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A Tale of Three Intersecting Lines

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the triangle as the most basic closed shape, focusing on its construction, properties, and classification. It explains how to construct triangles given side lengths or angles, introduces the triangle inequality, and proves the angle sum property. The one big idea a student must walk away with is that a triangle exists only when the sum of any two sides is greater than the third side, and the sum of its three angles is always 180°.
Key Concepts (16)
Triangle
A triangle is the most basic closed shape consisting of three corner points called vertices and three line segments called sides that join the pairs of vertices.
e.g. A slice of pizza is shaped like a triangle with three corners and three edges.
Vertex
A corner point of a triangle where two sides meet.
e.g. In triangle ABC, points A, B, and C are the vertices.
Side
A line segment joining two vertices of a triangle.
e.g. In triangle ABC, AB, BC, and CA are the sides.
Angle of a triangle
The angle formed at a vertex by the two sides meeting there.
e.g. In triangle ABC, ∠CAB (or ∠A) is the angle at vertex A.
Equilateral triangle
A triangle in which all three sides are of equal length.
e.g. A triangle with each side 4 cm is an equilateral triangle.
Isosceles triangle
A triangle in which two sides are of equal length.
e.g. A triangle with sides 5 cm, 5 cm, and 6 cm is an isosceles triangle.
Scalene triangle
A triangle in which all three sides are of different lengths.
e.g. A triangle with sides 3 cm, 4 cm, and 5 cm is a scalene triangle.
Triangle inequality
For any triangle, the sum of the lengths of any two sides is greater than the length of the third side. If a set of three lengths satisfies this condition, a triangle exists with those side lengths.
e.g. For sides 3 cm, 4 cm, and 5 cm: 3+4>5, 4+5>3, 5+3>4, so a triangle exists.
Included angle
The angle between two given sides of a triangle.
e.g. In triangle ABC with AB=5 cm and AC=4 cm, ∠A is the included angle between sides AB and AC.
Included side
The side that is common to two given angles of a triangle.
e.g. In triangle ABC with ∠A=45° and ∠B=80°, side AB is the included side.
Angle sum property of triangles
The sum of the three angles of any triangle is always 180°.
e.g. In a triangle with angles 50°, 60°, and 70°, the sum is 50+60+70=180°.
Exterior angle of a triangle
The angle formed between the extension of a side of a triangle and the other side.
e.g. In triangle ABC, if side BC is extended to D, then ∠ACD is an exterior angle.
Altitude of a triangle
A perpendicular line segment from a vertex to the opposite side (or its extension).
e.g. In triangle ABC, the line from A perpendicular to BC is an altitude.
Right-angled triangle
A triangle that has one right angle (90°).
e.g. A triangle with angles 90°, 45°, and 45° is a right-angled triangle.
Acute-angled triangle
A triangle in which all three angles are acute (less than 90°).
e.g. A triangle with angles 50°, 60°, and 70° is an acute-angled triangle.
Obtuse-angled triangle
A triangle that has one obtuse angle (greater than 90°).
e.g. A triangle with angles 100°, 40°, and 40° is an obtuse-angled triangle.
Formulas & Equations (3)
Sum of angles of a triangle = 180°
Reads as: The sum of the three angles of a triangle is equal to 180 degrees.
Variables: ∠A, ∠B, ∠C are the three angles of the triangle (measured in degrees).
Use when: Use this to find the third angle when two angles are given, or to check if a set of angles can form a triangle.
⚠ Students forget that the sum is exactly 180°, not 180° or 360°.
Triangle inequality: For sides a, b, c of a triangle, a + b > c, b + c > a, c + a > b
Reads as: For any triangle, the sum of any two sides is greater than the third side.
Variables: a, b, c are the lengths of the three sides (in any unit, e.g., cm, m, km).
Use when: Use this to check whether three given lengths can form a triangle.
⚠ Students only check one pair (e.g., a+b>c) and forget to check all three conditions.
Exterior angle = Sum of the two opposite interior angles
Reads as: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Variables: Exterior angle is the angle formed outside the triangle; opposite interior angles are the two angles not adjacent to it.
