Essential Triangle Theorems Every Student Should Know
Triangle is a plane figure consisting of three sides, and three angles opposite to the sides makes a sum of 180 degrees. It comes in the category of a polygon; polygon is a closed shape plane with line segments. Triangle is the simplest polygon in geometry.
Triangles theorems have significant use in geometry; they prove various properties associated with it. To give students in-depth knowledge about triangle theorems, Vedantu has uploaded the well-researched explanations on the website. Understanding the theorems is necessary to solve the problems linked with triangles.
Properties of a Triangle
The properties of a triangle include the followings:
It has three sides, angles, and vertices
The sum of three interior angles are always 180 degree
The sum of the two sides of this geometrical figure is greater than its third one
The area of the product of this figure’s height and the base is equal to twice its area.
Types of Triangle
There are different types of triangles, and here are the classifications –
According to the Measurement of Angles
Acute angle, where all interior angles are less than 90 degrees.
Right angle, where one of the three interior angles of a triangle is 90 degrees.
Obtuse angle, where one interior angle is greater than 90 degrees.
According to the Measurement of Sides
An equilateral triangle is where all 3 sides are equal.
An isosceles triangle is where 2 sides are equal.
A scalene triangle is where no sides are equal.
Since the definition of triangles and their types are now clear, students can now understand the theorems quicker.
Theorems of Triangle
Triangle theorems are based on various properties of this geometrical shape, here are some prominent theorems associated with this is that students must know –
1. Pythagoras Theorem
Probably the most popular and widely discussed triangle theorems are Pythagoras’ one.
Pythagoras theorem Class 10 states that ‘in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. According to this theorem, the sides of a triangle are named perpendicular, hypotenuse, and base. The hypotenuse is the longest one among these three sides, and it sits opposite to the right angle, i.e. 90 degrees.
Moreover, when sides of a Pythagoras theorem triangle have a positive integer value, and then squared and entered into an equation, they are known as a Pythagorean triple.
Formula of Pythagoras theorem: Hypotenuse2 = Perpendicular2 + Base2
2. Triangle Similarity Theorems
The focus of this theorem is to prove the similarity between two triangles. It specifies conditions under which more than one triangle can be regarded as similar. It considers sides and angles to conclude, and once every condition is met, triangles are considered the same.
There are three subtypes of this triangle theorem. These are –
AA Similarity Theorem:
This theorem suggests that if two angles of two triangles are similar, they have the same properties. Since the total value of three angles is 180 degrees, once the value of two angles is known, it is easy to find the third one by subtracting it from 180.
SSS Theorem:
This theorem suggests that when 3 sides of two triangles have the same value or they are proportionate to each other, they are identical or congruent. Moreover, for these two, three sides need to be proportional.
SAS Theorem:
When two sides of two triangles are proportional, and the angle between them have similar values, these two triangles will be similar.
3. Basic Proportionality Theorem
The Triangle proportionality theorem suggests that, when a line is drawn matching to one side of a triangle intersecting the other two at particular points, these other two sides are divided in the same ratio.
4. Triangle Sum Theorem
Probably the most basic among every triangle theorem, this one proves that all-three angles of this geometric figure constitute a total value of 180 degrees.
5. Triangle Inequality Theorem
The Triangle inequality theorem suggests that one side of a triangle must be shorter than the other two. Otherwise, they will not meet and create a triangle. These are some of the notable theorems associated with triangles. Students can learn more about them from the website of Vedantu – India’s leading e-learning platform.
At the website and mobile application of Vedantu, students will find relevant information and explanation regarding triangle theorems. Furthermore, there are various study materials to aid students in this regard. Also, they can sign up for online classes and doubt clearing sessions to better their preparations for this concept.
Learn Triangle Properties on Vedantu
Learn about the properties of the triangle explained by the subject matter expert of the Vedantu. Properties of a triangle help you understand different theorems and find a difference between the various types of them.
On Vedantu, you get a chance to learn from India’s best maths faculties. Join the online maths tuition class to clear your doubts from them and boost your preparation to score good marks in the exams.
Triangle Theorems Explanation
Theorems are a statement that has definite proof. For example, the Pythagoras theorem proves a square of the hypotenuse is equal to the sum of the squares of the other two sides.
