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Understanding Elementary Shapes - Classification of Triangles (by sides and angles)

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A triangle is a closed shape with three sides, three vertices, and three angles. It is the polygon with the fewest sides. Visually, it is formed by connecting three points that are not on the same line.

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Classification by Sides - Scalene Triangle: A triangle in which all three sides have different lengths. Visually, the triangle looks lopsided or irregular, and none of its angles are equal.

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Classification by Sides - Isosceles Triangle: A triangle with at least two sides of equal length. Visually, it often appears symmetrical, like a peaked roof, and the angles opposite the equal sides are also equal.

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Classification by Sides - Equilateral Triangle: A triangle where all three sides are equal in length. Visually, it is a perfectly balanced shape where all three sides and all three interior angles (60∘60^{\circ} each) are identical.

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Classification by Angles - Acute-angled Triangle: A triangle where every internal angle is less than 90∘90^{\circ}. Visually, all the corners look sharp and narrow.

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Classification by Angles - Right-angled Triangle: A triangle that has one angle of exactly 90∘90^{\circ}. Visually, it contains a perfect square corner (perpendicular), often indicated by a small square symbol.

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Classification by Angles - Obtuse-angled Triangle: A triangle that has one angle greater than 90∘90^{\circ}. Visually, one corner is spread open very wide, making the triangle look blunt or stretched out.

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Angle Sum Property: The sum of the three interior angles of any triangle is always 180∘180^{\circ}. This is a fundamental rule used to find a missing angle when the other two are known.

📐Formulae

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}

Perimeter=a+b+c\text{Perimeter} = a + b + c

a+b>ca + b > c

Each angle of an Equilateral Triangle=60∘\text{Each angle of an Equilateral Triangle} = 60^{\circ}

💡Examples

Problem 1:

In △PQR\triangle PQR, the angles are ∠P=35∘\angle P = 35^{\circ} and ∠Q=55∘\angle Q = 55^{\circ}. Find the third angle ∠R\angle R and classify the triangle by its angles.

Solution:

  1. Use the Angle Sum Property: ∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^{\circ}. 2. Substitute the known values: 35∘+55∘+∠R=180∘35^{\circ} + 55^{\circ} + \angle R = 180^{\circ}. 3. Simplify the sum: 90∘+∠R=180∘90^{\circ} + \angle R = 180^{\circ}. 4. Solve for ∠R\angle R: ∠R=180∘−90∘=90∘\angle R = 180^{\circ} - 90^{\circ} = 90^{\circ}.

Explanation:

Since one of the angles (∠R\angle R) is exactly 90∘90^{\circ}, the triangle is classified as a Right-angled triangle.

Problem 2:

A triangle has side lengths of 8 cm8\text{ cm}, 5 cm5\text{ cm}, and 8 cm8\text{ cm}. Classify this triangle based on its side lengths.

Solution:

  1. Observe the lengths of the three sides: a=8 cma = 8\text{ cm}, b=5 cmb = 5\text{ cm}, and c=8 cmc = 8\text{ cm}. 2. Compare the side lengths: We see that side aa and side cc are equal (8 cm=8 cm8\text{ cm} = 8\text{ cm}). 3. Identify that at least two sides are equal.

Explanation:

A triangle with two equal sides is called an Isosceles triangle. Visually, the two sides of 8 cm8\text{ cm} would meet at a vertex, creating a symmetrical appearance.