Mathematics Notes for Class 10 Triangles

Triangles

Welcome to the chapter on Triangles for Class 10. In this chapter, you will learn about the properties of triangles, different types of triangles, criteria for similarity and congruence, and important theorems related to triangles. By the end of this chapter, you will be able to solve problems involving triangles and apply these concepts in geometry.

Key Concepts

Triangle: A polygon with three sides and three angles.
Congruent Triangles: Triangles that are exactly equal in size and shape.
Similar Triangles: Triangles that have the same shape but not necessarily the same size.

Types of Triangles

By Sides: Equilateral, Isosceles, Scalene
By Angles: Acute, Right, Obtuse

Congruence of Triangles

Two triangles are congruent if their corresponding sides and angles are equal. The main criteria for congruence are:

SSS (Side-Side-Side): All three sides of one triangle are equal to all three sides of another triangle.
SAS (Side-Angle-Side): Two sides and the included angle are equal.
ASA (Angle-Side-Angle): Two angles and the included side are equal.
AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
RHS (Right angle-Hypotenuse-Side): For right triangles, the hypotenuse and one side are equal.

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. The main criteria for similarity are:

AAA (Angle-Angle-Angle): All three angles of one triangle are equal to all three angles of another triangle.
SAS (Side-Angle-Side): Two sides are in the same ratio and the included angle is equal.
SSS (Side-Side-Side): All three sides are in the same ratio.

Important Theorems

Basic Proportionality Theorem (Thales' Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those sides 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.
Area Theorem for Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Applications

Solving problems involving heights and distances.
Finding unknown sides or angles using similarity or congruence.
Using theorems to prove geometric results.

Practice Questions

State the criteria for congruence of triangles.
If two triangles have all their angles equal, are they similar? Why?
In △ABC, DE || BC and D, E are points on AB and AC. If AD/DB = 2/3, find AE/EC.
If the sides of two similar triangles are in the ratio 3:5, what is the ratio of their areas?
State and prove the Pythagoras theorem.

Challenge Yourself

Draw two triangles that are similar but not congruent. Explain why.
Given two right triangles with hypotenuses of 10 cm and 15 cm, and one side of 6 cm and 9 cm respectively, show that they are similar.

Did You Know?

The sum of the angles of any triangle is always 180°.
Triangles are the only polygons that are always rigid and cannot be deformed without changing the length of their sides.

Glossary

Congruent: Exactly equal in size and shape.
Similar: Same shape but not necessarily the same size.
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
Corresponding Sides: Sides that are in the same position in different triangles.

Answers to Practice Questions

SSS, SAS, ASA, AAS, RHS.
Yes, because equal angles mean the triangles are similar by AAA criterion.
AE/EC = AD/DB = 2/3 (by Basic Proportionality Theorem).
Ratio of areas = (3/5)² = 9/25.
Pythagoras theorem: In a right-angled triangle, (Hypotenuse)² = (Base)² + (Perpendicular)².
Proof: (Provide a step-by-step proof as per your syllabus.)

Master triangles to unlock many secrets of geometry and solve real-world problems!

Mathematics Class 10 - Triangles Notes