A Tale of Three Intersecting Lines — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "A Tale of Three Intersecting Lines" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "A Tale of Three Intersecting Lines" — 8 important questions with detailed answers f…
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NCERT Solutions for Class 7 Mathematics — A Tale of Three Intersecting Lines. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
A Tale of Three Intersecting Lines — All 8 Questions Solved
Q1. Define a triangle and state the sum of its interior angles.
Step 1: Definition of a triangle:
A triangle is a closed figure formed by three line segments (sides) connecting three non-collinear points (vertices).
Step 2: Parts of a triangle:
- Three sides (line segments)
- Three vertices (corners/angles)
- Three interior angles
Step 3: Angle sum property of a triangle:
The sum of all interior angles of a triangle is always 180°.
If the angles are A, B, and C, then: A + B + C = 180°
Answer: A triangle is a closed figure with three sides and three vertices. The sum of its interior angles is always 180°.
Q2. In a triangle, two angles are 50° and 70°. Find the third angle.
Step 1: Use the angle sum property of a triangle:
Angle 1 + Angle 2 + Angle 3 = 180°
Step 2: Substitute known values:
50° + 70° + Angle 3 = 180°
120° + Angle 3 = 180°
Step 3: Solve for Angle 3:
Angle 3 = 180° - 120° = 60°
Step 4: Verify:
50° + 70° + 60° = 180° ✓
Answer: The third angle is 60°.
Q3. What is an exterior angle of a triangle? State the exterior angle theorem.
Step 1: Definition of exterior angle:
An exterior angle of a triangle is formed when one side of the triangle is extended beyond a vertex.
Step 2: Exterior angle theorem:
An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles (also called remote interior angles).
Step 3: Example:
In triangle ABC, if side BC is extended to point D, the exterior angle ACD equals angle A + angle B.
Angle ACD = Angle A + Angle B
Step 4: Relationship to angle sum:
Exterior angle + Adjacent interior angle = 180° (linear pair)
Answer: An exterior angle is formed when a side is extended. It equals the sum of the two remote interior angles. (Exterior angle = sum of non-adjacent interior angles)
Q4. The exterior angle of a triangle is 110°. If one of the remote interior angles is 60°, find the other remote interior angle.
Step 1: Apply the exterior angle theorem:
Exterior angle = Remote interior angle 1 + Remote interior angle 2
Step 2: Substitute known values:
110° = 60° + Remote interior angle 2
Step 3: Solve:
Remote interior angle 2 = 110° - 60° = 50°
Step 4: Verify:
Sum of remote angles = 60° + 50° = 110° = Exterior angle ✓
Answer: The other remote interior angle is 50°.
Q5. Classify triangles based on sides and angles.
Step 1: Classification based on SIDES:
(a) Equilateral triangle: All three sides are equal
All angles = 60° each
(b) Isosceles triangle: Two sides are equal
The angles opposite the equal sides are equal
(c) Scalene triangle: All three sides are unequal
All angles are unequal
Step 2: Classification based on ANGLES:
(a) Acute triangle: All three angles are less than 90°
All angles are acute
(b) Right triangle (Right-angled): One angle is exactly 90°
The other two angles are acute and sum to 90°
(c) Obtuse triangle: One angle is greater than 90°
The other two angles are acute
Answer: By sides: Equilateral (all equal), Isosceles (two equal), Scalene (all unequal). By angles: Acute (all < 90°), Right (one = 90°), Obtuse (one > 90°).
Q6. In triangle ABC, angle A = 35°, angle B = 65°. Classify the triangle by angles. Also find the exterior angle at C.
Step 1: Find angle C using angle sum property:
Angle A + Angle B + Angle C = 180°
35° + 65° + Angle C = 180°
100° + Angle C = 180°
Angle C = 80°
Step 2: Classify by angles:
All angles (35°, 65°, 80°) are less than 90°.
Therefore, this is an ACUTE triangle.
Step 3: Find exterior angle at C:
Exterior angle at C = Angle A + Angle B (by exterior angle theorem)
Exterior angle at C = 35° + 65° = 100°
Step 4: Verify:
Exterior angle + Interior angle C = 100° + 80° = 180° ✓ (linear pair)
Answer: The triangle is an acute triangle. The exterior angle at C is 100°.
