Mathematics Notes for Class 11 Triangles

Triangles

Welcome to the chapter on Triangles for Class 11. In this chapter, you will learn about the properties of triangles, different types of triangles, important theorems, and methods to solve problems involving triangles. By the end of this chapter, you will be able to apply these concepts to geometry problems and real-life situations.

Key Concepts

Triangle: A polygon with three sides and three angles.
Types of Triangles: Based on sides (scalene, isosceles, equilateral) and angles (acute, right, obtuse).
Congruence: Two triangles are congruent if their corresponding sides and angles are equal.
Similarity: Two triangles are similar if their corresponding angles are equal and sides are in proportion.

Properties of Triangles

The sum of the angles of a triangle is always 180°.
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
The area of a triangle can be found using various formulas (base-height, Heron's formula, trigonometric formulas).

Congruence of Triangles

Triangles are congruent if they satisfy any of the following criteria:

SSS (Side-Side-Side): All three sides are equal.
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, hypotenuse and one side are equal.

Similarity of Triangles

Triangles are similar if:

AA (Angle-Angle): Two angles are equal.
SSS (Side-Side-Side): All sides are in the same ratio.
SAS (Side-Angle-Side): Two sides are in the same ratio and the included angle is equal.

Important Theorems

Pythagoras Theorem: In a right triangle, (Hypotenuse)² = (Base)² + (Height)².
Basic Proportionality Theorem (Thales' Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides those sides in the same ratio.
Angle Bisector Theorem: The angle bisector of a triangle divides the opposite side in the ratio of the other two sides.

Area of a Triangle

Base-Height Formula: Area = ½ × base × height
Heron's Formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
Trigonometric Formula: Area = ½ ab sin C

Practice Questions

Prove that the sum of the angles of a triangle is 180°.
If two triangles have sides in the ratio 2:3 and their included angles are equal, are the triangles similar? Why?
Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
State and prove the Basic Proportionality Theorem.
In a right triangle, if the base is 6 cm and the height is 8 cm, find the hypotenuse and area.

Challenge Yourself

Draw two triangles that are similar but not congruent. Explain why.
If the area of a triangle is 24 cm² and the base is 6 cm, find the height.
Use trigonometry to find the area of a triangle with sides 5 cm and 7 cm and included angle 60°.

Did You Know?

The triangle is the only polygon that is always rigid and cannot be deformed without changing the length of its sides.
Triangles are used in construction for strength and stability (e.g., bridges, roofs).

Glossary

Congruent: Exactly equal in shape and size.
Similar: Same shape but not necessarily the same size.
Hypotenuse: The longest side of a right triangle, opposite the right angle.
Angle Bisector: A line that divides an angle into two equal parts.

Answers to Practice Questions

Draw a triangle and use parallel lines to show the sum of angles is 180° (refer to textbook proof).
Yes, by SAS similarity criterion (sides in same ratio and included angle equal).
Use Heron's formula: s = (7+8+9)/2 = 12, Area = √[12×5×4×3] = √720 = 26.83 cm² (approx).
See textbook for proof: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
Hypotenuse = √(6²+8²) = 10 cm; Area = ½ × 6 × 8 = 24 cm².

Master triangles to solve complex geometry problems and understand the world of shapes!

Mathematics Class 11 - Triangles Notes