Understanding Elementary Shapes Ex 5.6 Class 6 NCERT

Classifying Triangles: By Sides and Angles

Triangles are fascinating shapes, and we can classify them into different categories based on two main characteristics: the lengths of their sides and the measures of their angles. Understanding these classifications is key to truly understanding elementary shapes.

Classification by Sides:

1. Scalene Triangle: Imagine a triangle where all three sides have different lengths. For example, a triangle with sides measuring 3 cm, 4 cm, and 5 cm. Because all sides are unequal, all its angles will also be unequal. Think of it as a "unique" triangle where no two parts are the same.

1. Isosceles Triangle: This is a special type of triangle where at least two of its sides are equal in length. For example, a triangle with sides measuring 5 cm, 5 cm, and 7 cm. An interesting property of an isosceles triangle is that the angles opposite the equal sides are also equal. If you see a triangle with two equal sides, you know it's an isosceles triangle!

1. Equilateral Triangle: The word 'equi' means equal, and 'lateral' refers to sides. So, an equilateral triangle is one where all three sides are equal in length. For instance, a triangle with all sides measuring 6 cm. Not only are its sides equal, but all its three angles are also equal, each measuring 60 degrees. It's the most symmetrical type of triangle.

Classification by Angles:

1. Acute-angled Triangle: If all three angles inside a triangle are less than 90 degrees, it's called an acute-angled triangle. For example, a triangle with angles 60°, 70°, and 50° is acute-angled. All angles are 'sharp'.

1. Right-angled Triangle: A right-angled triangle has exactly one angle that measures 90 degrees. This 90-degree angle is also called a 'right angle' and looks like a perfect corner or an 'L' shape. The side opposite the right angle is always the longest side and is called the hypotenuse. A triangle with angles 30°, 60°, and 90° is a right-angled triangle.

1. Obtuse-angled Triangle: If a triangle has one angle that is greater than 90 degrees (but less than 180 degrees), it's an obtuse-angled triangle. For example, a triangle with angles 20°, 30°, and 130° is an obtuse-angled triangle. This triangle will look 'wide' or 'blunt' at one corner. Remember, a triangle can have at most one obtuse angle, because the sum of all three angles must be 180 degrees.

Step-by-Step: How to Classify Any Triangle

1. Step 1: Examine the Side Lengths — Look at the lengths of all three sides of the triangle. Compare them to see if they are all different, if two are equal, or if all three are equal. This will help you classify it based on its sides.
2. Step 2: Classify by Sides — Based on your observations from Step 1: If all three sides are different, it's a Scalene Triangle. If two sides are equal, it's an Isosceles Triangle. * If all three sides are equal, it's an Equilateral Triangle.
3. Step 3: Examine the Angle Measures — Next, look at the measures of all three angles of the triangle. Pay close attention to whether any angle is 90 degrees, greater than 90 degrees, or if all are less than 90 degrees.
4. Step 4: Classify by Angles — Based on your observations from Step 3: If all three angles are less than 90°, it's an Acute-angled Triangle. If exactly one angle is equal to 90°, it's a Right-angled Triangle. * If exactly one angle is greater than 90° (but less than 180°), it's an Obtuse-angled Triangle.
5. Step 5: Combine the Classifications (Optional but good practice) — Often, you'll be asked for a full classification. Combine your classification from sides and angles. For example, a triangle could be a 'Right-angled Isosceles Triangle' or an 'Acute-angled Scalene Triangle'. This gives a complete picture of the triangle's properties.

