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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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Triangles are classified by their sides into three types: Scalene (no sides equal), Isosceles (at least two sides equal), and Equilateral (all three sides equal).

Visual comparison of Scalene, Isosceles, and Equilateral triangles.
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Triangles are classified by their angles: Acute-angled (all angles <90∘< 90^{\circ}), Right-angled (one angle =90∘= 90^{\circ}), and Obtuse-angled (one angle >90∘> 90^{\circ}).

A right-angled triangle showing a 90 degree corner.
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In an Equilateral triangle, all three interior angles are equal and each measures 60∘60^{\circ}.

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The sum of any two sides of a triangle must always be greater than the third side (a+b>ca + b > c).

📐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.

Problem 3:

Classify △ABC\triangle ABC where AB=7 cmAB = 7\text{ cm}, BC=2 cmBC = 2\text{ cm}, and AC=7 cmAC = 7\text{ cm} based on its sides.

Isosceles triangle ABC with two sides of 7cm and base of 2cm.

Solution:

  1. Identify side lengths: 7 cm7\text{ cm}, 2 cm2\text{ cm}, and 7 cm7\text{ cm}.
  2. Observe that two sides (ABAB and ACAC) are equal.
  3. A triangle with two equal sides is called an Isosceles triangle.

Explanation:

Since AB=AC=7 cmAB = AC = 7\text{ cm}, the triangle has two equal sides, making it isosceles.

Problem 4:

A triangle has angles measuring 120∘120^{\circ}, 35∘35^{\circ}, and 25∘25^{\circ}. Classify this triangle by its angles.

Obtuse-angled triangle with one angle clearly wider than 90 degrees.

Solution:

  1. Look at the measure of each angle.
  2. One angle measures 120∘120^{\circ}, which is greater than 90∘90^{\circ}.
  3. An angle greater than 90∘90^{\circ} is an obtuse angle.
  4. Therefore, the triangle is an Obtuse-angled triangle.

Explanation:

If any one angle of a triangle is greater than 90∘90^{\circ}, the triangle is classified as obtuse-angled.