Geometry Maths 6 T2
Class 6 | Maths | Unit 4
Class 6 Unit 4 Maths
CHAPTER 4: GEOMETRY
Learning Objectives
- To understand the formation of triangles and the basic elements of a triangle.
- To know the types of triangles and their properties.
- To draw parallel and perpendicular lines using a set square.
4.1 Introduction
We already studied the basic geometrical concepts such as angles and its types, drawing line segments, drawing and measuring angles in the first term. In this term, we will study triangles and their types, construction of parallel and perpendicular lines to a given line segment.
MATHEMATICS ALIVE – TRIANGLES IN REAL LIFE
The triangle is used in most types of construction work including bridges, buildings, cell-phone towers, aeroplane wings and pitched roofs. Its use in construction gives an object the quality of stiffness, resulting in rigid and strong structures.
Think about the situation:
A teacher distributes 2, 3, 4 and 5 sticks of equal lengths to four students and asks them to form a closed figure. Three students make the following figures. But one of the students who has 2 sticks with him creates the following figure. He is not able to create a closed figure.
Do you know why?
Can you guess the least number of sticks required to form a closed figure? Three sticks.
If you had formed a closed figure with three sticks, then what shape would you get? Is there any special name for it? Yes. It's a triangle.
A closed figure formed by three-line segments is called a triangle.
Activity Classify Shapes
Classify the given shapes into triangles and non-triangles.
4.2 Basic Elements of a Triangle
Mark 3 points A, B, C on a paper, such that they do not lie on a straight line. Consider the line segments AB, BC and CA. We see that they form a triangle ABC which is represented as △ABC or △BCA or △CAB.
In △ABC, the line segments AB, BC and CA are called the sides of the triangle and ∠CAB, ∠ABC and ∠BCA (∠A, ∠B & ∠C) are called the angles of the triangle. The point of intersection of two sides of the triangle is called the vertex. A, B and C are three vertices of triangle ABC.
Key Facts
Hence, a triangle has 3 sides, 3 angles and 3 vertices.
Think
Can a triangle be drawn using 3 points on a straight line?
4.3 Types and Properties of Triangles
Some triangles are drawn in the dotted sheet. Try to draw as many triangles as you can. Then, measure the sides and angles of all triangles and fill the table given below:
Activity Triangle Measurement Table
| Serial Number | Measure of Angles | Sum of the Measure of Angles | Nature of Angles | Measure of Sides | Nature of Sides |
| 1 | 60°, 60°, 60° | 180° | Three angles are equal | 3 cm, 3 cm, 3 cm | Three sides are equal |
| 2 | | | | | |
| 3 | | | | | |
| 4 | | | | | |
| 5 | | | | | |
| 6 | | | | | |
Observations from the Table
In a triangle:
- If the measure of all angles are different, then all sides are different.
- If the measure of two angles are equal, then two sides are equal.
- If the measure of three angles are equal, then three sides are equal and each angle measures 60°.
- Sum of three angles of a triangle is 180°.
Activity Triangle Formation with Sticks
Students are divided into groups and each group is given 3 sticks of length 9 units, 2 sticks of length 3 units, 2 sticks of length 2 units, 1 stick of length 5 units and 1 stick of length 4 units. Using the given sticks, they are asked to form three triangles, find the length of the sides of each triangle and tabulate them.
| Triangle | Length of Side 1 | Length of Side 2 | Length of Side 3 | All sides equal / 2 sides equal / 3 sides different |
| Triangle 1 | | | | |
| Triangle 2 | | | | |
| Triangle 3 | | | | |
Read the table and answer the following questions:
1. Was each group able to form 3 triangles?
2. In each of the triangle formed, how many sides are equal?
4.3.1 Types of Triangle Based on its Sides
1) Scalene Triangle
If three sides of a triangle are different in lengths, then it is called a Scalene Triangle.
2) Isosceles Triangle
If any two sides of a triangle are equal in length, then it is called an Isosceles Triangle.
3) Equilateral Triangle
If three sides of a triangle are equal in length, then it is called an Equilateral Triangle.
Thus, based on the sides of triangles, we can classify triangles into 3 types.
Try These Complete the following table
In any triangle:
| Serial Number | Side 1 | Side 2 | Side 3 | Type of Triangle |
| 1 | 6 cm | 7 cm | 8 cm | Scalene Triangle |
| 2 | 5 cm | 5 cm | 5 cm | |
| 3 | 2.2 cm | 2.5 cm | 3.2 cm | |
| 4 | 7 cm | 7 cm | 10 cm | |
| 5 | 10 cm | 10 cm | 10 cm | |
| 6 | 10 cm | 8 cm | 8 cm | |
4.3.2 Types of Triangles Based on its Angles
Write the given angles as acute, obtuse or right angle formed by two line segments AB and AC. Now, join the third side to form a triangle in each case and identify the kinds of angles and list them down.
