Theorems of Triangles Class 10

In Mathematics, a theorem is a representation of a general concept that belongs to a broader theory. Proof is the method of demonstrating the correctness of a theorem. In this article, we will discuss all the important theorems in Mathematics Class 10.

The Maths theorem Class 10 includes theorems from circles, triangles, Pythagoras theorem, fundamental theorems of Arithmetic etc.

A triangle is a three-sided polygon with three vertices and three sides. It is one of the most fundamental geometric shapes. The Triangles Theorem Class 10 will help us to understand the properties of triangles. By proving Triangles Class 10 theorems the students' logical thinking and reasoning skills will be improved and it will also help them to clearly understand the concepts of triangles.

The Class 10 Maths all Theorems PDF is available completely free on the Vedantu platform. This PDF explains all theorems of triangles Class 10 proofs in a step by step manner.

Maths Theorems Class 10 

Here let us have a look at some of the important theorems in Mathematics Class 10.

• Pythagoras Theorem

• Midpoint Theorem

• Remainder Theorem

• Fundamental Theorem of Arithmetic

• Angle Bisector Theorem

• Inscribed Angle Theorem

• Ceva’s Theorem

• Bayes Theorem

Pythagoras Theorem

The Pythagoras theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle in mathematics.

According to Pythagoras theorem “The area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides”.

This theorem can be written as the Pythagorean equation, which relates the lengths of the sides a, b, and c.

\[c^{2} = a^{2} + b^{2} \]

Where c is the hypotenuse and a and b are the other two sides of the triangle.

Midpoint Theorem

According to the midpoint theorem “The line segment in a triangle connecting the midpoints of two sides of the triangle is said to be parallel to its third side and is also half the length of the third side”.

The midpoint theorem formula is given as follows:

If \[P(x_{1},y_{1})\] and \[Q(x_{2},y_{2})\] are the coordinates of the two given endpoints of a line then the midpoint formula is given by

Midpoint \[= (x_{m},y_{m})\] = \[\frac{{(x_{1} + x_{2})}}{2}\] , \[\frac{{(y_{1} + y_{2})}}{2}\]

Remainder Theorem

The remainder theorem states that when a polynomial \[f(x)\] is divided by a linear polynomial \[(x - a)\], the remainder is the same as \[f(a)\].

The proof of the remainder theorem is as follows:

The proof for the polynomial remainder theorem is derived from the Euclidean division theorem. According to these two polynomials \[P(x)\] which is the dividend and \[g(x)\] which is the divisor, asserts the existence of a quotient \[Q(x)\] and a remainder \[R(x)\] such that

\[P(x) = Q(x) \times g(x) \times + R(x) and R (x) = 0\]

If the divisor \[g(x) = x - a\], where a is a constant then \[R(x) = 0\]

In both cases, \[R(x)\]  is independent of x that is \[R(x)\] is a constant. So we get 

\[P(x) = Q(x) \times (x - a) + R\]

Now let us make x equal to ‘a’ in this formula, we get

\[P(a) = Q(a) \times (a - a) + R\]

\[P(x) = Q(a) \times 0 +R\]

\[P(x) = R \]

Hence Proved.

Fundamental Theorem of Arithmetic

Apart from rearrangement as a product of one or more primes, the fundamental theorem of arithmetic states that any positive integer except 1 can be interpreted in exactly one way. The special factorization theorem is another name for this theorem.

Angle Bisector Theorem

According to the Angle Bisector Theorem, an angle bisector splits the opposite side of a triangle into two parts that are equal to the triangle's other two sides.

Inscribed Angle Theorem

According to the inscribed angle theorem “An angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. As a result, when the vertex of the angle is moved around the circle, the angle does not change.

Ceva’s Theorem

Ceva's theorem is an affine geometry theorem in the sense that it can be claimed and proven without using angles, regions, or lengths. As a result, triangles in an affine plane over any field are real.

Bayes Theorem

The Bayes theorem calculates the probability of an occurrence based on new knowledge related to that event. The formula can also be used to see how hypothetical new information affects the likelihood of an occurrence happening, assuming the new information is valid.

Apart from these theorems, the most important theorems of class 10 are from Triangles and Circles.

All Theorems of Triangles Class 10 PDF

The important triangles theorem Class 10 are as follows:

• All congruent triangles are similar, but it doesn’t mean that all similar triangles are congruent.

• If there are two triangles and if their corresponding angles are equal and also the corresponding sides are in the same ratio then the two triangles are similar triangles.

• If the sides of one triangle are proportional to the sides of the other, their corresponding angles are identical, and the two triangles are similar.

Circle Theorem Class 10

Here let us look at the important Circle Theorem Class 10.

• At the center of the circle, equal chords of a circle subtend equal angles.

• If taken from the center of the circle, the perpendicular to a chord bisects the chord.

• A circle's equal chords are equidistant from the circle's center.

• In a cyclic quadrilateral, the opposite angles are supplementary.

• Angles subtended by the same arc at every point on the circumference of the circle are equal to half of the angle subtended by the same arc at the center.

Conclusion

Theorems aid in the easy solution of mathematical problems, and their proofs aid in the development of a deeper understanding of the underlying concepts. Some theorems are important because they incorporate new proof methods or include a new lemma that is more useful than the proved theorem. This said the important theorems in Mathematics class 10 helps students to understand the fundamental concepts of geometry. The proof of these theorems of class 10 is created in such a way that the students will be able to grasp the concepts easily without any doubts.