Use when: Use this to find an exterior angle or one of the interior angles when the other is known.
⚠ Students confuse the exterior angle with the adjacent interior angle (they are supplementary, not equal).
How It Works
Constructing a triangle when all three sides are given (SSS construction)
  1. Step 1: Choose one side as the base and draw it using a ruler. For example, draw AB = 4 cm.
  2. Step 2: Using a compass, draw an arc of radius equal to the second side (e.g., 5 cm) from point A.
  3. Step 3: From point B, draw an arc of radius equal to the third side (e.g., 6 cm) so that it intersects the first arc.
  4. Step 4: Mark the intersection point as C. Join AC and BC to get the required triangle.
💡 The arcs must intersect for a triangle to exist. If they do not intersect, the given side lengths do not satisfy the triangle inequality.
Constructing a triangle when two sides and the included angle are given (SAS construction)
  1. Step 1: Draw the base side (one of the given sides). For example, draw AB = 5 cm.
  2. Step 2: At point A, construct the given included angle (e.g., 45°) using a protractor.
  3. Step 3: On the other arm of the angle, mark point C such that AC equals the second given side (e.g., 4 cm).
  4. Step 4: Join BC to complete the triangle.
💡 The angle must be the included angle — the angle between the two given sides. If the angle is not between them, the construction is different.
Constructing a triangle when two angles and the included side are given (ASA construction)
  1. Step 1: Draw the included side as the base. For example, draw AB = 5 cm.
  2. Step 2: At point A, construct one given angle (e.g., 45°) using a protractor.
  3. Step 3: At point B, construct the other given angle (e.g., 80°) using a protractor.
  4. Step 4: Extend the two arms until they meet at point C. This is the third vertex.
💡 The sum of the two given angles must be less than 180° for a triangle to exist. If the sum is 180° or more, the arms will never meet.
Constructing an altitude of a triangle using a set square
  1. Step 1: Place a ruler along the base (e.g., side BC).
  2. Step 2: Place a set square on the ruler such that one edge of the right angle touches the ruler.
  3. Step 3: Slide the set square along the ruler until the vertical edge touches the vertex (e.g., A).
  4. Step 4: Draw the altitude along the vertical edge of the set square from the vertex to the base.
💡 The altitude is always perpendicular to the base. If the vertex is above the base, the altitude falls inside the triangle; if the triangle is obtuse, the altitude may fall outside and you need to extend the base.
Proving the angle sum property using parallel lines
  1. Step 1: Draw a triangle ABC.
  2. Step 2: Through vertex A, draw a line XY parallel to side BC.
  3. Step 3: Since XY ∥ BC, ∠XAB = ∠B (alternate interior angles) and ∠YAC = ∠C (alternate interior angles).
  4. Step 4: On line XY, ∠XAB + ∠BAC + ∠YAC = 180° (straight angle).
  5. Step 5: Substitute: ∠B + ∠A + ∠C = 180°. Hence, the sum of the angles of a triangle is 180°.
💡 This proof uses the concept of parallel lines and alternate angles. Students often forget to state that XY is drawn parallel to BC.
Exam Traps — Don't Make These Mistakes
⚠ Triangle inequality — checking only one pair
✗ For sides 3, 4, 8: 3+4=7 < 8, so triangle is not possible.
→ The statement is correct for this example, but students often check only one pair (e.g., the largest side) and forget to check all three conditions. For sides 4, 5, 9, they might write 4+5=9, so triangle is possible, but actually 4+5=9 is not greater than 9, so no triangle exists.
✓ Check all three conditions: a+b>c, b+c>a, c+a>b. For a triangle to exist, each sum must be strictly greater than the third side.
⚠ Angle sum property — using 180° for exterior angle
✗ If ∠A=50° and ∠B=60°, then exterior angle ∠ACD = 180° - 70° = 110°.
→ The calculation is correct, but the reasoning is incomplete. Students often forget to state that ∠ACB = 70° (using angle sum property) and that ∠ACB and ∠ACD form a straight angle (180°).
✓ First, find ∠ACB using angle sum property: 50°+60°+∠ACB=180°, so ∠ACB=70°. Since ∠ACB and ∠ACD form a straight angle, ∠ACD=180°-70°=110°. Alternatively, use the exterior angle property: ∠ACD = ∠A+∠B = 50°+60°=110°.