1) Acute Angled Triangle
If three angles of a triangle are acute angles (between 0° and 90°), then it is called an Acute Angled Triangle.
2) Right-Angled Triangle
If an angle of a triangle is a right angle (90°), then it is called a Right-Angled Triangle.
3) Obtuse Angled Triangle
If an angle of a triangle is an obtuse angle (between 90° and 180°), then it is called an Obtuse Angled Triangle.
Thus, based on the angles of triangles, we can classify triangles into 3 types.
Try These Complete the table
| S.No | ∠A | ∠B | ∠C | Sum of Three Angles | Can △ABC be formed? | Type of Triangle |
| 1 | 60° | 60° | 60° | 180° | Yes | Acute angled triangle |
| 2 | 50° | 40° | 90° | | | |
| 3 | 60° | 30° | 90° | | | |
| 4 | 95° | 40° | 35° | | | |
| 5 | 110° | 40° | 30° | | | |
| 6 | 150° | 60° | 70° | | | |
Do You Know?
A triangle can have three acute angles, but cannot have more than one right angle or an obtuse angle.
Activity Identify and Name Triangles
In the given figure, there are some triangles. Measure their sides and angles and name them in two ways. (One is done for you!)
4.3.3 Triangle Inequality Property
Think about the situation:
Three students Kamala, Madhan and Sumathi are asked to form triangles with the given sticks of measure 6 cm, 8 cm, 5 cm; 4 cm, 10 cm, 5 cm and 10 cm, 6 cm, 4 cm respectively. All of them try to form a triangle. While Kamala is successful in forming a triangle, Madhan and Sumathi are struggling. Why?
When they are trying to join the ends of the two smaller sticks, they find that the two smaller sticks coincide with the longer stick or shorter than the longer stick and they are unable to form triangles.
From this, they understand that, to form a triangle the sum of two smaller sides must be greater than the third side.
Triangle Inequality Property
In a triangle, the sum of any two sides of a triangle is greater than the third side. This is known as Triangle Inequality Property.
Note
If three sides are equal in length, then definitely a triangle can be formed.
- AB + BC > CA
- BC + CA > AB
- CA + AB > BC
Do You Know?
If any two sides of the triangle are given, then the length of the third side will lie between the difference and sum of the lengths of two given sides.
Example 1: Can a triangle be formed with 7 cm, 10 cm and 5 cm as its sides?
Solution: Instead of checking triangle inequality by all the sides in the triangle, check only with two smaller sides.
Sum of two smaller sides = 5 + 7 = 12 cm, which is greater than 10 cm (the third side).
It is greater than the third side. So, a triangle can be formed with the given sides.
Example 2: Can a triangle be formed with 7 cm, 7 cm and 7 cm as its sides?
Solution: If three sides are equal, then definitely a triangle can be formed, as the triangle inequality is satisfied.
Example 3: Can a triangle be formed with 8 cm, 3 cm and 4 cm as its sides?
Solution: The sum of two smaller sides = 3 + 4 = 7 cm, which is less than 8 cm (the third side).
It is less than the third side. So, a triangle cannot be formed with the given sides.
Try These Can a triangle be formed with the given sides?
| S.No | AB | BC | CA | Can △ABC be formed? | Type of Triangle |
| 1 | 7 cm | 10 cm | 6 cm | | |
| 2 | 10 cm | 8 cm | 8 cm | | |
| 3 | 8.5 m | 7.3 m | 6.8 m | | |
| 4 | 4 cm | 5 cm | 12 cm | | |
| 5 | 15 m | 20 m | 20 m | | |
| 6 | 23 cm | 20 cm | 18 cm | | |
| 7 | 3.2 cm | 1.5 cm | 1.5 cm | | |
Example 4: Can a triangle be formed with the angles 80°, 30°, 40°?
Solution: In a triangle, the sum of three angles is 180°.
Here, sum of three angles = 80° + 30° + 40° = 150° (≠ 180°)
So, a triangle cannot be formed with the given angles.
Think
Can the difference between two larger sides be less than the third side?
Activity A Triangle Game
In each turn a student must draw one line connecting two dots. A line should not cross other lines or touch other dots than the two that are connected to. If a student closes a triangle with his line then he gets a point. Once there are no more lines that can be drawn the game is over and the student who gains more points wins the game.
