A Tale Of Three Intersecting Lines Important Notes - Class 7 Mathematics (Ganita Prakash) | Fully Solved Notes For Students

Table of Contents
1. Introduction
2. Understanding Triangles
3. Equilateral Triangles
4. Constructing a Triangle When Its Three Sides are Given
5. When is a Triangle Possible? Triangle Inequality Theorem
6. Constructing Triangles When Some Sides and Angles are Given
7. Angle Sum Property
8. Exterior Angle Theorem
9. Worked Example
10. Constructions Related to Altitudes (Heights) of Triangles
11. Types of Triangles
12. Summary
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Introduction

Have you ever looked closely at the world around you? From the slice of cheese you eat to the bridges you cross, straight lines and closed shapes are everywhere. The simplest closed shape made with straight lines is the triangle.

One of the most fundamental and fascinating shapes is the triangle. It might seem simple, just three lines meeting at three points, but this basic shape holds a surprising amount of depth and is the building block for many complex structures.

A triangle is a closed figure formed by three straight line segments joining three non-collinear points. A triangle has three vertices, three sides and three interior angles. In the sections that follow we will study these parts, how to name triangles, how to construct them, when a set of three lengths can form a triangle, and important properties used in solving geometric problems.

Understanding Triangles

A triangle is completely determined by its three vertices and the straight line segments that join them. The triangle with vertices labelled A, B and C is written as ∆ABC. The three sides are AB, BC and CA. The interior angles at vertices A, B and C are written as ∠A, ∠B and ∠C respectively.

Vertices, Sides and Angles

• Vertices: the three corner points where sides meet.
• Sides: the three straight line segments joining pairs of vertices.
• Angles: the three interior angles, each formed by two sides meeting at a vertex.

 These are the corner points where the sides intersect. Think of them as the 'tips' of the triangle.

 These are the straight line segments connecting each pair of vertices. They form the boundary of the triangle.

Triangles come in many shapes: tall and thin, short and wide, symmetric or completely unequal in their sides and angles.

Naming Triangles

We name a triangle by its three vertex letters. For example, the triangle with vertices A, B and C is written ∆ABC. The order of vertices may be changed but the same three letters represent the same triangle.

Angles of a Triangle

For ∆ABC, the interior angles are ∠A (or ∠CAB), ∠B (or ∠ABC) and ∠C (or ∠BCA). Often we use the single letter notation ∠A, ∠B and ∠C.

Equilateral Triangles

An equilateral triangle is a triangle with all three sides equal. Equilateral triangles are perfectly symmetric; each interior angle measures 60°.

Because all sides are equal, all angles in an equilateral triangle are also equal (each measuring 60 degrees).

Constructing an Equilateral Triangle

To construct an equilateral triangle with each side 4 cm exactly, use a ruler and a compass. A ruler alone would require trial and error; the compass gives an accurate construction.

Steps using ruler and compass:

• Draw the base: Draw a line segment AB of length 4 cm using a ruler.

• Draw the first arc: With the compass set to 4 cm, place the needle at A and draw an arc of radius 4 cm.

 Place the compass point on vertex A. Adjust the compass width to 4 cm (the desired side length). Draw a long arc above the base line. The third vertex, C, must lie somewhere on this arc because every point on this arc is exactly 4 cm away from A

• Draw the second arc: Without changing the compass width, place the needle at B and draw another arc that intersects the first arc. Let the intersection be C. That point is 4 cm from both A and B.

• Join the vertices: Use a ruler to join A to C and B to C. The triangle ∆ABC is equilateral with AB = BC = CA = 4 cm.

Constructing a Triangle When Its Three Sides are Given

Given three side lengths, you can construct the triangle (when it exists) using two circles whose radii are the given side lengths from the chosen base endpoints. For example, to construct a triangle with sides 4 cm, 5 cm and 6 cm:

Steps to construct the triangle with sides 4 cm, 5 cm and 6 cm

1. Draw the base: Choose one side to be the base; draw AB = 4 cm and label its endpoints A and B.

2. Draw an arc from A: With the compass set to 5 cm, place the needle at A and draw an arc showing all points 5 cm from A.