Vedantu has explained the types and theorems of triangles to help students understand the shape that will help them solve geometrical problems effortlessly. These theorems cover the basics of a triangle that stays with you for a long duration. And benefits you in solving numerical in the higher classes and also in the engineering or architecture stream.
FAQs on Triangle Theorems: Concepts, Properties & Examples
1. What are the fundamental criteria for proving two triangles are congruent as per the CBSE syllabus?
To prove that two triangles are congruent (identical in shape and size), you must satisfy one of the five main criteria specified in the NCERT syllabus. These are:
- SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle.
- SAS (Side-Angle-Side): If two sides and the included angle (the angle between them) of one triangle are equal to the corresponding sides and included angle of another.
- ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to the corresponding angles and included side of another.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to the corresponding angles and non-included side of another.
- RHS (Right-angle-Hypotenuse-Side): In two right-angled triangles, if the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other.
2. What is the difference between triangle congruence and similarity?
The key difference lies in size. Congruent triangles are exact copies of each other, meaning they have the same shape and the same size. Their corresponding angles and sides are all equal. In contrast, similar triangles have the same shape but can have different sizes. Their corresponding angles are equal, but their corresponding sides are in the same ratio (proportional). Think of it like a photograph and its enlargement – they are similar, not congruent.
3. What are the three main conditions to prove that two triangles are similar?
According to the triangle theorems in the Class 10 syllabus, two triangles can be proven similar if they meet any of the following conditions:
- AA (Angle-Angle) Similarity: If two angles of one triangle are equal to the two corresponding angles of another triangle, the third angles will also be equal, and the triangles are similar.
- SSS (Side-Side-Side) Similarity: If the lengths of the corresponding sides of two triangles are in the same ratio (proportional), then the triangles are similar.
- SAS (Side-Angle-Side) Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the triangles are similar.
4. What is the Triangle Inequality Theorem and why is it important?
The Triangle Inequality Theorem is a fundamental rule that states the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For example, if sides are a, b, and c, then a + b > c, a + c > b, and b + c > a. This theorem is important because it determines if a given set of three side lengths can actually form a triangle.
5. Which key triangle theorems are most important for the CBSE Class 10 Maths exam?
For the Class 10 board exams (2025-26), understanding the proofs and applications of the following theorems is crucial:
- Basic Proportionality Theorem (BPT) or Thales' Theorem: States that if a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, the other two sides are divided in the same ratio.
- Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Its proof and application are frequently asked.
- Areas of Similar Triangles Theorem: States that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
6. How does the Basic Proportionality Theorem (BPT) practically apply to geometry problems?
The Basic Proportionality Theorem (BPT) is a powerful tool for solving problems involving unknown lengths in triangles. Its main application is to establish a proportional relationship between segments. If a line parallel to a side divides two other sides, you can set up a ratio to find missing lengths. It is the foundation for proving the similarity of triangles and is also used in geometric constructions, such as dividing a line segment into a given ratio.
7. Beyond 'a² + b² = c²', what is the real-world significance of the Pythagoras Theorem?
The Pythagoras Theorem is more than just a formula for triangles; it's a cornerstone of geometry and measurement. Its significance extends to:
- Navigation and Surveying: It is used to calculate the shortest distance between two points (the hypotenuse).
- Coordinate Geometry: The distance formula, used to find the distance between two points on a plane, is a direct application of the Pythagorean theorem.
- Architecture and Construction: Builders use it to ensure that corners are perfectly square (90 degrees) by using measurements like the 3-4-5 ratio.
- Trigonometry: It forms the basis for the fundamental trigonometric identity sin²(θ) + cos²(θ) = 1.
8. Do you need to learn the proofs of theorems, or is just knowing the formula enough?
While you can solve many problems by just applying the formula, understanding the proof of a theorem is essential for deep conceptual clarity. For CBSE exams, proofs of key theorems like BPT and Pythagoras are often asked directly for marks. More importantly, understanding the proof helps you develop logical reasoning and know exactly why and when to apply a theorem, especially for complex or Higher Order Thinking Skills (HOTS) questions where the application isn't obvious.