Think
In a right-angled triangle, what measures can the other two angles have?
Exercise 4.1
Exercise 1. Fill in the blanks
a) Every triangle has at least acute angles.
b) A triangle in which none of the sides equal is called a .
c) In an isosceles triangle angles are equal.
d) The sum of three angles of a triangle is .
e) A right-angled triangle with two equal sides is called .
Exercise 2. Match the following
| 1. No sides are equal | |
| 2. One right angle | |
| 3. One obtuse angle | |
| 4. Two sides of equal length | |
| 5. All sides are equal | |
Options: Isosceles triangle, Scalene triangle, Right angled triangle, Equilateral triangle, Obtuse angled triangle
Exercise 3. In △ABC, name the following
a) Three sides: , ,
b) Three Angles: , ,
c) Three Vertices: , ,
Exercise 4. Classify based on sides
Classify the given triangles based on its sides as scalene, isosceles or equilateral.
Exercise 5. Classify based on angles
Classify the given triangles based on its angles as acute angled, right angled or obtuse angled.
Exercise 6. Classify based on sides and angles
Classify the following triangles based on its sides and angles.
Exercise 7. Triangle formation with given sides
Can a triangle be formed with the following sides? If yes, name the type of triangle.
(1) 8 cm, 6 cm, 4 cm
(2) 10 cm, 8 cm, 5 cm
(3) 6.2 cm, 1.3 cm, 3.5 cm
(4) 6 cm, 6 cm, 4 cm
(5) 3.5 cm, 3.5 cm, 3.5 cm
(6) 9 cm, 4 cm, 5 cm
Exercise 8. Triangle formation with given angles
Can a triangle be formed with the following angles? If yes, name the type of triangle.
(1) 60°, 60°, 60°
(2) 90°, 55°, 35°
(3) 60°, 40°, 42°
(4) 60°, 90°, 90°
(5) 70°, 60°, 50°
(6) 100°, 50°, 30°
Exercise 9. Find the third angle
Two angles of the triangles are given. Find the third angle.
(1) 80°, 60° → Third angle:
(2) 75°, 35° → Third angle:
(3) 52°, 68° → Third angle:
(4) 50°, 90° → Third angle:
(5) 120°, 30° → Third angle:
(6) 55°, 85° → Third angle:
Exercise 10. Riddle
I am a closed figure with each of my three angles is 60°. Who am I?
Exercise 11. Type of Triangle Table
Using the given information, write the type of triangle in the table given below:
| S.No | Angle 1 | Angle 2 | Angle 3 | Type of Triangle (Angles) | Type of Triangle (Sides) |
| 1 | 60° | 40° | 80° | Acute angled triangle | Scalene Triangle |
| 2 | 50° | 50° | 80° | | |
| 3 | 45° | 45° | 90° | | |
| 4 | 55° | 45° | 80° | | |
| 5 | 75° | 35° | 70° | | |
| 6 | 60° | 30° | 90° | | |
| 7 | 25° | 64° | 91° | | |
| 8 | 120° | 30° | 30° | | |
Objective Type Questions
Exercise 12. MCQ
12. The given triangle is ___.
a) a right-angled triangle b) an equilateral triangle c) a scalene triangle d) an obtuse angled triangle Exercise 13. MCQ
13. If all angles of a triangle are less than a right angle, then it is called ___.
a) an obtuse angled triangle b) a right-angled triangle c) an isosceles right angled triangle d) an acute angled triangle Exercise 14. MCQ
14. If two sides of a triangle are 5 cm and 9 cm, then the third side is ___.
a) 5 cm b) 3 cm c) 4 cm d) 14 cm Exercise 15. MCQ
15. The angles of a right-angled triangle are:
a) acute, acute, obtuse b) acute, right, right c) right, obtuse, acute d) acute, acute, right Exercise 16. MCQ
16. An equilateral triangle is:
a) an obtuse angled triangle b) a right-angled triangle c) an acute angled triangle d) a scalene triangle
4.4 Construction of Perpendicular Lines
4.4.1 Introduction
Have you ever noticed that the wall and floor are always perpendicular to each other? So, to measure our heights, we make use of scale represented on the walls as shown in the figure. In Geometry, to measure the height of figures, we use perpendicular lines. Using a set square, find the height of the given figures. Let us learn to construct perpendicular lines by using set square.
