Geometry (Term 2) — Exercise Solutions
How to Use This Page
This page covers every question from Exercise 4.1 of the TN Class 6 Maths Term 2 Geometry chapter, plus additional Samacheer practice problems. Each solution follows the same format:
- Problem statement is repeated so you can work without a textbook
Start with the Key Concepts recap if you need a quick refresher, then dive into the exercise solutions. Common mistakes and the exam checklist at the end will help you avoid losing marks.
Key Concepts (Quick Recap)
Classification by Sides
| Type | Condition | Example |
|---|---|---|
| Equilateral | All three sides equal | 5 cm, 5 cm, 5 cm |
| Isosceles | Exactly two sides equal | 6 cm, 6 cm, 4 cm |
| Scalene | No sides equal | 3 cm, 4 cm, 5 cm |
Classification by Angles
| Type | Condition | Example |
|---|---|---|
| Acute-angled | All three angles < 90° | 60°, 70°, 50° |
| Right-angled | One angle = 90° | 90°, 50°, 40° |
| Obtuse-angled | One angle > 90° | 110°, 40°, 30° |
Angle Sum Property
The sum of the three interior angles of any triangle is always 180°.
∠A + ∠B + ∠C = 180°
This is the single most-used property in this chapter. Every "find the third angle" question uses it.
Triangle Inequality
The sum of any two sides of a triangle must be greater than the third side.
- a + b > c
- b + c > a
- a + c > b
If even one inequality fails, the three lengths cannot form a triangle.
Perpendicular Lines
Two lines are perpendicular when they meet at a 90° angle. A perpendicular from a point to a line is the shortest distance from that point to the line.
Exercise 4.1 — Solutions
Fill in the Blanks
Q1 (a). Every triangle has at least \_\_\_\_ acute angles.
Q1 (b). A triangle in which none of the sides are equal is called \_\_\_\_.
Q1 (c). In an isosceles triangle, \_\_\_\_ angles are equal.
Q1 (d). The sum of three angles of a triangle is \_\_\_\_.
Q1 (e). A right-angled triangle with two equal sides is called \_\_\_\_.
Match the Following
Q2. Match the items in Column I with Column II:
| Column I | Column II | Match |
|---|---|---|
| (i) No sides are equal | Scalene triangle | ✓ |
| (ii) One right angle | Right-angled triangle | ✓ |
| (iii) One obtuse angle | Obtuse angled triangle | ✓ |
| (iv) Two sides of equal length | Isosceles triangle | ✓ |
| (v) All sides are equal | Equilateral triangle | ✓ |
- (i) No sides are equal → Scalene triangle — A triangle where all sides have different lengths is scalene.
- (ii) One right angle → Right-angled triangle — A triangle with one 90° angle is right-angled.
- (iii) One obtuse angle → Obtuse angled triangle — A triangle with one angle greater than 90° is obtuse angled.
- (iv) Two sides of equal length → Isosceles triangle — A triangle with exactly two equal sides is isosceles.
- (v) All sides are equal → Equilateral triangle — A triangle where all three sides are equal is equilateral.
Naming Triangle Components
Q3. In ∆ABC, name: (i) three sides (ii) three angles (iii) three vertices.
A triangle is named by its three vertices. From the name ∆ABC, the vertices are the letters A, B, and C. The sides connect pairs of vertices, and the angles are at each vertex.
(i) Three sides: AB, BC, CA
(ii) Three angles: ∠A, ∠B, ∠C
(iii) Three vertices: A, B, C
Classification by Sides
Q4. Classify the following triangles by their sides:
(i) Sides: 5 cm, 5 cm, 5 cm
(ii) Sides: 7 cm, 5 cm, 3 cm
(iii) Sides: 6 cm, 6 cm, 4 cm
(iv) Sides: 8 cm, 5 cm, 3 cm
Classification by Angles
Q5. Classify the following triangles by their angles:
(i) Angles: 60°, 60°, 60°
(ii) Angles: 90°, 45°, 45°
(iii) Angles: 120°, 30°, 30°
(iv) Angles: 80°, 55°, 45°
Classification by Both Sides and Angles
Q6. Classify the following triangles by both sides and angles:
(i) Sides: 7 cm, 7 cm, 5 cm; Angles: 70°, 70°, 40°
(ii) Sides: 8 cm, 6 cm, 5 cm; Angles: 90°, 55°, 35°
(iii) Sides: 6 cm, 6 cm, 9 cm; Angles: 100°, 40°, 40°
(iv) Sides: 5 cm, 5 cm, 5√2 cm; Angles: 90°, 45°, 45°
(v) Sides: 4 cm, 4 cm, 4 cm; Angles: 60°, 60°, 60°
(vi) Sides: 7 cm, 6 cm, 4 cm; Angles: 100°, 45°, 35°
Triangle Inequality (Can a Triangle Be Formed?)
Q7. Can a triangle be formed with the given side lengths? If yes, classify by sides.
(i) 8, 6, 4
- 6 + 4 = 10 > 8 ✓
- 8 + 4 = 12 > 6 ✓
- 8 + 6 = 14 > 4 ✓
All inequalities hold. All three sides are different.
(ii) 10, 8, 5
- 8 + 5 = 13 > 10 ✓
- 10 + 5 = 15 > 8 ✓
- 10 + 8 = 18 > 5 ✓
All inequalities hold. All three sides are different.
(iii) 6.2, 1.3, 3.5
- 1.3 + 3.5 = 4.8 < 6.2 ✗
The sum of the two smaller sides (4.8) is less than the largest side (6.2). The triangle inequality fails.
