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Geometry relies on triangles as one of its fundamental elements because they serve as foundations for multiple mathematical definitions, along with practical industrial uses. A triangle represents a fundamental closed polygon having three sides together with three angles, while also having three vertices, which have crucial significance in mathematics. This section discusses triangle classifications along with the Pythagoras theorem and the Basic Proportionality theorem, and describes the applications of triangle similarity in direct measurement calculations.
Analysis of triangles remains vital for solving advanced mathematical problems and enables applications throughout construction and physics, and navigation disciplines. Students must practice all the topics of Linear Equations in Two Variables and their examples from the NCERT Exemplar Solutions for Class 10 Maths Chapter 6 Triangles. Students must practice questions and check solutions from the NCERT Solutions for Class 10 Maths Chapter 6 Triangles.
A triangle exists as a polygon closed by three sides, three angles, and three vertices. Every triangle contains interior angle measures that total up to 180 degrees, and exterior angles total up to 360 degrees.
Triangles can be classified based on sides and angles:
Based on Sides:
There are different triangle types which depend on the length measurements of their sides. The classification system helps both understand properties and solve related problems more easily. Based on the sides triangle is divided into three types:
Based on Angles:
Relating to angles forms an additional way to categorize triangles. The classification of triangle shapes allows a better understanding of angle characteristics while solving geometric problems. On the basis of angles triangle is divided into 3 types:
In geometry, two shapes qualify as similar if they share equivalent shapes regardless of dimensional differences. In the case of polygons, two polygons are said to be similar if:
Therefore, this means that if two triangles are given ∆ABC and ∆DEF, they are said to be similar if:
1) ∠A =∠D, ∠B =∠E and ∠C =∠F
As like:-
OR
2)
As like:-
Two or more triangles are congruent if they have the same size and shape. Congruence is denoted by the symbol "≅" and read as "is congruent to". The shapes perfectly fit together when placed on top of each other. Congruent triangles can be flipped, rotated and mirrored and will still be congruent. Example: Two copies of the same triangle.
△ABC ≅ △DEF
Two similar figures have the same shape but not necessarily the same size. One figure may exist at a different scale than the other while retaining the same basic dimensions. Similarity is denoted by the sign “~”. Example: A small triangle inside a bigger triangle with the same angles but different side lengths.
△ABC ~ △DEF
Observation:
From the above, we can say that:
To check whether two triangles are similar, there are three conditions:
If two triangles have their corresponding two angles equal, then the triangles are said to be similar triangles. This happens because when two angles match, the third angle will automatically match. After all, every triangle features a 180° angle total.
Two triangles are similar when their corresponding sides maintain the same ratio because this proportionality results in equal corresponding angles. Example: If in one triangle, the sides are 3 cm, 6 cm, and 9 cm, and in another triangle, the sides are 6 cm, 12 cm, and 18 cm, then the ratio is the same (1:2), so they are similar.
Two triangles are similar when they share equal angles between corresponding sides, which also stand in an equal proportional relation. Example: If one triangle has sides 5 cm and 10 cm with an included angle of 50°, and another triangle has sides 10 cm and 20 cm with an included angle of 50°, they are similar.
The Basic Proportionality Theorem (BPT), also known as Thales’ Theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Consider ΔABC, where DE is a line parallel to BC, and it cuts AB at D and AC at E. Then, according to the theorem:
Proof:
As per Pythagoras' Theorem, “In a right-angled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”
This means:
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
When the two similar triangles are given, then the square of the ratio of their corresponding sides will be equal to the ratio of their area.
If ∆ABC ~ ∆PQR, then
Students must download the notes below for each chapter to ace the topics.
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.
To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.
The sixth chapter of NCERT Class 10 Maths textbook focuses on demonstrating the concepts of similar and congruent triangles. This chapter offers multiple theorems about triangle similarity which contain the Pythagoras theorem and the Basic Proportionality Theorem (Thales Theorem) and advanced methods to establish triangular similarity.
The Pythagoras theorem states that in a right-angled triangle:
(Hypotenuse)2 = (Base)2 + (Height)2
To solve the problems related to right-angled triangles, this theorem is used.
Two triangles demonstrate similarity when all pairs of angles match in size and the lengths of their sides maintain identical ratios. The proof of similarity requires the following:
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