4.4.2 Set Squares
The set squares are two triangle shaped instruments in the Geometry Box. Each of them has a right angle. One set square has the angles 30°, 60°, 90° and the other set square has the angles 45°, 45°, 90°. The perpendicular edges are graduated in centimetres.
Set Squares Have Several Uses:
- To construct the specific angles 30°, 45°, 60°, 90°
- To draw parallel and perpendicular lines
- To measure the height of the shapes
Do You Know?
If the perpendicular from P meets AB at Q, the point Q is called the foot of the perpendicular from P to AB and the symbol "⊥" means "is perpendicular to", that is, PQ ⊥ AB.
Example 5: Construct a line perpendicular to the given line at a point on the line.
Step 1: Draw a line AB and take a point P anywhere on the line.
Step 2: Place the set square on the line in such a way that the vertex which forms right angle coincides with P and one arm of the right angle coincides with the line AB.
Step 3: Draw a line PQ through P along the other arm of the right angle of the set square.
Step 4: The line PQ is perpendicular to the line AB at P. That is, PQ ⊥ AB and ∠APQ = ∠BPQ = 90°.
Example 6: Construct a line perpendicular to the given line through a point above it.
Step 1: Draw a line PQ. Take a point X anywhere above the line PQ.
Step 2: Place one of the arms of the right angle of a set square along the line PQ and the other arm of its right angle touches the point X.
Step 3: Draw a line through the point X meeting PQ at Y.
Step 4: The line XY is perpendicular to the line PQ at Y. That is, XY ⊥ PQ.
4.5 Construction of Parallel Lines
Place a scale on a paper and draw lines along both the edges of the scale as shown. Place the set square at two different points on l₁ and find the distance between l₁ and l₂. Are they equal? Yes. Thus, the perpendicular distance between a set of parallel lines remains the same.
Note
Parallel line segments need not be of equal length.
Think
Identify the parallel lines in English alphabets (Capital Letters) and list the letters.
Examples: E, W
Example 7: Draw a line segment AB equal to 6.5 cm and mark a point M above it. Through M draw a line parallel to AB.
Step 1: Draw a line. Mark two points A and B on the line such that AB = 6.5 cm. Mark a point M anywhere above the line.
Step 2: Place the set square below AB in such a way that one of the edges that form a right angle lies along AB. Place the scale along the other edge of the set square.
Step 3: Holding the scale firmly, slide the set square along the edge of the scale until the other edge of the set square reaches the point M. Through M draw a line as shown.
Step 4: The line MN is parallel to AB. That is, MN ∥ AB.
Example 8: Draw a line and mark a point R at a distance of 4.8 cm above the line. Through R draw a line parallel to the given line.
Step 1: Using a scale draw a line AB and mark a point Q on the line.
Step 2: Place the set square in such a way that the vertex of the right angle coincides with Q and one of the edges of right angle lies along AB. Mark the point R such that QR = 4.8 cm.
Step 3: Place the scale and the set square as shown in the figure.
Step 4: Hold the scale firmly and slide the set square along the edge of the scale until the other edge touches the point R. Draw a line RS through R.
Step 5: The line RS is parallel to AB. That is, RS ∥ AB.
Example 9: Draw a line segment PQ equal to 12 cm. Mark two points M, N at a distance of 5 cm above the line segment PQ. Through M and N draw a line parallel to PQ.
Step 1: Using a scale, draw a line segment PQ equal to 12 cm. Mark two points A and B on the line segments.
Step 2: Using the set square as shown, mark points M and N such that AM = BN = 5 cm.
Step 3: Using the scale, join M and N. MN is parallel to PQ. That is, MN ∥ PQ.
GEOMETRY ICT CORNER
Step 1: Open the Browser and type the URL Link given below (or) Scan the QR Code. GeoGebra worksheet named "Geometry" will open. The worksheet contains three activities: 1. Types of triangles, 2. Perpendicular line construction and 3. Parallel line construction. In the first activity move the sliders or enter the angle to change the angles of the triangle and check what type of triangle it is and compare with the angles.
Step 2: In the second and third activity you can learn how to draw Perpendicular and parallel lines through a video.
Browse in the link: Geometry: https://ggbm.at/dPXHSSTF or Scan the QR Code.
Exercise 4.2
Exercise 4.2 Q1. Draw perpendicular at a point
Draw a line segment AB equal to 7 cm and mark a point P on it. Draw a line perpendicular to the given line segment at P.
Exercise 4.2 Q2. Draw perpendicular through a point
Draw a line segment LM equal to 6.5 cm and take a point P not lying on it. Using a set square construct a line perpendicular to LM through P.