⚠ Constructing a triangle with two sides and a non-included angle
✗ Construct triangle ABC with AB=5 cm, AC=4 cm, and ∠B=45°.
→ Students assume the given angle is the included angle, but here ∠B is not between sides AB and AC (it is between AB and BC). This is an SSA case, which may have 0, 1, or 2 possible triangles, and is not covered in this chapter.
✓ Check if the given angle is the included angle (between the two given sides). If not, the construction method in this chapter does not apply. For SAS construction, the angle must be between the two given sides.
⚠ Altitude in an obtuse triangle
✗ Draw the altitude from vertex A to side BC by dropping a perpendicular inside the triangle.
→ In an obtuse triangle, the altitude from the vertex of the acute angle may fall outside the triangle. Students often draw it inside, which is incorrect.
✓ If the triangle is obtuse, extend the base (side BC) beyond the triangle. Then drop the perpendicular from vertex A to this extended line. The altitude is the length of this perpendicular segment.
⚠ Classifying triangles by angles — acute vs obtuse
✗ A triangle with one acute angle is an acute-angled triangle.
→ Every triangle has at least two acute angles. An acute-angled triangle requires all three angles to be acute (less than 90°).
✓ An acute-angled triangle has all three angles less than 90°. A right-angled triangle has one angle exactly 90°. An obtuse-angled triangle has one angle greater than 90°.
Compare & Contrast
Number of equal sides
Equilateral triangle
Isosceles triangle
An equilateral triangle has all three sides equal, while an isosceles triangle has exactly two sides equal.
Measure of angles
Acute-angled triangle
Obtuse-angled triangle
An acute-angled triangle has all three angles less than 90°, while an obtuse-angled triangle has one angle greater than 90°.
Definition and construction
Altitude
Median
An altitude is a perpendicular from a vertex to the opposite side (or its extension), while a median joins a vertex to the midpoint of the opposite side. They are not the same unless the triangle is isosceles or equilateral.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks) — total 10 marks typically
Most asked: Triangle inequality — checking if three given lengths can form a triangle · Angle sum property — finding the third angle when two are given · Construction of triangles (SSS, SAS, ASA) · Classification of triangles (equilateral, isosceles, scalene; acute, right, obtuse) · Exterior angle property
Don't skip: Construction of a triangle given three sides or two sides and included angle (5 marks); Application of triangle inequality to determine if a triangle exists (3 marks); Angle sum property with algebraic expressions for angles (3 marks)
One-Line Revision (read aloud before the exam)
  1. A triangle has three vertices, three sides, and three angles.
  2. An equilateral triangle has all three sides equal.
  3. An isosceles triangle has two sides equal.
  4. A scalene triangle has all sides of different lengths.
  5. The triangle inequality states that the sum of any two sides of a triangle is greater than the third side.
  6. If three lengths satisfy the triangle inequality, a triangle exists with those side lengths.
  7. The sum of the three angles of any triangle is always 180°.
  8. An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
  9. An altitude is a perpendicular from a vertex to the opposite side.
  10. A right-angled triangle has one angle of 90°.
  11. An acute-angled triangle has all angles less than 90°.
  12. An obtuse-angled triangle has one angle greater than 90°.
  13. To construct a triangle given three sides (SSS), draw arcs from the endpoints of the base with radii equal to the other two sides.
  14. To construct a triangle given two sides and the included angle (SAS), draw the base, then the angle, then mark the second side.
  15. To construct a triangle given two angles and the included side (ASA), draw the base, then both angles at its endpoints, and extend the arms to meet.
  16. A triangle does not exist if the sum of two given angles is 180° or more.
  17. In an obtuse triangle, an altitude may fall outside the triangle, requiring the base to be extended.

