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Class 7 Maths Ganita Prakash Chapter 7 A Tale of Three Intersecting Lines Practice Paper 21 (CBSE) has 20 questions worth 40 marks in Sections A to E, and every question comes with a clear worked solution. It tests angles in a given ratio such as 1 : 2 : 3, why the construction arcs meet at two points, where the altitudes of different triangles lie, right-angled, obtuse-angled and isosceles triangles, the exterior angle property, and constructing a triangle with sides 6 cm, 7 cm and 9 cm.
Section A: Multiple Choice Questions (1 Mark Each)
(a) Triangle can be formed
(b) Triangle cannot be formed
(c) Triangle is right-angled
(d) Triangle is equilateral
Show Answer & Explanation
Answer: (b) Triangle cannot be formed
Explanation:
1. Longest stick = 15 cm. Sum of the other two = 6 + 8 = 14 cm.
2. 14 cm is less than 15 cm, so the triangle inequality is not satisfied.
3. Hence a triangle cannot be formed.
Teacher's Note:
1. The sum of the two shorter sides must be greater than the longest side.
2. 6, 8, 10 would make a right-angled triangle, but 6, 8, 15 makes no triangle at all.
(a) No triangle is possible
(b) Exactly one point can be the third vertex
(c) Two mirror-image triangles are possible
(d) Only an equilateral triangle is possible
Show Answer & Explanation
Answer: (c) Two mirror-image triangles are possible
Explanation:
1. The two crossing points lie one above and one below the base.
2. Each crossing point is at the required distances from both ends of the base, so either one can be used as the third vertex.
3. This gives two triangles that are mirror images of each other across the base; they have exactly the same shape and size.
Teacher's Note:
1. The two triangles are copies of each other, so the side lengths fix the triangle's shape and size.
2. Arcs crossing at two points also tells us that the triangle inequality is satisfied.
(a) 30°
(b) 45°
(c) 60°
(d) 90°
Show Answer & Explanation
Answer: (d) 90°
Explanation:
1. Let the angles be x, 2x and 3x.
2. By the angle sum property, x + 2x + 3x = 180°, so 6x = 180° and x = 30°.
3. The angles are 30°, 60° and 90°, so the largest angle = 3x = 90°.
Teacher's Note:
1. Total parts = 1 + 2 + 3 = 6, so one part = 180° ÷ 6 = 30°.
2. This triangle is right-angled.
(a) Obtuse-angled triangle
(b) Right-angled triangle
(c) Acute-angled triangle
(d) Isosceles obtuse triangle
Show Answer & Explanation
Answer: (c) Acute-angled triangle
Explanation:
1. In an obtuse-angled triangle (including an isosceles obtuse one), two altitudes fall outside the triangle.
2. In a right-angled triangle, two altitudes coincide with the sides.
3. Only in an acute-angled triangle do all three altitudes lie inside.
Teacher's Note:
1. Draw a quick sketch of each type to see where the perpendiculars fall.
2. For an obtuse triangle, you must extend the sides to draw two of the altitudes.
(a) Acute-angled
(b) Right-angled
(c) Obtuse-angled
(d) Equilateral
Show Answer & Explanation
Answer: (c) Obtuse-angled
Explanation:
1. 100° is greater than 90°, so it is an obtuse angle.
2. A triangle with one obtuse angle is called obtuse-angled.
3. The other two angles together are only 80°, so both are acute.
Teacher's Note:
1. A triangle can have at most one obtuse angle.
2. An equilateral triangle has every angle 60°, so it can never have a 100° angle.
(a) Acute-angled scalene
(b) Right-angled scalene
(c) Obtuse-angled isosceles
(d) Right-angled isosceles
Show Answer & Explanation
Answer: (b) Right-angled scalene
Explanation:
1. All three sides are different, so the triangle is scalene (this rules out options c and d).
2. Squares: 8² + 15² = 64 + 225 = 289, and 17² = 289.
3. The squares of the two shorter sides add up exactly to the square of the longest side, so the angle between the 8 cm and 15 cm sides is 90°.
4. Hence the triangle is right-angled scalene.
Teacher's Note:
1. 8, 15, 17 is another well-known right-triangle set, like 3, 4, 5 and 5, 12, 13.
2. Construct it to scale and measure: the angle opposite the 17 cm side is a right angle.
(a) 75°
(b) 80°
(c) 85°
(d) 90°
Show Answer & Explanation
Answer: (c) 85°
Explanation:
1. Exterior angle = sum of the two interior opposite angles.
2. 60° + other angle = 145°.