Exercise 4.2 Q3. Check parallel lines
Find the distance between the given lines using a set square at two different points on each of the pairs of lines and check whether they are parallel.
Exercise 4.2 Q4. Draw parallel line at 5 cm
Draw a line segment measuring 7.8 cm. Mark a point B above it at a distance of 5 cm. Through B draw a line parallel to the given line segment.
Exercise 4.2 Q5. Draw parallel line through R
Draw a line and mark a point R below it at a distance of 5.4 cm. Through R draw a line parallel to the given line.
Exercise 4.3 – Miscellaneous Practice Problems
Exercise 4.3 Q1. Isosceles right angled triangle angles
1. What are the angles of an isosceles right angled triangle?
Exercise 4.3 Q2. MCQ – Describe the triangle
2. Which of the following correctly describes the given triangle?
a) It is a right isosceles triangle. b) It is an acute isosceles triangle. c) It is an obtuse isosceles triangle. d) It is an obtuse scalene triangle. Exercise 4.3 Q3. MCQ – Not possible
3. Which of the following is not possible?
a) An obtuse isosceles triangle b) An acute isosceles triangle c) An obtuse equilateral triangle d) An acute equilateral triangle Exercise 4.3 Q4. Find other angles
4. If one angle of an isosceles triangle is 124°, then find the other angles.
Exercise 4.3 Q5. Square diagonal triangles
5. The diagram shows a square ABCD. If the line segment joins A and C, then mention the type of triangles so formed.
Exercise 4.3 Q6. Perpendiculars at ends of AB
6. Draw a line segment AB of length 6 cm. At each end of this line segment AB, draw a line perpendicular to the line AB. Are these lines parallel?
Challenge Problems
Challenge Q7. Triangle with 90°, 90°, 0°
7. Is a triangle possible with the angles 90°, 90° and 0°? Why?
Challenge Q8. True statements
8. Which of the following statements is true? Why?
(a) Every equilateral triangle is an isosceles triangle.
(b) Every isosceles triangle is an equilateral triangle.
Challenge Q9. Isosceles triangle with 70°
9. If one angle of an isosceles triangle is 70°, then find the possibilities for the other two angles.
Challenge Q10. Sides of isosceles triangle
10. Which of the following can be the sides of an isosceles triangle?
a) 6 cm, 3 cm, 3 cm b) 5 cm, 2 cm, 2 cm c) 6 cm, 6 cm, 7 cm d) 4 cm, 4 cm, 8 cm Challenge Q11. Identify triangles from figure
11. Study the given figure and identify the following triangles.
(a) Equilateral triangle:
(b) Isosceles triangles:
(c) Scalene triangles:
(d) Acute triangles:
(e) Obtuse triangles:
(f) Right triangles:
Challenge Q12. Find the third side
12. Two sides of the triangle are given in the table. Find the third side of the triangle.
| S.No | Side 1 | Side 2 | The length of the third side (any three measures) |
| 1 | 7 cm | 4 cm | |
| 2 | 8 cm | 8 cm | |
| 3 | 7.5 cm | 3.5 cm | |
| 4 | 10 cm | 14 cm | |
Challenge Q13. Complete the triangle types table
13. Complete the following table:
| Types of Triangle / Its Angles | Acute Angled Triangle | Right Angled Triangle | Obtuse Angled Triangle |
| Any two angles | Always acute angles | | Always acute angles |
| Third angle | | Right angle | |
Summary
- A closed figure formed by three-line segments is called a triangle.
- A triangle has 3 sides, 3 angles and 3 vertices.
- Based on the sides of triangles, we can classify triangles into 3 types as scalene triangle, isosceles triangle and equilateral triangle.
- Based on the angles of triangles, we can classify triangles into 3 types as acute angled triangle, right angled triangle and obtuse angled triangle.
- In a triangle, the sum of any two sides is greater than the third side. This is known as Triangle Inequality property.
- Sum of three angles of a triangle is 180°.
- Parallel and Perpendicular lines can easily be drawn using set squares.
- The distance between a set of parallel lines always remains the same.
Book-back Exercises — Questions & Answers
Answers generated strictly from the Samacheer Kalvi textbook content. Attempt each question first, then click "Show Answer" to check.
Exercise 4.1
1. Fill in the Blanks
1a. Every triangle has at least ______ acute angles.
Answer: Two (2)
Explanation: A triangle can have three acute angles, but cannot have more than one right angle or an obtuse angle, so at least two angles are always acute.
1b. A triangle in which none of the sides equal is called ______.
