The Triangle And Its Properties Class 7 Notes

Key Definitions

Triangle

A simple closed figure formed by three line segments. It has three vertices, three sides, and three angles.

Median

A line segment joining a vertex of a triangle to the midpoint of the opposite side. A triangle has three medians.

Altitude

A line segment from a vertex of a triangle perpendicular to the opposite side (or its extension). It represents the height of the triangle. A triangle has three altitudes.

Interior Angles

The angles formed inside the triangle by its sides.

Exterior Angle

An angle formed by extending one side of a triangle. It is equal to the sum of the two opposite interior angles.

Equilateral Triangle

A triangle with all three sides equal in length and all three angles equal (each 60 degrees).

Isosceles Triangle

A triangle with at least two sides of equal length. The angles opposite to the equal sides are also equal.

Scalene Triangle

A triangle with all three sides of different lengths and all three angles of different measures.

Right-angled Triangle

A triangle in which one of the angles is a right angle (90 degrees).

Hypotenuse

The side opposite the right angle in a right-angled triangle; it is always the longest side.

Understanding Triangles: Types and Basic Elements

A triangle is a fundamental polygon with three sides, three vertices, and three angles. The sum of these three interior angles always adds up to 180 degrees – this is known as the Angle Sum Property. Triangles can be classified based on two main criteria: their sides and their angles.

Classification Based on Sides:

• Scalene Triangle: In a scalene triangle, all three sides have different lengths. Consequently, all three angles also have different measures. For example, a triangle with sides 3cm, 4cm, and 5cm is scalene.
• Isosceles Triangle: An isosceles triangle has at least two sides of equal length. The angles opposite to these equal sides are also equal. If a triangle has sides 5cm, 5cm, and 7cm, it's isosceles, and the angles opposite the 5cm sides will be equal.
• Equilateral Triangle: An equilateral triangle is a special type of isosceles triangle where all three sides are equal in length. This also means that all three angles are equal, and since the sum is 180°, each angle in an equilateral triangle is 60 degrees.

Classification Based on Angles:

• Acute-angled Triangle: An acute-angled triangle is a triangle where all three interior angles are acute (less than 90 degrees).
• Right-angled Triangle: A right-angled triangle has exactly one right angle (equal to 90 degrees). The side opposite the right angle is called the hypotenuse, which is always the longest side. The other two sides are called legs.
• Obtuse-angled Triangle: An obtuse-angled triangle has exactly one obtuse angle (greater than 90 degrees but less than 180 degrees).

Median vs. Altitude: Key Differences

Important Properties of Triangles

Understanding these properties is crucial for solving various geometry problems involving triangles:

1. Angle Sum Property of a Triangle:

• The sum of the measures of the three interior angles of any triangle is always 180 degrees. This is a fundamental property.
• Formula: ∠A + ∠B + ∠C = 180°
• Memory Tip: Imagine tearing off the three corners of a paper triangle and arranging them side-by-side; they will form a straight line (180°).

1. Exterior Angle Property of a Triangle:

• If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two opposite interior angles.
• Formula: Exterior Angle = Sum of its interior opposite angles.
• For example, if side BC of triangle ABC is extended to D, then ∠ACD (exterior angle) = ∠A + ∠B.

1. Triangle Inequality Property:

• The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
• This property determines if three given line segments can actually form a triangle.
• Formulas:
• AB + BC > AC
• BC + CA > AB
• CA + AB > BC

1. Pythagoras Property (for Right-angled Triangles):

• In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
• Formula: (Hypotenuse)² = (Leg 1)² + (Leg 2)²
• If 'c' is the hypotenuse and 'a' and 'b' are the legs, then c² = a² + b².
• This property is named after the Greek mathematician Pythagoras and is one of the most famous theorems in geometry.

Key Points to Remember

• A triangle has 3 vertices, 3 sides, and 3 angles.
• The sum of interior angles of any triangle is always 180°.
• An exterior angle of a triangle is equal to the sum of its two interior opposite angles.
• The three medians of a triangle intersect at a single point called the centroid.
• The three altitudes of a triangle intersect at a single point called the orthocenter.
• For any triangle to exist, the sum of any two sides must be greater than the third side.
• In an isosceles triangle, angles opposite to equal sides are equal.
• In a right-angled triangle, the side opposite the 90° angle is the hypotenuse and is the longest side.
• Pythagoras property: In a right-angled triangle, a² + b² = c² (where c is hypotenuse).
• An equilateral triangle is also an equiangular triangle (all angles are 60°).

Solved Examples

• {"problem":"Q1: Find the value of x in the triangle below if the angles are x, 2x, and 3x.","solution":"A1: According to the Angle Sum Property, x + 2x + 3x = 180°. So, 6x = 180°, which means x = 30°."}
• {"problem":"Q2: In a right-angled triangle, the two legs are 3 cm and 4 cm. Find the length of the hypotenuse.","solution":"A2: Using Pythagoras Property, Hypotenuse² = 3² + 4² = 9 + 16 = 25. So, Hypotenuse = √25 = 5 cm."}
• {"problem":"Q3: An exterior angle of a triangle is 110°. Its interior opposite angles are 50° and x°. Find x.","solution":"A3: By Exterior Angle Property, 110° = 50° + x°. So, x = 110° - 50° = 60°."}

Exam Tip for The Triangle And Its Properties

Always start by drawing and labeling the triangle accurately if it's not provided in the question. This helps visualize the problem, especially for medians, altitudes, and exterior angles. For questions involving the Pythagoras property, ensure you correctly identify the hypotenuse (the side opposite the right angle) before applying the formula. Don't confuse medians with altitudes; remember medians go to the midpoint, while altitudes are perpendicular. Pay close attention to keywords like 'midpoint' or 'perpendicular' in the problem statement.

Practice Questions with Solutions

• Q: What is the sum of the angles in any triangle? A: 180 degrees.
• Q: If two sides of a triangle are 6 cm and 8 cm, what is a possible length for the third side? A: Any length 'x' such that (8-6) < x < (8+6), i.e., 2 < x < 14 cm. For example, 10 cm.
• Q: In a triangle ABC, if ∠A = 70° and ∠B = 60°, what is ∠C? A: ∠C = 180° - (70° + 60°) = 180° - 130° = 50°.
• Q: Can a triangle have two obtuse angles? A: No, because the sum of two obtuse angles would already exceed 180°, violating the angle sum property.

Frequently Asked Questions