectively.

*Example 7*

The three angles of a certain triangle are such that the first angle is 20 % less than the second angle, and the third is 20% more than the second angle. Find the size of the three angles.

__Solution__

Let the second angle be x

First angle = x – 20x/100 = x – 0.2x

Third angle = x + 20x/100 = x + 0.2x

Sum of the three angles = 180 degrees.

x + x – 0. 2x + x + 0.2x = 180°

Simplify.

3x = 180°

x = 60°

Therefore,

2^{nd} second angle = 60°

1^{st} angle =48°

3^{rd} angle = 72°

So, the three angles of a triangle are 60°, 48°, and 72°.

*Example 8*

Calculate the size of angle p, q, r and s in the diagram below.

__Solution__

exterior angle = sum of the two non-adjacent interior angles.

140° = p + r …………. (i)

This is an isosceles triangle, so,

q = r

Angles on a straight line = 180°

140° + q = 180°

subtract 140 from both sides to get.

q = 40°

But q = r, so r is also 40°

r + s = 180° (linear angles)

40° + s =180°

s = 140°

Sum of interior angles = 180°

p + q + r = 180°

p + 40° + 40° = 180°

p = 180° – 80°

p = 100°