Class 7 Maths Ganita Prakash Part 1 Chapter 7 MCQs – A Tale of Three Intersecting Lines

A triangle needs three vertices that do not all lie on one straight line.
Here, joining the points only traces parts of the same line. No enclosed triangular region is formed, so this is not a triangle.

The third vertex C must be 4 cm from A and 4 cm from B.
An arc of radius 4 cm from each centre meets at such a point.
Thus, AB = AC = BC = 4 cm, giving an equilateral triangle.

The arc centred at A must contain points 5 cm from A, while the arc centred at B must contain points 6 cm from B.
Their intersection satisfies both distances and locates the required third vertex C.

OA and OB both join the centre to points on the circle.
Therefore, OA = OB, since all radii of a circle have equal length.
Triangle OAB has two equal sides and is isosceles.

The two shorter lengths add to 4 + 4 = 8 cm, which is greater than 6 cm, so a triangle exists.
Exactly two sides are equal. Therefore, it is an isosceles triangle.

For 3 cm, 4 cm and 6 cm:
3 + 4 = 7 > 6.
The sum of the two shorter sides is greater than the longest side, so a triangle exists. Each other set fails this comparison.

3 + 6 = 9.
For a triangle, the sum must be strictly greater than the third side. Equality would place the three vertices on one straight line.
Therefore, these lengths do not form a triangle.

The longest side c is the one that might be too long.
The other comparisons already hold because all lengths are positive and c is at least as large as a and b.
Thus, the necessary check is a + b > c.

The triangle inequalities give:
1 + x > 100, so x > 99.
1 + 100 > x, so x < 101. Together, 99 < x < 101. The endpoints are excluded because equality produces a straight-line arrangement.

The third side x must satisfy:
7 − 3 < x < 7 + 3 4 < x < 10. The whole-number possibilities are 5, 6, 7, 8 and 9. Therefore, there are five possible lengths.

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.

The radii satisfy 4 + 5 > 8 and 5 − 4 < 8. The circles intersect at two points, one on each side of AB. Either point gives a triangle with side lengths 4 cm, 5 cm and 8 cm.

Sum of the radii = 3 + 6 = 9 cm.
This equals the distance between the centres, so the circles just touch.
The centres and touching point are collinear. They do not form an enclosed triangle.

AC must form the 45° angle with AB, so C lies on the new ray.
The required length is AC = 4 cm.
Marking C 4 cm from A on that ray satisfies both the side and included-angle conditions.

The three angles of a triangle add up to 180°.
∠C = 180° − (45° + 80°)
= 180° − 125° = 55°.
The included side AB determines the size of the triangle, not this angle sum.

90° + 85° = 175°, leaving 5° for the third angle.
This is positive, so a triangle is possible.
The other pairs total 180° or more and leave no positive third angle.

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.

The third angle is 180° − 40° − B = 140° − B.
It must be positive, so B < 140°. B must also be positive. Therefore, the complete condition is 0° < B < 140°.

Third angle = 180° − (60° + 70°)
= 180° − 130° = 50°.
Changing the included side changes the size of the triangle. The two fixed angles still leave exactly 50°

Sum of the known angles = 36° + 72° = 108°.
Third angle = 180° − 108° = 72°.
The three angles are therefore 36°, 72° and 72°, and their total is 180°.

Let each equal angle be x.
x + x + x = 180°
3x = 180°
x = 60°.
Three angles of 70° would total 210°, so equal angles in a triangle cannot each measure 70°.

∠B + ∠C = 180° − 50° = 130°.
The two angles are equal, so each is half of 130°.
∠B = ∠C = 130° ÷ 2 = 65°.
Check: 50° + 65° + 65° = 180°.

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.

AB and AC act as transversals of the parallel lines.
Alternate angles reproduce ∠B and ∠C beside ∠A on the straight line through A.
The three angles then make a straight angle, giving ∠A + ∠B + ∠C = 180°.

Interior angle at C = 180° − 50° − 60° = 70°.
∠ACB and ∠ACD form a linear pair.
Exterior angle ∠ACD = 180° − 70° = 110°.
This also equals the sum of the two opposite interior angles.

The exterior angle and its adjacent interior angle total 180°.
The three interior angles also total 180°.
Removing the same adjacent interior angle from both sums leaves the exterior angle equal to the other two interior angles together.

An altitude measures the perpendicular height from a vertex to its opposite side.
Therefore, the segment starts at A and meets the line containing BC at 90°.
The meeting point may lie on BC or on its extension.

Since ∠B = 90°, AB is perpendicular to BC.
AB already joins vertex A to the opposite side BC at B.
Therefore, AB is the altitude from A when BC is taken as the base.

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.

The definition of an altitude uses a perpendicular from a vertex to the line containing the opposite side.
In an obtuse triangle, that perpendicular may meet an extension. Its position outside the triangle does not stop it from being an altitude.

The lengths form a triangle because 3 + 4 > 5.
All three side lengths are different.
A triangle with three different side lengths is scalene. This classification uses side lengths rather than angle measures.

An acute angle is smaller than 90°. An acute-angled triangle has all three angles smaller than 90°.
Having only one acute angle is not enough, because right-angled and obtuse-angled triangles also contain acute angles.

The remaining angles must satisfy ∠A + ∠C = 90°.
They can be 30° and 60°, 45° and 45°, or many other positive pairs.
Keeping AC at 5 cm still allows these different angle choices, so infinitely many triangles are possible.

Unfolding faces turns a possible surface route into a path on a flat plane.
A straight segment is the shortest route between its endpoints on that plane.
Comparing valid unfolded routes helps find a short surface path without allowing the spider to travel through the box.

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.