Answer: Scalene Triangle
Explanation: If three sides of a triangle are different in lengths, then it is called a Scalene Triangle.
1c. In an isosceles triangle ______ angles are equal.
Answer: Two (2)
Explanation: If the measure of two angles are equal, then two sides are equal — this is the property of an isosceles triangle.
1d. The sum of three angles of a triangle is ______.
Answer: 180°
Explanation: From the observations in the chapter: Sum of three angles of a triangle is 180°.
1e. A right-angled triangle with two equal sides is called ______.
Answer: Isosceles right-angled triangle
Explanation: A right-angled triangle that also has two equal sides is an isosceles right-angled triangle.
2. Match the Following
2. Match the Following
| Column A | Column B |
|---|
| (1) No sides are equal | Isosceles triangle |
| (2) One right angle | Scalene triangle |
| (3) One obtuse angle | Right angled triangle |
| (4) Two sides of equal length | Equilateral triangle |
| (5) All sides are equal | Obtuse angled triangle |
Answers: 1 – Scalene triangle, 2 – Right angled triangle, 3 – Obtuse angled triangle, 4 – Isosceles triangle, 5 – Equilateral triangle
3. In Triangle ABC, name the
a) Three sides: ___, ___, ___
b) Three Angles: ___, ___, ___
c) Three Vertices: ___, ___, ___
Answer:
a) Three sides: AB, BC, CA
b) Three Angles: ∠A (∠CAB), ∠B (∠ABC), ∠C (∠BCA)
c) Three Vertices: A, B, C
4. Classify the given triangles based on its sides as scalene, isosceles or equilateral.
Answer: Triangles are classified as follows based on sides:
• If all three sides are equal → Equilateral Triangle
• If any two sides are equal → Isosceles Triangle
• If all three sides are different → Scalene Triangle
(Refer to the figures in the textbook and apply these definitions.)
5. Classify the given triangles based on its angles as acute angled, right angled or obtuse angled.
Answer: Triangles are classified based on angles as follows:
• If all three angles are acute (between 0° and 90°) → Acute Angled Triangle
• If one angle is exactly 90° → Right-Angled Triangle
• If one angle is obtuse (between 90° and 180°) → Obtuse Angled Triangle
(Apply these definitions to the figures given in the textbook.)
6. Classify the following triangles based on its sides and angles.
Answer: Each triangle should be classified both by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). Measure the sides and angles of each figure in the textbook and apply the respective definitions from the chapter.
7. Can a triangle be formed with the following sides? If yes, name the type of triangle.
(1) 8 cm, 6 cm, 4 cm (2) 10 cm, 8 cm, 5 cm (3) 6.2 cm, 1.3 cm, 3.5 cm
(4) 6 cm, 6 cm, 4 cm (5) 3.5 cm, 3.5 cm, 3.5 cm (6) 9 cm, 4 cm, 5 cm
Answer: Using Triangle Inequality property (sum of two smaller sides must be greater than the third side):
(1) 6 + 4 = 10 > 8 → Yes, Scalene triangle
(2) 8 + 5 = 13 > 10 → Yes, Scalene triangle
(3) 1.3 + 3.5 = 4.8 < 6.2 → No, triangle cannot be formed
(4) 6 + 4 = 10 > 6 → Yes, Isosceles triangle
(5) All sides equal → Yes, Equilateral triangle
(6) 4 + 5 = 9, which is NOT greater than 9 → No, triangle cannot be formed
8. Can a triangle be formed with the following angles? If yes, name the type of triangle.
(1) 60°, 60°, 60° (2) 90°, 55°, 35° (3) 60°, 40°, 42°
(4) 60°, 90°, 90° (5) 70°, 60°, 50° (6) 100°, 50°, 30°
Answer: Sum of three angles must equal 180°:
(1) 60 + 60 + 60 = 180° → Yes, Acute angled (Equilateral) triangle
(2) 90 + 55 + 35 = 180° → Yes, Right angled triangle
(3) 60 + 40 + 42 = 142° ≠ 180° → No, triangle cannot be formed
(4) 60 + 90 + 90 = 240° ≠ 180° → No, triangle cannot be formed
(5) 70 + 60 + 50 = 180° → Yes, Acute angled triangle
(6) 100 + 50 + 30 = 180° → Yes, Obtuse angled triangle
9. Two angles of the triangles are given. Find the third angle.
(1) 80°, 60° (2) 75°, 35° (3) 52°, 68° (4) 50°, 90° (5) 120°, 30° (6) 55°, 85°
Answer: Since the sum of three angles of a triangle is 180°, the third angle = 180° − (sum of two given angles):
(1) 180 − (80 + 60) = 180 − 140 = 40°
(2) 180 − (75 + 35) = 180 − 110 = 70°
(3) 180 − (52 + 68) = 180 − 120 = 60°
(4) 180 − (50 + 90) = 180 − 140 = 40°
(5) 180 − (120 + 30) = 180 − 150 = 30°
(6) 180 − (55 + 85) = 180 − 140 = 40°
10. I am a closed figure with each of my three angles is 60°. Who am I?
Answer: Equilateral Triangle.