3. Other angle = 145° − 60° = 85°.
Teacher's Note:
1. Check: the adjacent interior angle is 180° − 145° = 35°, and 60° + 85° + 35° = 180°.
2. Do not use 180° − 145° as the answer; that is the angle next to the exterior angle.
(a) All angles are equal
(b) All sides are unequal
(c) Two sides are equal
(d) One angle is always 90°
Show Answer & Explanation
Answer: (c) Two sides are equal
Explanation:
1. By definition, an isosceles triangle has (at least) two equal sides.
2. Its angles need not all be equal, and it need not have a right angle.
3. So the statement true for every isosceles triangle is two sides are equal.
Teacher's Note:
1. The angles opposite the two equal sides are also equal.
2. An isosceles triangle can be acute, right or obtuse.
Assertion (A): If the sum of two sides of a triangle is less than the third side, no triangle can be formed.
Reason (R): The circles drawn during construction do not intersect.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation
Answer: (a) Both A and R are true, and R is the correct explanation of A.
Explanation:
1. Assertion: if two sides together are shorter than the third, they cannot reach each other over it. A is true.
2. Reason: draw circles from the ends of the third side with the two shorter lengths as radii; they are too small to meet, so they do not intersect. R is true.
3. With no meeting point there is no position for the third vertex, so no triangle forms. R explains A.
4. Answer: (a).
Teacher's Note:
1. Draw it once with lengths like 3 cm, 4 cm and 9 cm to see the gap between the circles.
2. The construction is a picture of the triangle inequality.
Assertion (A): The exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Reason (R): The sum of interior angles of a triangle is 180°.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation
Answer: (a) Both A and R are true, and R is the correct explanation of A.
Explanation:
1. Both statements are true properties of triangles.
2. Let the angles be ∠A, ∠B, ∠C and the exterior angle at C be ∠x. Then ∠x + ∠C = 180° (linear pair).
3. Also ∠A + ∠B + ∠C = 180° (angle sum property). So ∠x = 180° − ∠C = ∠A + ∠B.
4. The exterior angle property is proved using R, so R correctly explains A. Answer: (a).
Teacher's Note:
1. The proof needs two facts: the linear pair and the angle sum.
2. Example: angles 50° and 60° give exterior angle 110° at the third vertex.
Section B: Short Answer Questions (2 Marks Each)
Question 11. State one condition necessary to construct a triangle using three given sides.
Answer:
The sum of any two of the given sides must be greater than the third side (triangle inequality).
It is enough to check that the two shorter sides together are longer than the longest side.
Example: 4 cm, 5 cm, 8 cm works since 4 + 5 = 9 > 8; but 3 cm, 4 cm, 8 cm does not since 3 + 4 = 7 < 8.
In simple words: The two smaller sides together must be longer than the biggest side.
Teacher's Note:
1. Use the word 'greater'; an equal sum gives no triangle.
2. Giving a short example earns full marks.
Question 12. Find the third angle if two angles of a triangle are 52° and 68°.
Answer:
Angle sum property: the three angles add up to 180°.
52° + 68° = 120°.
Third angle = 180° − 120° = 60°.
In simple words: 180° minus (52° + 68°) gives the missing angle, 60°.
Teacher's Note:
1. Check: 52° + 68° + 60° = 180°.
2. All angles are below 90°, so it is an acute-angled triangle.
Question 13. Name the triangle in which one angle is greater than 90°.
Answer:
An angle greater than 90° (and less than 180°) is an obtuse angle.
A triangle with one obtuse angle is called an obtuse-angled triangle.
In simple words: A triangle with a corner wider than a right angle is obtuse-angled.
Teacher's Note:
1. Only one angle of a triangle can be obtuse.
2. Example: a triangle with angles 110°, 40° and 30°.
Question 14. Why is a right-angled triangle always scalene or isosceles but never equilateral?
Answer:
Every angle of an equilateral triangle is 60° (180° ÷ 3).
A right-angled triangle must have one angle of 90°, which cannot be 60°.
Also, the side opposite the 90° angle is the longest side, so all three sides can never be equal.
Hence a right-angled triangle can be isosceles (angles 90°, 45°, 45°) or scalene, but never equilateral.
In simple words: An equilateral triangle has only 60° angles, so it can never contain a 90° angle.
Teacher's Note:
1. Give an example of each type: 90°, 45°, 45° (isosceles) and 90°, 30°, 60° (scalene).
2. The reason must mention the 60° angles of an equilateral triangle.
Section C: Short Answer Questions (3 Marks Each)
Question 15. Verify whether a triangle is possible with sides 7 cm, 9 cm and 17 cm.