3. Draw an arc from B: With the compass set to 6 cm, place the needle at B and draw an arc showing all points 6 cm from B.

4. Locate their intersection: The intersection point of the two arcs is C. That point is 5 cm from A and 6 cm from B.

5. Join the points: Draw AC and BC to complete triangle ∆ABC with sides 4 cm, 5 cm and 6 cm.

You have constructed a triangle whose sides are exactly 4 cm, 5 cm and 6 cm.

Note: the construction succeeds only if the three given lengths satisfy the condition needed for a triangle to exist (see the Triangle Inequality Theorem below).

When is a Triangle Possible? Triangle Inequality Theorem

Not every three positive lengths can be the sides of a triangle. The Triangle Inequality Theorem states:

For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Examples to test

• Try to make a triangle with side lengths 3 cm, 4 cm and 8 cm.
• Try to make a triangle with side lengths 2 cm, 3 cm and 6 cm.
• Try to make a triangle with side lengths 10 cm, 15 cm and 30 cm.

In each of these examples the sum of the two shorter sides is equal to or less than the longest side, so a triangle cannot be formed.

Intuitive illustration: imagine walking from a tent to a tree directly, or walking from the tent to a pole and then to the tree. The direct route is always shorter than any roundabout path. Thus, if one side is as long as or longer than the sum of the other two, the three points will lie on a straight line or cannot close to form a triangle.

Triangle Inequality Example

Checking a proposed triangle

Consider side lengths 10 cm, 15 cm and 30 cm. Check whether a triangle is possible:

• 10 + 15 = 25 < 30 → the sum of two shorter sides is less than the third side.

Because 25 < 30, a triangle is not possible with these sides. In such a case one side is longer than the sum of the other two, so the supposed triangle cannot exist.

Circle construction viewpoint (proof by construction)

Another way to see the triangle inequality is by the circle method used in constructions. Take the longest side AB as fixed. Draw a circle with centre A and radius equal to one of the other given lengths, and a circle with centre B and radius equal to the remaining length. The two circles will intersect in two points only when the sum of the two radii is greater than AB. The three possibilities are:

• Case 1 - circles touch each other externally: radius1 + radius2 = AB. The "triangle" collapses to a straight line (no area), so a triangle does not form.

• Case 2 - circles are separate (do not meet): radius1 + radius2 < AB. There is no common point and therefore no possible third vertex C; a triangle cannot be formed.

• Case 3 - circles intersect at two points: radius1 + radius2 > AB. The intersection points give possible positions for the third vertex C, and a triangle can be formed.

Conclusion: triangles exist only when the sum of any two side lengths is greater than the third side.

Constructing Triangles When Some Sides and Angles are Given

Two sides and the included angle (SAS)

When two sides of a triangle and the angle between them are given, the triangle can often be constructed using ruler, compass and protractor. This is called the "two sides and the included angle" (SAS) condition.

Example: Construct a triangle ABC given AB = 5 cm, AC = 4 cm and ∠A = 45°.

Steps:

1. Draw AB = 5 cm.
2. At A, draw an angle of 45°. Use a protractor to mark the angle.
3. On the arm of the 45° angle from A, mark a point C such that AC = 4 cm.
4. Join B and C. The triangle ABC is complete.

Note: Sometimes even when two sides and their included angle are specified, the triangle may not be constructible if the given data forces the arms to miss each other. You must check that the third side can actually join the two points.

Two angles and the included side (ASA)

If two angles and the side between them are given, the triangle can usually be constructed because the third angle is fixed by the angle sum property. This case is called ASA (angle-side-angle).

Example: Given AB = 5 cm, ∠A = 45°, ∠B = 80°:

1. Draw AB = 5 cm.
2. At A draw ∠A = 45° using a protractor.
3. At B draw ∠B = 80° using a protractor.
4. The two lines will meet at C; join AC and BC to form ∆ABC.