Q7. The angles of a triangle are in the ratio 1:2:3. Find all three angles.
Step 1: Let the angles be x, 2x, and 3x (in ratio 1:2:3)
Step 2: Use angle sum property:
x + 2x + 3x = 180°
6x = 180°
Step 3: Solve for x:
x = 30°
Step 4: Find all three angles:
First angle = x = 30°
Second angle = 2x = 2(30°) = 60°
Third angle = 3x = 3(30°) = 90°
Step 5: Verify:
30° + 60° + 90° = 180° ✓
Ratio: 30:60:90 = 1:2:3 ✓
Answer: The three angles are 30°, 60°, and 90°. (This is a right triangle.)
Q8. An equilateral triangle has one side of length 5 cm. Find the perimeter and classify it.
Step 1: Definition of equilateral triangle:
All three sides are equal in length.
All three angles are equal (each 60°).
Step 2: Given information:
One side = 5 cm
Since all sides are equal: Each side = 5 cm
Step 3: Calculate perimeter:
Perimeter = Side 1 + Side 2 + Side 3
Perimeter = 5 + 5 + 5 = 15 cm
Step 4: Classification:
By sides: Equilateral triangle (all sides equal)
By angles: Acute triangle (all angles = 60°, which is less than 90°)
Answer: Perimeter = 15 cm. It is an equilateral triangle and also an acute triangle.
A Tale of Three Intersecting Lines — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. When two distinct lines intersect, how many points of intersection do they have?
- A. Zero
- B. One
- C. Two
- D. Infinitely many
Answer: B. One — Two distinct lines, if they intersect, can only cross at a single common point. If they had more than one common point, they would be the same line.
MCQ 2. What is the special name given to two lines that intersect to form a right angle (90 degrees)?
- A. Parallel lines
- B. Skew lines
- C. Perpendicular lines
- D. Transversal lines
Answer: C. Perpendicular lines — Lines that intersect at a 90-degree angle are specifically called perpendicular lines. This is a fundamental definition in geometry.
MCQ 3. If two lines never intersect, no matter how far they are extended, what are they called?
- A. Intersecting lines
- B. Perpendicular lines
- C. Parallel lines
- D. Concurrent lines
Answer: C. Parallel lines — Parallel lines are defined as lines in the same plane that never meet or intersect. They maintain a constant distance from each other.
MCQ 4. What is the sum of angles in a linear pair?
- A. 90 degrees
- B. 180 degrees
- C. 360 degrees
- D. 45 degrees
Answer: B. 180 degrees — A linear pair consists of two adjacent angles whose non-common sides are opposite rays, forming a straight line. The sum of angles on a straight line is always 180 degrees.
MCQ 5. When two lines intersect, the angles opposite to each other at the intersection point are called?
- A. Adjacent angles
- B. Corresponding angles
- C. Vertically opposite angles
- D. Alternate interior angles
Answer: C. Vertically opposite angles — Vertically opposite angles are formed when two lines intersect, and they are always equal. They share a common vertex but no common arm.
MCQ 6. In the context of two lines cut by a transversal, which of the following angle pairs are always equal if the two lines are parallel?
- A. Interior angles on the same side of the transversal
- B. Consecutive interior angles
- C. Vertically opposite angles
- D. Corresponding angles
Answer: D. Corresponding angles — When a transversal intersects two parallel lines, corresponding angles are equal. This is one of the key properties of parallel lines.
MCQ 7. If two lines L and M are intersected by a transversal T, and the alternate interior angles are equal, what can be concluded about lines L and M?
- A. They are perpendicular
- B. They are intersecting but not parallel
- C. They are parallel
- D. They are concurrent
Answer: C. They are parallel — This is the converse of a property for parallel lines. If alternate interior angles formed by a transversal are equal, then the two lines must be parallel.
MCQ 8. If three or more lines pass through the same point, what are they called?
- A. Parallel lines
- B. Perpendicular lines
- C. Concurrent lines
- D. Transversal lines
Answer: C. Concurrent lines — When three or more lines intersect at a single common point, they are known as concurrent lines. The common point is called the point of concurrency.
