The Triangle and its Properties
Overview
A triangle is a simple closed figure made of three line segments. This chapter explores the different parts of a triangle, types of triangles, special properties like the angle sum property, exterior angle property, and the Pythagorean theorem. Students learn how the sides and angles of a triangle relate to each other, which forms the foundation for understanding geometry in higher classes. These properties help solve many real-life measurement problems.
Typed Notes
Parts and Types of Triangles
A triangle has three sides, three angles, and three vertices. Special line segments include medians (vertex to midpoint of opposite side) and altitudes (perpendicular from vertex to opposite side). Triangles are classified by sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal). By angles: Acute (all angles less than 90°), Right (one angle exactly 90°), Obtuse (one angle greater than 90°). Every triangle has exactly three medians and three altitudes, all meeting at a single point inside or outside the triangle.
Angle Sum and Exterior Angle Properties
The Angle Sum Property states that the sum of the three interior angles of any triangle is 180°. For example, if two angles are 60° and 70°, the third is 180° − 60° − 70° = 50°. The Exterior Angle Property states that when a side of a triangle is extended, the exterior angle formed equals the sum of the two interior opposite angles. For example, if interior opposite angles are 55° and 65°, the exterior angle is 55° + 65° = 120°. These two properties are very useful for finding unknown angles in triangles.
Triangle Inequality Property
The Triangle Inequality Property states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This must hold true for all three combinations of sides. For example, sides 3 cm, 4 cm, and 5 cm form a triangle because 3+4>5, 3+5>4, and 4+5>3. But sides 2 cm, 3 cm, and 6 cm do NOT form a triangle because 2+3=5, which is not greater than 6. This property helps us check whether a given set of lengths can form a triangle.
Pythagoras Theorem
In a right-angled triangle, the side opposite the right angle is called the hypotenuse and is the longest side. Pythagoras Theorem states: (Hypotenuse)² = (Base)² + (Perpendicular)². For example, if base = 3 cm and perpendicular = 4 cm, then hypotenuse² = 9 + 16 = 25, so hypotenuse = 5 cm. The converse is also true: if the square of one side equals the sum of squares of the other two sides, the triangle is right-angled. Common Pythagorean triplets are (3,4,5), (5,12,13), and (8,15,17).
Key Concepts
Revision ke liye ek nazar — exam se pehle yahi padhna hai.
- 01The sum of all three interior angles of any triangle is always 180°.
- 02An exterior angle of a triangle equals the sum of the two non-adjacent interior angles (remote interior angles).
- 03In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides (Pythagoras theorem).
- 04The sum of any two sides of a triangle is always greater than the third side (Triangle Inequality).
- 05A median of a triangle connects a vertex to the midpoint of the opposite side; an altitude is the perpendicular from a vertex to the opposite side.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
The two interior angles of a triangle are 65° and 75°. Find the third angle.
An exterior angle of a triangle is 110° and one of the interior opposite angles is 45°. Find the other interior opposite angle.
Check whether sides of length 5 cm, 8 cm, and 11 cm can form a triangle.
In a right-angled triangle, the two legs (perpendicular sides) are 6 cm and 8 cm. Find the hypotenuse.
In an isosceles triangle, the vertex angle is 50°. Find the two base angles.
A triangle has three sides, angles, and vertices. Key properties include: angle sum = 180°, exterior angle = sum of two remote interior angles, sum of any two sides > third side, and Pythagoras theorem for right-angled triangles. Triangles are classified by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse). Medians and altitudes are important line segments inside a triangle.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.What is the sum of all interior angles of a triangle?
Q2.Which of the following sets of sides CANNOT form a triangle?
Q3.In a right-angled triangle with legs 5 cm and 12 cm, what is the hypotenuse?
Q4.An exterior angle of a triangle is always __________.
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