Samacheer Kalvi Class 6 Maths Solutions Term 2 Chapter 4 Geometry Exercise 4.3

Miscellaneous Practice Problems

 

Question 1. What are the angles of an isosceles right-angled triangle?
Answer: For a right-angled triangle, one angle is always \( 90^\circ \). Since it is also an isosceles triangle, its other two angles must be equal. Because the sum of angles in a triangle is \( 180^\circ \), the remaining \( 90^\circ \) must be split equally between the two other angles. So, each of these angles will be \( 45^\circ \). Thus, the angles of an isosceles right-angled triangle are \( 90^\circ \), \( 45^\circ \), and \( 45^\circ \).
In simple words: An isosceles right-angled triangle has one angle of \( 90^\circ \), and the other two angles are both \( 45^\circ \) because they must be equal.

🎯 Exam Tip: Remember that "isosceles" means two sides and their opposite angles are equal, and "right-angled" means one angle is \( 90^\circ \).

 

Question 2. Which of the following correctly describes the given triangle?

(a) It is a right isosceles triangle
(b) It is an acute isosceles triangle
(c) It is an obtuse isosceles triangle
(d) It is an obtuse scalene triangle
Answer: (c) It is an obtuse isosceles triangle
In simple words: The triangle has two sides of 6 cm, so it is isosceles. Since \( 6^2 + 6^2 = 36 + 36 = 72 \), which is less than \( 10^2 = 100 \), the angle opposite the 10 cm side is greater than \( 90^\circ \), making it an obtuse triangle.

🎯 Exam Tip: To classify a triangle by its angles when given sides, use the Pythagorean theorem: if \( a^2 + b^2 = c^2 \), it's a right triangle; if \( a^2 + b^2 > c^2 \), it's acute; if \( a^2 + b^2 < c^2 \), it's obtuse (where c is the longest side).

 

Question 3. Which of the following is not possible?
(a) An obtuse isosceles triangle.
(b) An acute isosceles triangle.
(c) An obtuse equilateral triangle.
(d) An acute equilateral triangle.
Answer: (c) An obtuse equilateral triangle.
In simple words: An equilateral triangle always has all three angles equal to \( 60^\circ \), which means all its angles are acute. So, it's impossible for an equilateral triangle to have an angle larger than \( 90^\circ \) (obtuse).

🎯 Exam Tip: Remember the properties of triangles: equilateral triangles always have \( 60^\circ \) angles (acute), isosceles triangles have two equal angles, and the sum of all angles in any triangle is \( 180^\circ \).

 

Question 4. If one angle of an isosceles triangle is \( 124^\circ \), then find the other angles.
Answer: In an isosceles triangle, two sides are equal, and the angles opposite these equal sides are also equal. The sum of all three angles in any triangle is \( 180^\circ \). If one angle is \( 124^\circ \), the sum of the other two angles is \( 180^\circ - 124^\circ = 56^\circ \). Since these two angles must be equal (because it's an isosceles triangle), each of them will be \( \frac { 56^\circ }{ 2 } = 28^\circ \). So, the other two angles are \( 28^\circ \) and \( 28^\circ \).
In simple words: When one angle of an isosceles triangle is \( 124^\circ \), the two other angles are both \( 28^\circ \). This is because the other two angles must be equal, and their total sum must be \( 56^\circ \).

🎯 Exam Tip: When one angle in an isosceles triangle is obtuse (greater than \( 90^\circ \)), it must be the unique angle, and the other two equal angles must be acute.

 

Question 5. The diagram shows a square ABCD. If the line segment joins A and C, then mention the type of triangle so formed.

Answer: When a line segment joins A and C in a square ABCD, it forms two triangles: \( \triangle ABC \) and \( \triangle ADC \). For \( \triangle ABC \), sides AB and BC are equal (since all sides of a square are equal). Also, the angle \( \angle B \) is \( 90^\circ \) (since all angles of a square are \( 90^\circ \)). Therefore, \( \triangle ABC \) is an isosceles right-angled triangle. Both triangles formed, \( \triangle ABC \) and \( \triangle ADC \), are isosceles right-angled triangles.
In simple words: The triangle formed by joining A and C in a square is an isosceles right-angled triangle. This is because two of its sides are equal (the sides of the square), and it has one \( 90^\circ \) angle.

🎯 Exam Tip: Remember that a square has four equal sides and four \( 90^\circ \) angles. A diagonal splits it into two identical right-angled isosceles triangles.

 

Question 6. Draw a line segment AB of length 6 cm. At each end of this line segment AB, draw a line perpendicular to the line segment AB. Are these lines parallel?

Answer: Yes, these lines are parallel. When two lines are drawn perpendicular to the same line segment (AB in this case), they will always be parallel to each other. This is a fundamental property in geometry: lines perpendicular to the same line are always parallel.
In simple words: Yes, the lines drawn perpendicular to line AB at its ends will be parallel. If two lines both make a \( 90^\circ \) angle with a third line, they will never meet.