MCQ 9. Two adjacent angles that form a straight line are always:
- A. Complementary
- B. Supplementary
- C. Equal
- D. Right angles
Answer: B. Supplementary — Angles that form a straight line add up to 180 degrees, which is the definition of supplementary angles. A linear pair is an example of supplementary angles.
MCQ 10. If an angle is 70 degrees, what is the measure of its vertically opposite angle?
- A. 20 degrees
- B. 110 degrees
- C. 70 degrees
- D. Cannot be determined
Answer: C. 70 degrees — Vertically opposite angles are always equal. Therefore, if one angle is 70 degrees, its vertically opposite angle will also be 70 degrees.
MCQ 11. A line that intersects two or more lines at distinct points is called a:
- A. Parallel line
- B. Perpendicular line
- C. Transversal
- D. Ray
Answer: C. Transversal — A transversal is a line that crosses two or more other lines, often creating various angle pairs that are studied in geometry.
MCQ 12. If two parallel lines are cut by a transversal, and one of the interior angles on the same side of the transversal is 60 degrees, what is the measure of the other interior angle on the same side?
- A. 30 degrees
- B. 60 degrees
- C. 120 degrees
- D. 180 degrees
Answer: C. 120 degrees — When a transversal intersects two parallel lines, the sum of interior angles on the same side of the transversal is 180 degrees. So, 180 - 60 = 120 degrees.
MCQ 13. How many pairs of corresponding angles are formed when a transversal intersects two lines?
- A. 2
- B. 4
- C. 6
- D. 8
Answer: B. 4 — When a transversal intersects two lines, four pairs of corresponding angles are formed. Each pair consists of angles in the same relative position at each intersection.
MCQ 14. If lines A and B are parallel, and an angle formed by a transversal with line A is 100 degrees (exterior angle), what is the measure of the corresponding angle with line B?
- A. 80 degrees
- B. 100 degrees
- C. 180 degrees
- D. Cannot be determined
Answer: B. 100 degrees — For parallel lines intersected by a transversal, corresponding angles are equal. Thus, the corresponding angle with line B will also be 100 degrees.
MCQ 15. What is the relationship between alternate exterior angles when two lines are intersected by a transversal and are parallel?
- A. They are supplementary
- B. They are complementary
- C. They are equal
- D. They add up to 90 degrees
Answer: C. They are equal — When two parallel lines are cut by a transversal, alternate exterior angles are equal. This is a fundamental property of parallel lines.
MCQ 16. If angle X and angle Y form a linear pair, and angle X = 3Y, what is the value of angle Y?
- A. 45 degrees
- B. 90 degrees
- C. 135 degrees
- D. 180 degrees
Answer: A. 45 degrees — Since X and Y form a linear pair, X + Y = 180°. Substituting X = 3Y, we get 3Y + Y = 180°, so 4Y = 180°, which means Y = 45°.
MCQ 17. Consider three distinct lines L, M, and N. If L is parallel to M, and M is parallel to N, what is the relationship between L and N?
- A. L is perpendicular to N
- B. L intersects N
- C. L is parallel to N
- D. L and N are concurrent
Answer: C. L is parallel to N — This is a transitive property of parallel lines: if two lines are parallel to the same line, then they are parallel to each other. So, L is parallel to N.
MCQ 18. If two lines are intersected by a transversal such that the sum of interior angles on the same side is 180 degrees, then the lines are:
- A. Perpendicular
- B. Intersecting
- C. Parallel
- D. Skew
Answer: C. Parallel — This is a condition for parallel lines. If the sum of consecutive interior angles is 180 degrees, the lines must be parallel.
MCQ 19. An angle is 45 degrees. What is the measure of its adjacent angle if they form a linear pair?
- A. 45 degrees
- B. 90 degrees
- C. 135 degrees
- D. 180 degrees
Answer: C. 135 degrees — Angles in a linear pair sum up to 180 degrees. So, if one angle is 45 degrees, the other angle is 180 - 45 = 135 degrees.
MCQ 20. When three lines intersect at three different points, how many points of intersection are there?
- A. One
- B. Two
- C. Three
- D. Zero
Answer: C. Three — If three distinct lines intersect at three different points, it means each pair of lines intersects at a unique point. This often forms a triangle.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.