CBSE Important Questions Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines

CBSE Important Questions Class 7 Maths Chapter 7 – A Tale of Three Intersecting Lines

Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines is a chapter of the new NCERT (National Council of Educational Research and Training) textbook Ganita Prakash. Three lines that cross each other make a triangle. Students learn the angle sum property (the angles of a triangle add up to 180°), the triangle inequality (the sum of any two sides is greater than the third side), the types of triangles by their angles, exterior angles, and how to construct a triangle with a ruler and compass.

These CBSE important questions for Class 7 Maths Chapter 7 include checking whether a triangle can be drawn, identifying triangle types from figures, finding the range of the third side, counting possible triangles, and arranging construction steps in the right order. Every question has a clear step-by-step answer in simple English, so students can practise on their own, parents can check the working, and teachers can use them in class. A free printable PDF of these important questions is also available on this page.

CBSE Important Questions Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines

A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct triangles having the following sidelengths (all the units are in cm):
(a) 4, 4, 6
(b) 3, 4, 5
(c) 1, 5, 5
(d) 4, 6, 8
(e) 3.5, 3.5, 3.5
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Here is a triangle in which we know ∠B = ∠C and ∠A = 50°. Can you find ∠B and ∠C?
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct a right-angled triangle ∆ABC with ∠B = 90°, AC = 5 cm. How many different triangles exist with these measurements?
[Hint: Note that the other measurements can take any values. Take AC as the base. What values can ∠A and ∠C take so that the other angle is 90°?]
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct a triangle TRY with RY = 4 cm, TR = 7 cm, ∠R = 140°. Construct an altitude from T to RY.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct a triangle ABC with BC = 5 cm, AB = 6 cm, CA = 5 cm. Construct an altitude from A to BC.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
What could an acute-angled triangle be? Can we define it as a triangle with one acute angle? Why not?
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Cut out a paper triangle. Fix one of the sides as the base. Fold it in such a way that the resulting crease is an altitude from the top vertex to the base. Justify why the crease formed should be perpendicular to the base.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
​Find the exterior angle for different measures of ∠A and ∠B. 
Do you see any relation between the exterior angle and these two angles?
[Hint: From the angle sum property, we have ∠A + ∠B + ∠ACB = 180°.]We also have ∠ACD + ∠ACB = 180°, since they form a straight angle.​
What does this show?
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
The angle formed between the extension of a side of a triangle and the other side is called an exterior angle of the triangle. In this figure, ∠ACD is an exterior angle.
​Find ∠ACD, if ∠A = 50°, and ∠B = 60°.​
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Can you construct a triangle all of whose angles are equal to 70°? If two of the angles are 70°, what would the third angle be? If all the angles in a triangle have to be equal, then what must its measure be? Explore and find out.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any side length? Justify your answer.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Find the third angle of a triangle (using a parallel line) when two of the angles are:
(a) 36°, 72°
(b) 150°, 15°
(c) 90°, 30°
(d) 75°, 45°
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Let us take two angles, say 60° and 70°, whose sum is less than 180°. Let the included side be 5 cm.
What could the measure of the third angle be? Does this measure change if the base length is changed to some other value, say 7 cm? Construct and find out.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Determine which of the following pairs can be the angles of a triangle and which cannot:
(a) 35°, 150°
(b) 70°, 30°
(c)90°, 85°
(d) 50°, 150°
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category:
(a) 30°
(b) 70°
(c) 54°
(d) 144°
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct triangles for the following measurements:
(a) 75°, 5 cm, 75°
(b) 25°, 3 cm, 60°
(c) 120°, 6 cm, 30°
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Construct triangles for the following measurements, where the angle is included between the sides:
(a) 3 cm, 75°, 7 cm
(b) 6 cm, 25°, 3 cm
(c) 3 cm, 120°, 8 cm
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
For each of the following, give at least 5 possible values for the third length so there exists a triangle having these as sidelengths (decimal values could also be chosen):
(а) 1, 100
(b) 5, 5
(c) 3, 7
....continued one more part
See if you can describe all possible lengths of the third side in each case, so that a triangle exists with those side lengths.
For example, in case (a), all numbers strictly between 99 and 101 would be possible.
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
For each of the following, give at least 5 possible values for the third length so there exists a triangle having these as sidelengths (decimal values could also be chosen):
(а) 1, 100
(b) 5, 5
(c) 3, 7
A Tale of Three Intersecting Lines (New NCERT)
Medium
Q.
Through construction, explore if it is possible to construct an equilateral triangle that is (i) right-angled, (ii) obtuse-angled.
Also construct an isosceles triangle that is (i) right-angled, (ii) obtuse-angled.

