NCERT MCQ Solutions for Class 7 Maths Ganita Prakash Chapter 7 A Tale of Three Intersecting Lines

Using a parallel line through the opposite vertex or folding a triangular paper cut-out shows that the sum of the three interior angles of any triangle is always 180°. This is known as the angle sum property of triangles and holds for all triangle types.

The triangle inequality states that for a triangle to exist, the sum of any two side lengths must be greater than the third side. If this rule is not satisfied, the lines cannot enclose a triangle and the figure cannot be constructed.

Equilateral triangles have all three sides of equal length. As a result, each of the interior angles measures 60°. These triangles are the most symmetric and can be efficiently constructed using a compass and arcs of equal radius from each vertex.

When all side lengths are known, a compass helps in accurately marking arcs from two fixed points. The point where the arcs intersect gives the third vertex of the triangle. This ensures the constructed sides match the required lengths without trial and error.

If the sum of the two shorter sides equals the third side, the lines lie on a straight line and cannot form a closed shape. In such cases, the triangle degenerates into a straight line and does not satisfy the triangle inequality.

Using the angle sum property of triangles, we subtract the known angles from 180°:
∠C = 180° – (∠A + ∠B) = 180° – (50° + 60°) = 70°. This helps in calculating the third angle when two angles are given.

Equilateral triangles have all sides equal. Since the angles are formed by equal sides, all interior angles are also equal. Each angle measures 60°, making this triangle both angle- and side-symmetric, the most balanced type of triangle.

A triangle can only be formed when the sum of any two side lengths is greater than the third side. This ensures the lines can enclose a space and meet at a common point to form a closed triangle.

An altitude is a line segment from a vertex of a triangle that meets the opposite side at a right angle (90°). It helps in calculating area and is essential for classifying triangle types in terms of height and base.

A scalene triangle has no sides equal and all angles different. It is the least symmetric among triangle types and does not have lines of symmetry. This classification helps in identifying triangle properties and internal measurements.

A triangle has three vertices which are the corner points where two sides meet. These points define the triangle’s shape and are labeled using capital letters like A, B and C when naming a triangle such as ∆ABC.

A triangle is a closed figure made of three intersecting lines. If all vertices lie on a straight line, they form a single straight segment and not a triangle. The figure fails to be enclosed and has no internal area.

An exterior angle of a triangle is formed when a side is extended. Its measure equals the sum of the two interior angles not adjacent to it. This property provides a useful way to calculate unknown interior angles.

An obtuse triangle has one angle greater than 90°. The other two angles are always acute to ensure the total sum remains 180°. This classification is based on the angle properties and not on side lengths.

If ∠B = ∠C and ∠A = 50°, then the remaining 130° must be equally divided between ∠B and ∠C. So each measures 65°. This makes the triangle isosceles since two angles and their opposite sides are equal.

A triangle with all angles less than 90° is called acute-angled. Since 75°, 45° and 60° are all acute, the triangle fits this category. This classification is based solely on angle measurement and not on side lengths.

If the sum of interior angles is less than 180°, the lines cannot meet to form a closed triangle. This violates the fundamental angle sum property of triangles which must always total 180° for any triangle to exist.

To construct a triangle given two angles and a side, we start by drawing the base. Then we use a protractor to mark the angles at the base’s ends. The intersection of the two rays completes the triangle.

This set of sides fails the triangle inequality. The longest side (30 cm) is not smaller than the sum of the other two (5 + 10 = 15 cm). Hence, these sides cannot enclose a triangle and no triangle can be formed.

For circles drawn from endpoints of the base to intersect and form a triangle, the sum of the radii (the other two side lengths) must be greater than the base length. This condition matches the triangle inequality and confirms triangle formation.