🎯 Exam Tip: Parallel lines never intersect and always maintain the same distance from each other. Two lines that are perpendicular to the same transversal (like AB) are always parallel.

Challenge Problems

 

Question 7. Is a triangle possible with the angles \( 90^\circ \), \( 90^\circ \), and \( 0^\circ \), Why?
Answer: No, a triangle is not possible with angles \( 90^\circ \), \( 90^\circ \), and \( 0^\circ \). The sum of the interior angles of any triangle must always be exactly \( 180^\circ \). If a triangle had two angles of \( 90^\circ \), their sum alone would be \( 180^\circ \). This would mean the third angle has to be \( 0^\circ \), which is impossible for a polygon with three distinct vertices to form a closed shape. A \( 0^\circ \) angle would mean two sides are perfectly aligned, forming a line, not a triangle.
In simple words: No, you cannot make a triangle with angles \( 90^\circ \), \( 90^\circ \), and \( 0^\circ \). This is because the angles of any triangle must add up to \( 180^\circ \), and a \( 0^\circ \) angle means it is not a proper triangle.

🎯 Exam Tip: The fundamental rule for any triangle is that its three interior angles must sum up to exactly \( 180^\circ \). If the sum is anything else, it's not a triangle.

 

Question 8. Which of the following statements is true? Why?
(a) Every equilateral triangle is an isosceles triangle.
(b) Every isosceles triangle is an equilateral triangle
Answer: The true statement is (a) "Every equilateral triangle is an isosceles triangle."
This is true because an isosceles triangle is defined as a triangle having at least two equal sides. An equilateral triangle has all three sides equal, which automatically means it has at least two equal sides. Therefore, an equilateral triangle meets the definition of an isosceles triangle. Statement (b) is false because an isosceles triangle only needs two equal sides, while an equilateral triangle needs all three sides to be equal.
In simple words: Statement (a) is true because an equilateral triangle has three equal sides, which means it definitely has at least two equal sides, fitting the definition of an isosceles triangle. Statement (b) is false because an isosceles triangle only needs two equal sides, not all three.

🎯 Exam Tip: Think of it like categories: All equilateral triangles are a type of isosceles triangle, but not all isosceles triangles are equilateral. It's a subset relationship.

 

Question 9. If one angle of an isosceles triangle is \( 70^\circ \), then find the possibilities for the other two angles.
Answer: In an isosceles triangle, there are two possibilities if one angle is \( 70^\circ \):
(i) **Case 1: The \( 70^\circ \) angle is one of the two equal angles.**
If one of the equal angles is \( 70^\circ \), then the other equal angle must also be \( 70^\circ \). The sum of these two angles is \( 70^\circ + 70^\circ = 140^\circ \). Since the total sum of angles in a triangle is \( 180^\circ \), the third angle will be \( 180^\circ - 140^\circ = 40^\circ \). So, the angles could be \( 70^\circ \), \( 70^\circ \), and \( 40^\circ \).
(ii) **Case 2: The \( 70^\circ \) angle is the unique angle (not one of the equal angles).**
If the \( 70^\circ \) angle is the unique angle, then the sum of the other two equal angles is \( 180^\circ - 70^\circ = 110^\circ \). Since these two angles are equal, each of them will be \( \frac { 110^\circ }{ 2 } = 55^\circ \). So, the angles could be \( 70^\circ \), \( 55^\circ \), and \( 55^\circ \).
In simple words: There are two ways an isosceles triangle can have a \( 70^\circ \) angle. It could be \( 70^\circ \), \( 70^\circ \), \( 40^\circ \) or it could be \( 70^\circ \), \( 55^\circ \), \( 55^\circ \). This happens because either the \( 70^\circ \) is one of the two equal angles, or it's the unique angle.

🎯 Exam Tip: Always consider both scenarios for an isosceles triangle when only one angle is given: either the given angle is one of the two equal angles, or it's the unique angle. This ensures you find all possible solutions.

 

Question 10. Which of the following can be the sides of an isosceles triangle?
(a) 6 cm, 3 cm, 3 cm
(b) 5 cm, 2 cm, 2 cm
(c) 6 cm, 6 cm, 7 cm
(d) 4 cm, 4 cm, 8 cm
Answer: (c) 6 cm, 6 cm, 7 cm
In simple words: For a triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side. In option (c), \( 6+6 > 7 \), \( 6+7 > 6 \), so it forms a valid triangle with two equal sides. Options (a), (b), and (d) do not satisfy this rule. For example, in (d), \( 4+4 \) is not greater than 8, so it cannot form a triangle.

🎯 Exam Tip: To check if three lengths can form a triangle, use the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side (a + b > c, a + c > b, b + c > a).