CBSE Important Questions Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines

Q1. Is it possible to draw a triangle having the following sides or angles? Support your answer with a reason.
(a) 120°, 60°, 30°
(b) 50°, 50°, 80°
(c) 10 cm, 4 cm, 5 cm
(d) 12 cm, 6 cm, 6 cm
(e) 30°, 60°, 90°

Answer: (a) No. The angles of a triangle add up to 180°, but 120° + 60° + 30° = 210°, which is more than 180°. So this triangle cannot be drawn.
(b) Yes. 50° + 50° + 80° = 180°, so a triangle with these angles can be drawn.
(c) No. In a triangle, the sum of any two sides must be greater than the third side. Here 4 cm + 5 cm = 9 cm, which is less than 10 cm. So this triangle cannot be drawn.
(d) No. 6 cm + 6 cm = 12 cm, which is equal to the third side (12 cm), not greater. So this triangle cannot be drawn.
(e) Yes. 30° + 60° + 90° = 180°, so a triangle with these angles can be drawn.

Q2. Identify the following triangles on the basis of their angles.

Three triangles: ABC, LMN and PQR

Answer: (a) ΔABC – all three angles are less than 90°, so it is an acute-angled triangle.
(b) ΔLMN – it has a right angle (90°) at N, so it is a right-angled triangle.
(c) ΔPQR – the angle at Q is more than 90°, so it is an obtuse-angled triangle.

Q3. In the following figure, what is the value of x + y + z?
(a) 90°
(b) 180°
(c) 270°
(d) 360°

Triangle ABC with exterior angles x, y and z

Answer: (d) 360°
x, y and z are the exterior angles of the triangle, one at each vertex. Each exterior angle is equal to the sum of the two opposite interior angles. Adding all three, each interior angle is counted twice, so x + y + z = 2 × 180° = 360°.

Q4. Two sides of a triangle are of lengths 7.2 cm and 4.2 cm. The length of the third side of the triangle cannot be ___________.
(a) 10.8 cm
(b) 3.2 cm
(c) 2.9 cm
(d) 11.0 cm

Answer: (c) 2.9 cm
The third side must be more than the difference of the two sides (7.2 − 4.2 = 3 cm) and less than their sum (7.2 + 4.2 = 11.4 cm). So it must lie between 3 cm and 11.4 cm. 10.8 cm, 3.2 cm and 11.0 cm all lie in this range, but 2.9 cm is less than 3 cm. So the third side cannot be 2.9 cm.

Q5. The perimeter of a triangle is 15 cm. If two of its sides are 5 cm and 7 cm, find its third side and verify the triangle inequality.

Answer: Perimeter = first side + second side + third side
15 cm = (5 cm + 7 cm) + third side
Third side = 15 cm − 12 cm = 3 cm
Verifying the triangle inequality (the sum of any two sides must be greater than the third side):
3 + 5 = 8 > 7 ✓, 5 + 7 = 12 > 3 ✓, 3 + 7 = 10 > 5 ✓
All three conditions hold, so a triangle with sides 3 cm, 5 cm and 7 cm is possible.

Triangle with sides 3 cm, 5 cm and 7 cm

Q6. Two sides of a triangle are 7 cm and 11 cm. If the third side is an integer, how many possible triangles can be formed?
(a) 10
(b) 11
(c) 12
(d) 13

Answer: (d) 13
The third side must be greater than 11 − 7 = 4 cm and less than 11 + 7 = 18 cm. The integers between 4 and 18 (not including 4 and 18) are 5, 6, 7, …, 17. That is 13 values, so 13 triangles are possible.