Worked Examples: Identifying Triangle Types

• Example 1: A triangle has side lengths of 6 cm, 8 cm, and 10 cm. Its angles are 30°, 60°, and 90°. By Sides: All three side lengths (6, 8, 10) are different. So, it's a Scalene Triangle. By Angles: One angle is exactly 90°. So, it's a Right-angled Triangle. * Complete Classification: Right-angled Scalene Triangle.
• Example 2: A triangle has side lengths of 7 cm, 7 cm, and 7 cm. Its angles are 60°, 60°, and 60°. By Sides: All three side lengths (7, 7, 7) are equal. So, it's an Equilateral Triangle. By Angles: All three angles (60°, 60°, 60°) are less than 90°. So, it's an Acute-angled Triangle. * Complete Classification: Acute-angled Equilateral Triangle.
• Example 3: A triangle has side lengths of 5 cm, 5 cm, and 9 cm. Its angles are 25°, 25°, and 130°. By Sides: Two side lengths (5, 5) are equal. So, it's an Isosceles Triangle. By Angles: One angle (130°) is greater than 90°. So, it's an Obtuse-angled Triangle. * Complete Classification: Obtuse-angled Isosceles Triangle.

YoLearn.ai Exam Tip: Remember These Triangle Facts!

When classifying triangles, always keep these crucial points in mind to avoid common mistakes:

1. Sum of Angles: The sum of all three interior angles of ANY triangle is always 180 degrees. You can use this to find a missing angle if two are given.
2. One or Many: While all three angles in an acute-angled triangle must be acute, a triangle can only have one right angle or one obtuse angle. It cannot have more than one because that would make the sum of angles exceed 180 degrees!
3. Visual vs. Measurement: Don't just guess by looking at a diagram! Always rely on the given measurements of sides and angles to classify the triangle accurately. Diagrams can sometimes be misleading if not drawn to scale.
4. Complete Classification: For a complete answer, always try to classify a triangle using both its sides and its angles if enough information is provided. For example, instead of just 'Isosceles', say 'Acute-angled Isosceles'.

Practice Questions with Solutions

• Q: Classify a triangle with sides 12 cm, 5 cm, 13 cm and angles 23°, 67°, 90°. A: Step 1: Examine the side lengths. The sides are 12 cm, 5 cm, and 13 cm. All three are different. Step 2: Classify by sides. Since all sides are different, it is a Scalene Triangle. Step 3: Examine the angle measures. The angles are 23°, 67°, and 90°. One angle is exactly 90°. Step 4: Classify by angles. Since one angle is 90°, it is a Right-angled Triangle. Final answer: Right-angled Scalene Triangle.
• Q: Classify a triangle with sides 9 cm, 9 cm, 9 cm and angles 60°, 60°, 60°. A: Step 1: Examine the side lengths. The sides are 9 cm, 9 cm, and 9 cm. All three are equal. Step 2: Classify by sides. Since all sides are equal, it is an Equilateral Triangle. Step 3: Examine the angle measures. The angles are 60°, 60°, and 60°. All angles are less than 90°. Step 4: Classify by angles. Since all angles are less than 90°, it is an Acute-angled Triangle. Final answer: Acute-angled Equilateral Triangle.
• Q: A triangle has sides 7 cm, 7 cm, and 4 cm. Its angles are approximately 75°, 75°, and 30°. Classify the triangle. A: Step 1: Examine the side lengths. The sides are 7 cm, 7 cm, and 4 cm. Two sides are equal. Step 2: Classify by sides. Since two sides are equal, it is an Isosceles Triangle. Step 3: Examine the angle measures. The angles are 75°, 75°, and 30°. All angles are less than 90°. Step 4: Classify by angles. Since all angles are less than 90°, it is an Acute-angled Triangle. Final answer: Acute-angled Isosceles Triangle.
• Q: Classify a triangle with sides 10 cm, 10 cm, 15 cm and angles 29°, 29°, 122°. A: Step 1: Examine the side lengths. The sides are 10 cm, 10 cm, and 15 cm. Two sides are equal. Step 2: Classify by sides. Since two sides are equal, it is an Isosceles Triangle. Step 3: Examine the angle measures. The angles are 29°, 29°, and 122°. One angle (122°) is greater than 90°. Step 4: Classify by angles. Since one angle is greater than 90°, it is an Obtuse-angled Triangle. Final answer: Obtuse-angled Isosceles Triangle.

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