CBSE Notes Class 7 Maths Chapter 6 Triangles and Its Properties

Triangles are one of the most essential shapes in geometry, with three sides, three vertices, and three angles. They appear in various forms, each with unique properties that play a crucial role in geometry. This chapter introduces the concept of triangles and explores their classification based on sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse).

We'll dive into key properties, such as the angle sum property (the sum of interior angles is always 180° the triangle inequality theorem (sum of any two sides is greater than the third side), and specific properties of right-angled triangles, including the hypotenuse. These properties are fundamental for understanding and solving a wide range of geometric problems.

1.0Triangle

A closed figure formed by joining three non-collinear points is called a triangle. The three sides and three angles of a triangle are collectively known as elements of the triangle.

2.0Classification of Triangles 

According to sides 

1. Scalene Triangle: A triangle with three unequal sides is called a scalene triangle. (a ≠ b ≠ c) 

2. Isosceles Triangle: A triangle with any two pair of equal/same sides is called an isosceles triangle. PQ = PR, so a ΔPQR is an isosceles triangle. 

3. Equilateral Triangle: A triangle with all of its three sides equal/same is called an equilateral triangle. DE = EF = DF, so a ΔDEF is an equilateral triangle. 

According to Angles 

1. Acute Angled Triangle: A triangle with all three acute angles is an acute angled triangle. 

0° < (∠A, ∠B, ∠C) < 90° so ΔABC is an acute angled triangle. 

2. Obtuse Angled Triangle: A triangle with one angle is obtuse and two angles are acute is called an obtuse angled triangle. (90° < ∠Q < 180°) and 0° < (∠P, ∠R) < 90°, so Δ PQR is an obtuse angled triangle. 

3. Right Angled Triangle: A triangle with a right angle and two acute angles is called a right-angled triangle. ∠E = 90° and 0° < (∠D, ∠F) < 90° so ΔDEF is a right-angled triangle.

3.0Median of a Triangle 

A median of a triangle is a line segment that connects a vertex to the mid-point of the side opposite to that of vertex.

In ΔABC, D is the midpoint of BC and AD is a median. 

Likewise, points E and F are the midpoints of sides AC and AB, respectively and BE and CF are the medians of the triangle. 

The point where the three medians of a triangle meet is called the centroid of a triangle. 

The centroid of a triangle splits each median in a specific ratio 2: 1, 

i.e. AG: GD = BG: GE = CG: GF = 2: 1

4.0Altitude of a Triangle

In a triangle, the line segment drawn perpendicularly from any vertex to the opposite side is known as an altitude. In triangle ABC, the segments AD, BE, and CF represent the three altitudes. The point where these three altitudes intersect is called the orthocenter. Here, point P is the orthocenter of triangle ABC.

5.0Angle Bisector of a Triangle 

In a triangle, a line segment that divides an interior angle into two equal parts is known as an angle bisector. In triangle ABC, BD bisects ∠ABC, making BD the angle bisector, so ∠ABD = ∠CBD. Similarly, AF and CE are angle bisectors of ∠BAC and ∠BCA, respectively. The point where the three angle bisectors AF, BD, and CE meet is called the incenter, denoted by I, of triangle ABC.

6.0Angle Sum Property of a Triangle

Draw a triangle. Cut on three angles. Rearrange them as shown in figure. The three angles now constitute one angle. This angle is a straight angle and so it has measure 180°. 

Thus, the sum of the measures of the three angles of the triangle is equal to 180°.

7.0Angle Sum Property

The angle sum property of a triangle states that the sum of the interior angles of any triangle is always 180°.

Explanation:

Consider a triangle ABC with interior angles ∠A, ∠B, and ∠C. According to the angle sum property, we have:

∠A + ∠B + ∠C = 180°.

This fundamental property holds true for all types of triangles, whether they are scalene, isosceles, or equilateral.

8.0Exterior Angles of a Triangle 

Consider a triangle ABC. If one side of ΔABC, say BC, is produced and X is any point on the ray BC, then ∠ACX is called an exterior angle of the ΔABC at C. ∠ACB is the interior adjacent angle of ∠ACX while the other two angles, i.e., ∠A and ∠B are not the interior adjacent angles. Angles A and B are called the interior opposite angles corresponding to exterior angle ACX. 

Similarly, if AC is produced and Y is a point on ray AC, then ∠BCY is also an exterior angle of ΔABC at C. ∠A and ∠B are interior opposite angles of ∠BCY also.

Likewise, ∠BAL is the exterior angle of ΔABC at A and ∠B and ∠C are its interior opposite angles. 

∠ABM is the exterior angle of ΔABC at B and ∠A and ∠C are its interior opposite angles.

9.0Exterior Angle Property of a Triangle 

If any side of a triangle is extended, the resulting exterior angle is equal to the sum of the two opposite interior angles.

10.0Triangle Inequality Property 

In any triangle, the combined length of any two sides is always greater than the length of the third side. For a triangle PQR with side lengths a, b, and c opposite vertices P, Q, and R respectively, this can be expressed as:

• a + b > c
• b + c > a
• a + c > b

These inequalities must hold true for PQR to form a valid triangle.

11.0Pythagoras Theorem for Right-Angled Triangles

A right-angled triangle features one right angle and two acute angles. The side opposite the right angle is called the hypotenuse, whereas the other two sides are known as the triangle's legs. In a right-angled triangle, the sides that form the right angle are commonly called the base and the height (or perpendicular). 

12.0Sample Questions on Triangles and Its Properties

1. What is the area of a triangle?

Ans: The area of a triangle can be calculated using the formula:

$\text{ Area }=\frac{1}{2}\times \text{ base }\times \text{ height }$

Alternatively, for any triangle with sides a, b, and c, the area can be calculated using Heron's formula.

2. What is Heron's formula?

Ans: Heron's formula allows the area of a triangle to be calculated when the lengths of all three sides are known:

$s=\frac{a+b+c}{2}\text{ (semi-perimeter) }$

$\text{ Area }=\sqrt{s(s-a)(s-b)(s-c)}$