Q7. Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle cannot be
(a) 3.4 cm
(b) 3.6 cm
(c) 3.8 cm
(d) 4.1 cm

Answer: (a) 3.4 cm
The third side must lie between 5 − 1.5 = 3.5 cm and 5 + 1.5 = 6.5 cm. 3.6 cm, 3.8 cm and 4.1 cm are in this range, but 3.4 cm is less than 3.5 cm. So the third side cannot be 3.4 cm.

Q8. What is the correct order of steps to draw a triangle PQR with QR = 5 cm, ∠PQR = 30° and ∠PRQ = 105°?
(i) Let the two rays drawn in steps (ii) and (iii) intersect at point P.
(ii) Draw the base QR = 5 cm.
(iii) At Q, construct an angle of 30°.
(iv) At R, construct an angle of 105°.
(a) (iv), (iii), (ii), (i)
(b) (iii), (ii), (iv), (i)
(c) (ii), (iii), (iv), (i)
(d) (i), (iii), (iv), (ii)

Answer: (c) (ii), (iii), (iv), (i)
First draw the base QR = 5 cm. Then construct 30° at Q and 105° at R. The two rays meet at P, which completes the triangle PQR.

Q9. What is the correct order of steps to draw a triangle, when the measure of two of its angles and the length of the side between them is given as BC = 6 cm, ∠ABC = 75° and ∠ACB = 45°?
(a) (v), (iv), (i), (iii), (ii)
(b) (iii), (ii), (i), (v), (iv)
(c) (iii), (ii), (iv), (i), (v)
(d) (v), (iii), (ii), (iv), (i)

Steps (i) to (v) of constructing triangle ABC

Answer: (d) (v), (iii), (ii), (iv), (i)
Start with a rough sketch (v). Draw BC = 6 cm (iii). Construct 75° at B (ii). Construct 45° at C (iv). The two rays meet at A; join A to B and C to complete the triangle (i).

Q10. Sudha is drawing a triangle. The shortest side of the triangle is 2/3 of its longest side and the third side is 2 cm longer than the shortest side. The sum of the lengths of all three sides is at least 72 cm. What is the minimum length of the longest side?
(a) 20
(b) 22
(c) 30
(d) 42

Answer: (c) 30
Let the longest side be L cm. Then the shortest side = 2L/3 and the third side = 2L/3 + 2.
Perimeter = L + 2L/3 + 2L/3 + 2 = 7L/3 + 2
7L/3 + 2 ≥ 72, so 7L/3 ≥ 70 and L ≥ 30.
The minimum length of the longest side is 30 cm (sides 30 cm, 20 cm and 22 cm, perimeter 72 cm).

Q11. In which of the following conditions can triangle PQR be constructed?
(a) PQ = 6 cm, QR = 7 cm and RP = 4 cm
(b) PQ = 10 cm, QR = 1 cm and RP = 8 cm
(c) PQ = 3 cm, QR = 6 cm and RP = 2 cm
(d) PQ = 10 cm, QR = 3 cm and RP = 4 cm

Answer: (a) PQ = 6 cm, QR = 7 cm and RP = 4 cm
A triangle can be made only if the sum of any two sides is greater than the third side. In (a), 6 + 4 = 10 > 7, 6 + 7 > 4 and 7 + 4 > 6, so it works. In (b) 1 + 8 = 9 < 10, in (c) 3 + 2 = 5 < 6, and in (d) 3 + 4 = 7 < 10, so these triangles cannot be constructed.

Q12. If two angles and one side are given in a triangle, then is it possible to construct a triangle?
(a) Never
(b) Always
(c) Data is insufficient
(d) Sometimes

Answer: (b) Always
To construct a triangle we need three of its parts. Two angles and one side are three parts, and the third angle can be found because the angles of a triangle add up to 180°. So the triangle can be constructed (the two given angles must add up to less than 180°).

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