If the measure of three angles are equal and each angle measures 60°, then three sides are also equal — this is an Equilateral Triangle.
11. Using the given information, write the type of triangle in the table given below.
Answer:
1. 60°, 40°, 80° → Acute angled triangle / Scalene triangle
2. 50°, 50°, 80° → Acute angled triangle / Isosceles triangle
3. 45°, 45°, 90° → Right angled triangle / Isosceles triangle
4. 55°, 45°, 80° → Acute angled triangle / Scalene triangle
5. 75°, 35°, 70° → Acute angled triangle / Scalene triangle
6. 60°, 30°, 90° → Right angled triangle / Scalene triangle
7. 25°, 64°, 91° → Obtuse angled triangle / Scalene triangle
8. 120°, 30°, 30° → Obtuse angled triangle / Isosceles triangle
Objective Type Questions
12. The given triangle is ______.
a) a right-angled triangle b) an equilateral triangle c) a scalene triangle d) an obtuse angled triangle
Answer: b) an equilateral triangle
Explanation: If three sides of a triangle are equal in length, then it is called an Equilateral Triangle, and each angle measures 60°.
13. If all angles of a triangle are less than a right angle, then it is called ______.
a) an obtuse angled triangle b) a right-angled triangle c) an isosceles right angled triangle d) an acute angled triangle
Answer: d) an acute angled triangle
Explanation: If three angles of a triangle are acute angles (between 0° and 90°), then it is called an Acute Angled Triangle.
14. If two sides of a triangle are 5 cm and 9 cm, then the third side is ______.
a) 5 cm b) 3 cm c) 4 cm d) 14 cm
Answer: a) 5 cm
Explanation: The third side must be greater than the difference (9 − 5 = 4) and less than the sum (9 + 5 = 14) of the two sides. Only 5 cm satisfies this condition among the options (3 cm and 4 cm are not greater than 4, and 14 cm is not less than 14).
15. The angles of a right-angled triangle are
a) acute, acute, obtuse b) acute, right, right c) right, obtuse, acute d) acute, acute, right
Answer: d) acute, acute, right
Explanation: A triangle cannot have more than one right angle or an obtuse angle. A right-angled triangle has exactly one right angle (90°) and the other two angles are acute.
16. An equilateral triangle is
a) an obtuse angled triangle b) a right-angled triangle c) an acute angled triangle d) a scalene triangle
Answer: c) an acute angled triangle
Explanation: In an equilateral triangle, each angle measures 60°, which is an acute angle (between 0° and 90°), so it is an acute angled triangle.
Exercise 4.2
1. Draw a line segment AB equal to 7 cm and mark a point P on it. Draw a line perpendicular to the given line segment at P.
Answer: Step 1: Draw a line segment AB = 7 cm and mark a point P on it. Step 2: Place the set square so that the vertex forming the right angle coincides with P and one arm lies along AB. Step 3: Draw a line PQ along the other arm. Step 4: PQ is perpendicular to AB at P, i.e., PQ ⊥ AB and ∠APQ = ∠BPQ = 90°.
2. Draw a line segment LM equal to 6.5 cm and take a point P not lying on it. Using a set square construct a line perpendicular to LM through P.
Answer: Step 1: Draw a line segment LM = 6.5 cm and mark point P above it. Step 2: Place one arm of the right angle of the set square along LM and adjust so the other arm touches point P. Step 3: Draw a line through P meeting LM at Y. Step 4: The line PY is perpendicular to LM, i.e., PY ⊥ LM.
3. Find the distance between the given lines using a set square at two different points on each of the pairs of lines and check whether they are parallel.
Answer: Place the set square at two different points between the given lines and measure the perpendicular distance at each point. If the perpendicular distance between the two lines remains the same at both points, then the lines are parallel. The perpendicular distance between a set of parallel lines always remains the same.
4. Draw a line segment measuring 7.8 cm. Mark a point B above it at a distance of 5 cm. Through B draw a line parallel to the given line segment.