MCQs — 10 Questions with Answers

Q1. How many vertices does a triangle have?

  • A. Three ✓
  • B. Four
  • C. Two
  • D. Five

Answer: A — By definition, a triangle is formed by three corner points called vertices.

Q2. A triangle with all three sides equal is called a/an _____ triangle.

  • A. isosceles
  • B. equilateral ✓
  • C. scalene
  • D. right-angled

Answer: B — An equilateral triangle has all three sides of equal length.

Q3. What tool is most efficient for constructing triangles when sides are given?

  • A. Only a marked ruler
  • B. A compass and a ruler ✓
  • C. A protractor only
  • D. A set square only

Answer: B — A compass allows you to draw arcs of specific radii to locate the third vertex accurately without trial-and-error.

Q4. Which of the following is an isosceles triangle?

  • A. Sides 4 cm, 4 cm, 4 cm
  • B. Sides 3 cm, 4 cm, 5 cm
  • C. Sides 5 cm, 5 cm, 8 cm ✓
  • D. Sides 2 cm, 3 cm, 4 cm

Answer: C — An isosceles triangle has exactly two equal sides; here 5 cm and 5 cm are equal.

Q5. Meera wants to construct a triangle with sides 6 cm, 8 cm, and 14 cm. Will she be able to construct this triangle?

  • A. Yes, because all sides are different
  • B. No, because 14 is not less than 6 + 8 = 14 ✓
  • C. Yes, because 14 > 6 + 8
  • D. No, because the sides are too long

Answer: B — The triangle inequality requires each side to be strictly less than the sum of the other two; here 14 = 6 + 8, not less than, so no triangle exists.

Q6. A farmer wants to create a triangular vegetable plot with sides 5 m, 12 m, and 13 m. Can such a plot be created?

  • A. No, because 13 > 5 + 12
  • B. Yes, because 13 < 5 + 12 ✓
  • C. No, because the sides are too long
  • D. Yes, but only if the farmer uses special tools

Answer: B — Since 5 < 12 + 13, 12 < 5 + 13, and 13 < 5 + 12, the triangle inequality is satisfied, so the plot can be created.

Q7. When constructing triangle ABC with AB = 5 cm, AC = 6 cm, and BC = 7 cm, after drawing AB, what is the radius of the arc drawn from point A?

  • A. 5 cm
  • B. 7 cm
  • C. 6 cm ✓
  • D. 12 cm

Answer: C — The arc from A has radius AC = 6 cm because point C must be 6 cm away from A.

Q8. Why is the direct path between two points always shorter than any roundabout path through a third point?

  • A. Because the third point is always far away
  • B. Because a straight line is the shortest distance between two points ✓
  • C. Because we always walk in zigzag patterns
  • D. Because the roundabout path is always longer by definition

Answer: B — This is a fundamental principle of geometry: a straight line represents the shortest distance between any two points.

Q9. If a triangle has sides 9 cm, 10 cm, and 18 cm, which comparison fails the triangle inequality?

  • A. 9 < 10 + 18 ✓
  • B. 10 < 9 + 18 ✓
  • C. 18 < 9 + 10 ✗ ✓
  • D. All comparisons satisfy the inequality

Answer: C — Since 18 is not less than 9 + 10 = 19 (it equals 19), this side fails the triangle inequality condition.

Q10. Ajay constructs triangle PQR where PQ = QR = 8 cm and PR = 10 cm. What type of triangle is this, and why can it be constructed?

  • A. Equilateral, because all sides are whole numbers
  • B. Isosceles, and it satisfies triangle inequality since 10 < 8 + 8 = 16 ✓
  • C. Scalene, because PR is different from PQ
  • D. Right-angled, because 8 + 8 = 16

Answer: B — It is isosceles because PQ = QR = 8 cm, and it can be constructed because 10 < 8 + 8, and the other two inequalities also hold.

Flashcards

What are the three basic parts of a triangle?

A triangle has three vertices (corner points), three sides (line segments), and three angles where the sides meet.

Define an equilateral triangle.

An equilateral triangle is a triangle in which all three sides have equal length.

What is an isosceles triangle?

An isosceles triangle is a triangle with exactly two sides of equal length.

Why do we use a compass to construct triangles instead of just a ruler?

A compass allows us to mark points at specific distances from given points without trial-and-error, making construction efficient and accurate.

State the triangle inequality condition.

For any triangle with sides a, b, and c, each side must be smaller than the sum of the other two: a < b + c, b < a + c, and c < a + b.

Can a triangle be constructed with sides 5 cm, 7 cm, and 15 cm? Why or why not?

No, because 15 is not less than 5 + 7 = 12, so the triangle inequality is not satisfied.

What does it mean when the three vertices of a triangle lie on a straight line?

When all three vertices are collinear (on the same straight line), no triangle is formed because the three sides collapse into a single line.

In triangle construction using arcs, why must both arcs intersect?

Both arcs must intersect because the third vertex must lie at a distance equal to the first radius from point A and equal to the second radius from point B simultaneously.

How many ways can you name triangle ABC?

A triangle can be named in six ways because the three vertices can be arranged in any order: ABC, ACB, BAC, BCA, CAB, or CBA.

