RS Aggarwal Solutions for Class 6 Chapter 12 Parallel Lines

Exercise 12.1

Question 1. Take three non-collinear points A, B and C on a page of your notebook. Join AB, BC and CA. What figure do you get? Name the triangle. Also, name (i) the side opposite to ∠B (ii) the angle opposite to side AB (iii) the vertex opposite to side BC (iv) the side opposite to vertex B
Answer: When you take three non-collinear points A, B and C and join them together, you form a Triangle. The triangle formed is named as ΔABC.
(i) The side opposite ∠B is AC
(ii) The angle opposite side AB is ∠C
(iii) The vertex opposite side BC is A
(iv) The side opposite vertex B is AC

Exam Tip: Always identify opposite elements correctly - a vertex opposite a side is the point not touching that side, and a side opposite an angle is the side that does not form that angle.

Question 2. Take three collinear points A, B and C on a page of your note book. Join AB, BC and CA. Is the figure a triangle? If not why
Answer: When you take three collinear points A, B and C (all lying on the same straight line) and join them together, the figure formed is not a triangle. This is because the shape created becomes a single straight line with just one side, rather than a closed shape with three sides. A triangle must always be a closed figure made up of three distinct sides, which cannot happen when the three points are collinear.

Exam Tip: Remember that the fundamental requirement for a triangle is that its three vertices must be non-collinear - if they lie on the same line, it cannot form a triangle.

Question 3. Distinguish between a triangle and its triangular region.
Answer: A triangle is a closed polygon that consists of three straight line segments (the sides only). In contrast, a triangular region is the area that lies inside the three sides of the triangle - it includes both the sides and all the space enclosed within them. In other words, the triangle refers to just the boundary or outline, while the triangular region includes the boundary plus the entire interior space.

Exam Tip: Keep clear: triangle = boundary only (three sides); triangular region = boundary + interior (shaded area included).

Question 4. In fig 12.11, D is a point on side BC of a ΔABC. AD is joined. Name all the triangles that you can observe in the figure. How many are they?
Answer: The figure shows three distinct triangles: ΔADC, ΔADB and ΔABC. Therefore, a total of three triangles can be identified in the figure.

Exam Tip: When a line is drawn inside a triangle, always count all possible combinations systematically - include the smaller triangles formed and the original large triangle.

Question 5. In fig 12.12, A, B, C and D are four points, and no three points are collinear. AC and BD intersect at O. There are eight triangles that you can observe. Name all the triangles.
Answer: The figure contains the following eight triangles: ΔODC, ΔODA, ΔOBC, ΔOAB, ΔADB, ΔACB, ΔDAC and ΔDBC. Thus, there are exactly eight triangles present in the figure.

Exam Tip: For intersecting line segments, count both the small triangles formed around the intersection point and the larger triangles created by combining regions.

Question 6. What is the difference between triangle and a triangular region?
Answer: A triangle is a closed polygon consisting of three straight line segments forming the boundary. A triangular region, on the other hand, refers to the area that lies inside those three sides of the triangle. The key distinction is that a triangle is just the perimeter or outline, while a triangular region encompasses both the boundary and all the space contained within it.

Exam Tip: This is a core distinction - always remember that triangle refers to the sides themselves, not the interior space.

Question 7. Explain the following terms:
(i) Triangle
(ii) Parts or elements of a triangle
(iii) Scalene triangle
(iv) Isosceles triangle
(v) Equilateral triangle
(vi) Acute triangle
(vii) Right triangle
(viii) Obtuse triangle
(ix) Interior of a triangle
(x) Exterior of a triangle
Answer:
(i) Triangle - A closed polygon comprised of three straight line segments as its sides.
(ii) Parts or elements of a triangle - A triangle has three sides, three angles and three vertices as its main components.
(iii) Scalene triangle - A triangle in which all three sides have different lengths.
(iv) Isosceles triangle - A triangle in which two of the sides have equal length.
(v) Equilateral triangle - A triangle in which all three sides have the same length.
(vi) Acute triangle - A triangle in which all three angles are smaller than 90°.
(vii) Right triangle - A triangle that has one angle measuring exactly 90°.
(viii) Obtuse triangle - A triangle in which one of the angles is greater than 90°.
(ix) Interior of a triangle - The region that is contained within the boundaries or sides of a triangle.
(x) Exterior of a triangle - The region that is located outside the boundaries or sides of a triangle.

Exam Tip: Organize your understanding by grouping triangles by sides (scalene, isosceles, equilateral) separately from grouping by angles (acute, right, obtuse) - this helps you answer classification questions quickly.

Question 8. In fig 12.13, the length (in cm) of each side has been indicted along the side. State for each triangle whether it is a scalene, isosceles or equilateral:
Answer:
(i) This is a scalene triangle, since each of the three sides has a different length.
(ii) This is an equilateral triangle, because all the sides share the same length of 5 cm.
(iii) This is an isosceles triangle, since two of the sides have equal length (5.6 cm).
(iv) This is an isosceles triangle, since two of the sides have equal length (6.2 cm).
(v) This is a scalene triangle, since each of the three sides has a different length.
(vi) This is an acute angle triangle, since all angles measure less than 90°.

Exam Tip: Carefully check the side or angle measurements provided - compare them to determine whether sides are equal or angles are acute/obtuse/right.