Answer: Step 1: Draw a line segment AB = 7.8 cm. Mark a point Q on it. Step 2: Place the set square with the right angle vertex at Q and one edge along AB; mark point B such that QB = 5 cm. Step 3: Place the scale along the set square's other edge, hold it firmly and slide the set square until it touches B. Step 4: Draw a line through B parallel to the given line segment.
5. Draw a line and mark a point R below it at a distance of 5.4 cm. Through R draw a line parallel to the given line.
Answer: Step 1: Draw a line AB and mark a point Q on it. Step 2: Using the set square, mark point R below the line such that QR = 5.4 cm. Step 3: Place the scale and set square as shown, hold the scale firmly and slide the set square until its edge reaches point R. Step 4: Draw a line RS through R. RS is parallel to the given line AB.
Exercise 4.3 — Miscellaneous Practice Problems
1. What are the angles of an isosceles right angled triangle?
Answer: The angles of an isosceles right-angled triangle are 90°, 45°, and 45°. One angle is a right angle (90°) and the other two angles are equal and acute, each measuring 45°, since the sum of all three angles must be 180°.
2. Which of the following correctly describes the given triangle?
(a) It is a right isosceles triangle.
(b) It is an acute isosceles triangle.
(c) It is an obtuse isosceles triangle.
(d) It is an obtuse scalene triangle.
Answer: (c) It is an obtuse isosceles triangle.
A triangle with one obtuse angle and two equal sides is an obtuse isosceles triangle.
3. Which of the following is not possible?
(a) An obtuse isosceles triangle
(b) An acute isosceles triangle
(c) An obtuse equilateral triangle
(d) An acute equilateral triangle
Answer: (c) An obtuse equilateral triangle is not possible.
In an equilateral triangle, each angle is 60°, which is always acute. A triangle cannot have more than one obtuse angle, and an equilateral triangle can never have an obtuse angle.
4. If one angle of an isosceles triangle is 124°, then find the other angles.
Answer: Since the sum of three angles of a triangle is 180°, and the triangle is isosceles with one angle = 124° (obtuse), the remaining two angles are equal. Remaining = 180° − 124° = 56°. Each of the two equal angles = 56° ÷ 2 = 28°. The other two angles are 28° and 28°.
5. The diagram shows a square ABCD. If the line segment joins A and C, then mention the type of triangles so formed.
Answer: When the diagonal AC is drawn in square ABCD, two triangles ABC and ACD are formed. Each triangle has angles 45°, 45°, and 90°. These are isosceles right-angled triangles (right-angled and isosceles).
6. Draw a line segment AB of length 6 cm. At each end of this line segment AB, draw a line perpendicular to the line AB. Are these lines parallel?
Answer: Yes, the two perpendicular lines drawn at each end of AB are parallel to each other. The perpendicular distance between a set of parallel lines always remains the same, and both perpendiculars are at 90° to AB maintaining equal distance throughout.
Challenge Problems
7. Is a triangle possible with the angles 90°, 90° and 0°? Why?
Answer: No, a triangle is not possible with angles 90°, 90°, and 0°. The sum of these angles = 90 + 90 + 0 = 180°, but 0° is not a valid angle for a triangle. Also, a triangle cannot have more than one right angle, as stated in the chapter: "A triangle can have three acute angles, but cannot have more than one right angle or an obtuse angle."
8. Which of the following statements is true? Why?
(a) Every equilateral triangle is an isosceles triangle.
(b) Every isosceles triangle is an equilateral triangle.
Answer: Statement (a) is true. Every equilateral triangle is an isosceles triangle because an equilateral triangle has all three sides equal, which satisfies the condition that at least two
📝 Summary
- A triangle is a closed figure formed by three line segments, and it has 3 sides, 3 angles, and 3 vertices. At least 3 sticks are required to form a closed figure.
- The basic elements of a triangle ABC are its sides (AB, BC, CA), its angles (∠A, ∠B, ∠C), and its vertices (A, B, C), where the vertices are the points of intersection of two sides.
- The sum of the three angles of any triangle is always 180°, and if all three angles are equal, each angle measures 60°.
- Triangles can be classified based on their sides and angles: if all angles are different then all sides are different; if two angles are equal then two sides are equal; and if all three angles are equal then all three sides are equal.
- Triangles are widely used in real-life construction such as bridges, buildings, cell-phone towers, aeroplane wings, and pitched roofs, because the triangular shape provides stiffness, resulting in rigid and strong structures.