What is the most symmetric triangle among all triangles?

The equilateral triangle is the most symmetric triangle because all its sides are equal and all its angles are equal.

Important Board Questions

What is a triangle? [1 mark]

Define using vertices and sides; mention the symbol used to denote it.

Construct an equilateral triangle with side length 5 cm. List the steps you would follow. [2 marks]

First draw the base AB = 5 cm. Then use compass to draw arcs of radius 5 cm from both A and B; mark their intersection as C.

Can a triangle be constructed with sides 7 cm, 8 cm, and 20 cm? Check using the triangle inequality and explain your answer. [3 marks]

Check if each side is less than the sum of the other two. Since 20 > 7 + 8, the triangle inequality fails; no triangle can exist.

Construct a triangle ABC with sides AB = 6 cm, BC = 7 cm, and AC = 8 cm. Show all construction steps and verify that it satisfies the triangle inequality. [5 marks]

Draw AB = 6 cm as base. From A, draw arc of radius 8 cm (for AC). From B, draw arc of radius 7 cm (for BC). Mark intersection as C. Verify: 6 < 7 + 8, 7 < 6 + 8, 8 < 6 + 7 — all true.

Assertion-Reason Questions for A Tale of Three Intersecting Lines

Q1. Assertion (A):

An equilateral triangle is a special case of an isosceles triangle.

Reason (R):

An equilateral triangle has all three sides equal, while an isosceles triangle has at least two sides equal.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. An equilateral triangle, having all three sides equal, automatically has at least two sides equal, making it satisfy the definition of an isosceles triangle. The reason correctly explains why the assertion is true.

Q2. Assertion (A):

If the sum of two sides of a triangle equals the third side, the three points will lie on a straight line.

Reason (R):

The triangle inequality states that the sum of any two sides must be strictly greater than the third side.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true, but the reason does not directly explain the assertion. The reason restates the triangle inequality rule, while the assertion specifically addresses what happens when the inequality becomes an equality. The assertion describes the geometric consequence (collinear points), and the reason gives the algebraic condition, but they don't have a direct cause-and-effect relationship.

Q3. Assertion (A):

A triangle with sides 5 cm, 5 cm, and 10 cm can be constructed.

Reason (R):

The sum 5+5=10 equals the third side, not greater than it.

Answer: D — A is false but R is true — The assertion is false because the triangle cannot be constructed when the sum of two sides equals (rather than exceeds) the third side. The reason is also false in its reasoning — it correctly identifies that 5+5=10, but this is precisely why the triangle CANNOT exist, not why it can. Both parts are incorrect.

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Case-Based Questions (CBQ) for A Tale of Three Intersecting Lines

Case Study:

A construction worker is hired to create a triangular warning sign for a construction site. The supervisor specifies that the sign must have side lengths of 60 cm, 80 cm, and 100 cm. The worker is asked to first verify if such a triangle is geometrically possible before proceeding with the construction. The worker uses three wooden rods of the specified lengths and tries to arrange them to form a triangle by connecting the endpoints. To make this efficient, the worker decides to use the triangle inequality principle instead of physically trying to arrange the rods repeatedly.

Q1. According to the triangle inequality, which of the following conditions must be true for the triangle to exist?

  • A. 60+80>100, 60+100>80, and 80+100>60 ✓
  • B. 60+80=100, 60+100=80, and 80+100=60
  • C. 60+80<100, 60+100<80, and 80+100<60
  • D. 60×80>100, 60×100>80, and 80×100>60

Answer: A

Q2. Let the worker verify: Is 60+80>100?

  • A. No, 60+80=140, which is not greater than 100
  • B. Yes, 60+80=140, which is greater than 100 ✓
  • C. No, 60+80=140, which is less than 100
  • D. Yes, 60+80=100, so they are equal

Answer: B

Q3. After verifying all three conditions of the triangle inequality are satisfied, what can the worker conclude?

  • A. The triangle is impossible to construct
  • B. The triangle can be constructed with these side lengths ✓
  • C. The triangle must be equilateral
  • D. Only two of the three sides can be used

Answer: B

Q4. What type of triangle is formed by sides 60 cm, 80 cm, and 100 cm?

  • A. Equilateral triangle
  • B. Isosceles triangle
  • C. Scalene triangle ✓
  • D. Right-angled triangle (because 60²+80²=100²)

Answer: C

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