Answer:
Longest side = 17 cm.
Sum of the other two sides = 7 + 9 = 16 cm.
16 cm is less than 17 cm, so the triangle inequality is not satisfied.
Hence no triangle is possible with these sides.
In simple words: 7 cm and 9 cm together are still 1 cm short of 17 cm, so they cannot meet.
Teacher's Note:
1. One failed check is enough to say 'not possible'.
2. Arcs of 7 cm and 9 cm drawn from the ends of a 17 cm base would not meet.
Question 16. Find the exterior angle of a triangle if the interior opposite angles are 49° and 51°.
Answer:
Exterior angle = sum of the two interior opposite angles.
Exterior angle = 49° + 51° = 100°.
Check: the third angle = 180° − 100° = 80°, and 49° + 51° + 80° = 180°.
In simple words: Add the two opposite inside angles: 49° + 51° = 100°.
Teacher's Note:
1. The exterior angle is always greater than each interior opposite angle.
2. Exterior angle + its adjacent interior angle = 180°.
Question 17. Classify a triangle with angles 45°, 45° and 90°.
Answer:
By angles: one angle is 90°, so it is right-angled.
By sides: two angles are equal (45° each), so the sides opposite them are equal; it is isosceles.
Hence it is a right-angled isosceles triangle.
In simple words: It has a 90° corner and two equal angles, so it is a right-angled isosceles triangle.
Teacher's Note:
1. Check: 45° + 45° + 90° = 180°.
2. This is the shape you get by cutting a square along its diagonal.
Section D: Long Answer Question (5 Marks)
Question 18. Construct a triangle with sides 6 cm, 7 cm and 9 cm. Describe the steps clearly.
Answer:
First check: 6 + 7 = 13 > 9, so the triangle can be constructed.
Steps of construction:
Step 1: Draw a line segment AB = 7 cm.
Step 2: With A as centre and radius 6 cm, draw an arc above AB.
Step 3: With B as centre and radius 9 cm, draw another arc cutting the first arc at C.
Step 4: Join AC and BC.
△ABC is the required triangle with AB = 7 cm, AC = 6 cm and BC = 9 cm.
In simple words: Draw the 7 cm base, cut arcs of 6 cm and 9 cm from its ends, and join the crossing point to both ends.
Teacher's Note:
1. Measure the radius on the ruler carefully before drawing each arc.
2. The angle at A comes out a little less than 90°; you can check with a protractor.
3. Label the vertices and side lengths on the finished figure.
Section E: Case Study Based Questions (4 Marks Each)
Question 19. Case Study 1: Stage Support Structure
For a school annual function, a triangular metal support is designed for the stage backdrop. The rods available are of lengths 8 m, 11 m and 13 m. Before installation, students are asked to verify whether these rods can form a triangle using mathematical reasoning to ensure safety and stability.
Answer the following questions:
(a) Identify the longest rod.
(b) Find the sum of the other two rods.
(c) Can a triangle be formed?
(d) Name the mathematical rule used.
Answer:
(a) The rods are 8 m, 11 m and 13 m, so the longest rod is 13 m.
(b) Sum of the other two rods = 8 m + 11 m = 19 m.
(c) 19 m is greater than 13 m, so yes, a triangle can be formed.
(d) The triangle inequality: the sum of any two sides of a triangle is greater than the third side.
In simple words: 8 m and 11 m together (19 m) are longer than 13 m, so the rods can form a triangle.
Teacher's Note:
1. Testing the longest rod against the other two is the quickest check.
2. All three rods are different, so the frame would be a scalene triangle.
Question 20. Case Study 2: Architectural Angle Design
An architect designs a triangular window frame where two angles measure 54° and 66°. To complete the design accurately, the third angle is calculated using a basic geometric property. This avoids measurement errors and ensures symmetry in construction.
Answer the following questions:
(a) Name the property used.
(b) Find the third angle.
(c) Classify the triangle by angles.
(d) State whether the triangle is scalene or isosceles.
Answer:
(a) The angle sum property of a triangle: the three angles add up to 180°.
(b) Third angle = 180° − (54° + 66°) = 180° − 120° = 60°.
(c) The angles 54°, 66° and 60° are all less than 90°, so it is an acute-angled triangle.
(d) No two angles are equal, so no two sides are equal: it is a scalene triangle.
In simple words: The missing angle is 60°; all three angles are sharp and different, so the frame is an acute-angled scalene triangle.
Teacher's Note:
1. Check: 54° + 66° + 60° = 180°.
2. Three different angles always mean three different sides.
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