 

Question 11. Study the given figure and identify the following triangles,

(a) equilateral triangle
(b) isosceles triangle
(c) scalene triangles
(d) acute triangle
(e) obtuse triangle
(f) right triangle
Answer: From the given figure, we can identify the following types of triangles:
(a) **Equilateral triangle:** A triangle where all three sides are equal. In the figure, BC = 1 + 1 + 1 + 1 = 4 cm, and it is shown that AB = AC = 4 cm. Therefore, \( \triangle ABC \) is an equilateral triangle.
(b) **Isosceles triangle:** A triangle where at least two sides are equal. From the figure, we can see that \( \triangle ABC \) has AB = AC = 4 cm, making it isosceles. Also, \( \triangle AEF \) has AE = AF, so it is an isosceles triangle.
(c) **Scalene triangles:** A triangle where all three sides have different lengths. Based on the lengths in the figure, \( \triangle AEB \), \( \triangle AED \), \( \triangle ADF \), \( \triangle AFC \), \( \triangle ABD \), \( \triangle ADC \), \( \triangle ABF \), and \( \triangle AEC \) are all scalene triangles.
(d) **Acute triangle:** A triangle where all three angles are less than \( 90^\circ \). From the figure, \( \triangle ABC \), \( \triangle AEF \), \( \triangle ABF \), and \( \triangle AEC \) are acute-angled triangles.
(e) **Obtuse triangle:** A triangle where one angle is greater than \( 90^\circ \). From the figure, \( \triangle AEB \) and \( \triangle AFC \) are obtuse-angled triangles.
(f) **Right triangle:** A triangle where one angle is exactly \( 90^\circ \). From the figure, \( \triangle ADB \), \( \triangle ADC \), \( \triangle ADE \), and \( \triangle ADF \) are right-angled triangles. These triangles contain a \( 90^\circ \) angle.
In simple words: Looking at the picture, \( \triangle ABC \) is equilateral. \( \triangle ABC \) and \( \triangle AEF \) are isosceles. Many triangles like \( \triangle AEB \) and \( \triangle ADC \) are scalene. \( \triangle ABC \) and \( \triangle AEF \) are acute. \( \triangle AEB \) and \( \triangle AFC \) are obtuse. Finally, \( \triangle ADB \), \( \triangle ADC \), \( \triangle ADE \), and \( \triangle ADF \) are right-angled triangles because they have a \( 90^\circ \) angle.

🎯 Exam Tip: Accurately identifying triangle types requires knowing the definitions: equilateral (all sides equal), isosceles (at least two sides equal), scalene (all sides different), acute (all angles < \( 90^\circ \)), obtuse (one angle > \( 90^\circ \)), and right (one angle = \( 90^\circ \)).

 

Question 12. Two sides of the triangle are given in the table. Find the third side of the triangle.

Answer: To find the possible range for the third side of a triangle, we use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the absolute difference between the lengths of any two sides must be less than the length of the third side. If the third side is 'x', then \( |a - b| < x < a + b \).
(i) For sides 7 cm and 4 cm:
Difference = \( 7 - 4 = 3 \) cm
Sum = \( 7 + 4 = 11 \) cm
So, the third side must be between 3 cm and 11 cm.
(ii) For sides 8 cm and 8 cm:
Difference = \( 8 - 8 = 0 \) cm
Sum = \( 8 + 8 = 16 \) cm
So, the third side must be between 0 cm and 16 cm.
(iii) For sides 7.5 cm and 3.5 cm:
Difference = \( 7.5 - 3.5 = 4 \) cm
Sum = \( 7.5 + 3.5 = 11 \) cm
So, the third side must be between 4 cm and 11 cm.
(iv) For sides 10 cm and 14 cm:
Difference = \( 14 - 10 = 4 \) cm
Sum = \( 14 + 10 = 24 \) cm
So, the third side must be between 4 cm and 24 cm.
In simple words: The third side of a triangle must be shorter than the sum of the other two sides, but longer than their difference. For example, if two sides are 7 cm and 4 cm, the third side must be between 3 cm and 11 cm long. This rule makes sure the sides can actually form a triangle.

🎯 Exam Tip: Always remember the Triangle Inequality Theorem (the sum of any two sides must be greater than the third side) as it's crucial for determining if a triangle can be formed and the possible range of a missing side.

 

Question 13. Complete the following table:

Answer: Here is the completed table explaining the angle properties of different triangle types:

In simple words: An acute triangle has all three angles smaller than \( 90^\circ \). A right triangle has one angle exactly \( 90^\circ \), with the other two being acute. An obtuse triangle has one angle larger than \( 90^\circ \), and the other two are always acute.

🎯 Exam Tip: This table summarizes key definitions. Remember that in a right or obtuse triangle, there can only be one angle that is \( 90^\circ \) or greater than \( 90^\circ \); the remaining two angles must always be acute.