Question 9. In fig 12.14, there are five triangles. The measures of some of their angles have been indicated. State for each triangle whether it is acute, right or obtuse.
Answer:
(i) This is an obtuse angled triangle, since one angle (120°) is greater than 90° and less than 180°.
(ii) This is a right angle triangle, since it contains a 90° angle.
(iii) This is an acute angle triangle, since all angles are smaller than 90°.
(iv) This is an obtuse angled triangle, since one angle (110°) is greater than 90° and less than 180°.

Exam Tip: When classifying by angles, identify just one angle - if it's 90°, it's right; if greater than 90°, it's obtuse; if all are less than 90°, it's acute.

Question 10. Fill in the blanks with the correct word/symbol to make it a true statement:
(i) A triangle has ________.
(ii) A triangle has ________.
(iii) A triangle has ________.
(iv) A triangle has ________.
(Angles and sides are part of a triangle. So, three angles and three sides make six parts.)
(v) A triangle whose no two sides are equal is known as ________.
(A triangle whose lengths of all sides are different is called scalene triangle).
(vi) A triangle whose two sides are equal is known as ________.
(A triangle whose lengths of two sides are equal is called an equilateral triangle).
(vii) A triangle whose one angle is a right angle is known as ________.
(A triangle whose one angle is 90° is called a right angle triangle).
(viii) A triangle whose all angles are less than 90° is known as ________.
(A triangle whose all angle are less than 90° is known as Acute triangle).
(x) A triangle whose one side angle is more than 90° is known as ________.
(A triangle whose one angle is more than 90° is called Obtuse triangle).
Answer:
(i) A triangle has three sides.
(ii) A triangle has three vertices.
(iii) A triangle has three angles.
(iv) A triangle has six parts.
(v) A triangle whose no two sides are equal is known as Scalene triangle.
(vi) A triangle whose two sides are equal is known as Equilateral triangle.
(vii) A triangle whose one angle is a right angle is known as Right angled triangle.
(viii) A triangle whose all angles are less than 90° is known as an Acute triangle.
(x) A triangle whose one side angle is more than 90° is known as an Obtuse triangle.

Exam Tip: These are foundational definitions - memorize them precisely as they form the basis for all triangle classification and problem-solving.

Question 11. In each of the following, state if the statement is true or false:
Answer:
(i) True
(ii) False; a triangle consists of three vertices only.
(iii) False; three line segments joined by three non-collinear points can only form a triangle.
(iv) False; it lies on the triangle.
(v) True
(vi) False; the vertices of a triangle are three non-collinear points.
(vii) True
(viii) False; it can also be an isosceles triangle.
(ix) False; it can be an obtuse triangle.

Exam Tip: For true/false questions, always identify which part makes a statement false and provide a brief reason - this shows complete understanding.

Exercise 12.2

Question 1. Total number of parts of a triangle is
Answer: Six: Three sides and three angles

Exam Tip: A triangle always has exactly six parts - students sometimes forget to count angles, remembering only the three sides.

Question 2. A perpendicular drawn from a vertex to the opposite side of a triangle is known as
Answer: An Altitude: An altitude is defined as the perpendicular line drawn from a vertex to the opposite side of a triangle.

Exam Tip: The altitude must be perpendicular - a line from a vertex that is not perpendicular is not an altitude, even if it reaches the opposite side.

Question 3. A triangle
Answer: has three altitudes

Exam Tip: Since a triangle has three vertices, it must have exactly three altitudes - one from each vertex.

Question 4. Line segment joining the vertices to the mid - points of the opposite side of a triangle is known as
Answer: Medians: A median is defined as the line segment joining the vertex to the midpoint of the opposite side of a triangle.

Exam Tip: Do not confuse median with altitude - a median goes to the midpoint of the opposite side, while an altitude is perpendicular to the opposite side.

Question 5. A triangle whose no two sides are equal is called
Answer: A scalene triangle: A Scalene triangle is defined as the triangle in which no sides are equal.

Exam Tip: Scalene comes from the word meaning "unequal" - this helps remember that all three sides must be different lengths.

Question 6. A triangle whose two sides are equal is known as
Answer: An isosceles triangle: An isosceles triangle is a triangle that has two equal sides.

Exam Tip: Isosceles means "equal legs" - so always look for two equal sides in an isosceles triangle.

Question 7. A triangle whose two sides are equal is called
Answer: An equilateral triangle is defined as a triangle whose all sides are equal.

Exam Tip: Equilateral means "all sides equal" - this is different from isosceles which requires only two equal sides.

Question 8. The sum of the lengths of side of a triangle is known as its
Answer: Perimeter: Perimeter is defined as the sum of the length of all the sides of a triangle

Exam Tip: Perimeter is always measured in linear units (cm, m, etc.) - not in square units.

Question 9. A triangle having all sides of different length is known as its
Answer: Scalene triangle: A Scalene triangle is defined as a triangle having all sides of different length.

Exam Tip: This reinforces the definition of scalene - when you see "all different lengths," immediately think scalene triangle.

Question 10. A triangle whose one angle is more than 90° is called
Answer: An obtuse triangle: An obtuse triangle is a triangle whose one angle is more than 90°

Exam Tip: A triangle can have at most one obtuse angle - if it had two or more, the angle sum would exceed 180°.