Do triangles always exist with any two angles and the included side? No. If the two given angles sum to 180° or more, the lines will not meet to form a triangle. For example, if one angle is 40°, the other must be less than 140° for a triangle (because 40° + 140° = 180° leaves no angle for the third vertex).

Angle Sum Property

The sum of the interior angles of any triangle is 180°.

Proof idea using parallel lines: draw a line through one vertex parallel to the opposite side. Using alternate interior angles and corresponding angles, the three interior angles of the triangle line up to form a straight angle, which measures 180°.

Example: If ∠B = 50° and ∠C = 70°, then ∠A = 180° - (50° + 70°) = 60°.

A simple paper fold verification: cut out any triangle and fold its three corners so their tips meet; the three angles together form a straight line, showing the sum is 180°.

Exterior Angle Theorem

An exterior angle of a triangle is formed by extending one of its sides. The exterior angle is equal to the sum of the two interior opposite (non-adjacent) angles.

Example: If ∠A = 50° and ∠B = 60°, then the exterior angle ∠ACD (formed by extending BC or BA appropriately) equals 50° + 60° = 110°.

Worked Example

Example: Can we make a triangle with sides of length 3 cm, 4 cm, and 8 cm?

Sol:

We will use the Triangle Inequality Theorem, which requires that the sum of any two side lengths be greater than the third side.

Check the two shorter sides:

3 cm + 4 cm = 7 cm.

Compare with the longest side:

7 cm < 8 cm.

Since 3 cm + 4 cm is less than 8 cm, the three lengths do not satisfy the triangle inequality. Therefore, it is impossible to form a triangle with sides 3 cm, 4 cm and 8 cm.

Constructions Related to Altitudes (Heights) of Triangles

An altitude of a triangle is a perpendicular drawn from a vertex to the opposite side (or its extension). The length of this perpendicular is called the height of the triangle from that vertex.

To draw an altitude from vertex A to side BC:

• Place a ruler along BC.
• Use a set square against the ruler to get a perfect 90° edge.
• Slide the set square until its vertical edge passes through A, and draw the line from A perpendicular to BC. Mark the foot of the perpendicular as D; AD is the altitude.

Altitudes behave differently in different types of triangles:

• In acute triangles (all angles < 90°), all altitudes lie inside the triangle.
• In a right triangle, the altitude from the right-angled vertex coincides with one of the sides.
• In obtuse triangles (one angle > 90°), some altitudes fall outside the triangle; you must extend the opposite side and drop the perpendicular to that extension.

Types of Triangles

Based on sides

1. Equilateral triangle:

• All three sides equal.
• All three angles equal, each 60°.

2. Isosceles triangle:

• Two sides equal.
• The angles opposite the equal sides are equal.

3. Scalene triangle:

• All three sides of different lengths.
• All three angles are different.

4. Right-angled triangle:

• One angle is exactly 90°.
• The side opposite the right angle is the hypotenuse; the other two sides are legs.

Right-angled Triangle

Based on angles

1. Acute-angled triangle:
   • All three angles are less than 90°.
2. Right-angled triangle:
   • One angle is exactly 90°.
3. Obtuse-angled triangle:
   • One angle is greater than 90° but less than 180°; the other two are acute.
   • 

Connections between classifications

There is overlap between the two ways of classifying triangles. For example, every equilateral triangle is also an acute-angled triangle because each angle is 60°. Isosceles and scalene triangles may be acute, right or obtuse depending on their angle measures.

Summary

This chapter explained what triangles are, how to name and classify them, methods to construct triangles from given elements (sides and angles), the Triangle Inequality Theorem that decides when a triangle is possible, the angle sum property (sum of interior angles is 180°), the exterior angle theorem, and the construction and meaning of altitudes (heights). The geometry tools most commonly used are ruler, compass and the protractor. Always check the triangle inequality and angle conditions before attempting a construction.