(iv) 6, 6, 4
- 6 + 4 = 10 > 6 ✓
- 6 + 4 = 10 > 6 ✓
- 6 + 6 = 12 > 4 ✓
All inequalities hold. Two sides are equal (6, 6).
(v) 3.5, 3.5, 3.5
- 3.5 + 3.5 = 7 > 3.5 ✓
- 3.5 + 3.5 = 7 > 3.5 ✓
- 3.5 + 3.5 = 7 > 3.5 ✓
All inequalities hold. All three sides are equal.
(vi) 9, 4, 5
- 4 + 5 = 9
The sum of the two smaller sides equals the largest side (9), but the inequality requires the sum to be strictly greater than the third side. Since 4 + 5 = 9 (not > 9), the triangle inequality fails.
Angle Sum Verification (Can a Triangle Be Formed with Given Angles?)
Q8. Can a triangle be formed with the given angles? If yes, classify by angles.
(i) 60°, 60°, 60°
All three angles are less than 90°.
(ii) 90°, 55°, 35°
One angle equals 90°.
(iii) 60°, 40°, 42°
The sum is not 180°, so these angles cannot form a triangle.
(iv) 60°, 90°, 90°
The sum is not 180°. Also, a triangle cannot have two right angles.
(v) 70°, 60°, 50°
All three angles are less than 90°.
(vi) 100°, 50°, 30°
One angle is greater than 90° (100°).
Finding the Third Angle
Q9. Two angles of a triangle are given. Find the third angle.
(i) 80°, 60°, ?
(ii) 52°, 68°, ?
(iii) 75°, 35°, ?
(iv) 50°, 90°, ?
(v) 120°, 30°, ?
(vi) 55°, 85°, ?
Riddle
Q10. I am a closed figure with three sides. All my angles are 60°. Who am I?
Multiple Choice Questions
Q12. The triangle shown has all three sides equal. What type of triangle is it?
(a) Isosceles (b) Equilateral (c) Scalene (d) Right-angled
Q13. If all the angles of a triangle are less than a right angle, the triangle is called:
(a) Obtuse-angled (b) Right-angled (c) Scalene (d) Acute-angled
Q14. Two sides of a triangle are 5 cm and 9 cm. The third side cannot be:
(a) 5 cm (b) 8 cm (c) 4 cm (d) 13 cm
- 5 + 9 > x → x < 14
- 5 + x > 9 → x > 4
- 9 + x > 5 → x > −4 (always true)
So x must satisfy: 4 < x < 14.
Now check the options:
- x = 4 cm: 5 + 4 = 9, which is NOT > 9. Fails the inequality.
- x = 5 cm: 5 + 5 = 10 > 9 ✓, 5 + 9 = 14 > 5 ✓. Valid.
- x = 8 cm: 5 + 8 = 13 > 9 ✓, 8 + 9 = 17 > 5 ✓. Valid.
- x = 13 cm: 5 + 9 = 14 > 13 ✓, 5 + 13 = 18 > 9 ✓. Valid.
The value x = 4 cm fails the triangle inequality.
Q15. The angles of a right-angled triangle are:
(a) One right angle, two obtuse (b) One right angle, two acute (c) One right angle, one acute, one obtuse (d) Acute, acute, right
Q16. An equilateral triangle is also:
(a) A right-angled triangle (b) An obtuse-angled triangle (c) An acute-angled triangle (d) None of these
Additional Questions
AQ1. Identify the type of triangle:
(a) ∆PQR with ∠Q = 90°
(b) ∆ABC with ∠B = 90° and AB = BC
AQ2. Classify each triangle:
(a) AB = BC
(b) PQ = QR = RP
(c) ∠B = 90°
(d) EF = 3 cm, FG = 4 cm, GE = 3 cm
AQ3. ∆ABC has AB = BC = CA = 5 cm. Find all angles.
By the Angle Sum Property: ∠A + ∠B + ∠C = 180°
Since ∠A = ∠B = ∠C (equal sides imply equal opposite angles): 3 × ∠A = 180° ∠A = 180° / 3 = 60°
Therefore, ∠A = ∠B = ∠C = 60°
AQ4. ∆PQR has ∠P = ∠Q = ∠R = 60°. What type of triangle is it?
AQ5. In ∆ABC, AB = BC and ∠A = 50°. Find ∠C.
The equal sides are AB and BC. The angles opposite these sides are:
- Angle opposite AB is ∠C
- Angle opposite BC is ∠A
Therefore, ∠C = ∠A = 50°
AQ6. What kind of triangle can you form with matchsticks of the following counts?
(a) 3 sticks of equal length
(b) 4 sticks
(c) 5 sticks (two of length 2, one of length 1)
2 + 1 = 3 > 2 ✓, 2 + 2 = 4 > 1 ✓
(d) 6 sticks of equal length
2 + 2 = 4 > 2 ✓
AQ7. A shopkeeper bought an item for ₹4,500 and sold it for ₹4,800. Find the profit.
Profit = SP − CP = 4,800 − 4,500 = ₹300
AQ8. Draw a perpendicular from a point P to a line segment AB.
1. Draw line segment AB. 2. Mark point P outside the line AB. 3. With P as center and a radius greater than the distance from P to AB, draw an arc that cuts AB at two points. Label them C and D. 4. With C as center and radius more than half of CD, draw an arc below AB. 5. With D as center and the same radius, draw another arc below AB cutting the first arc at point Q. 6. Join P and Q. PQ is perpendicular to AB.
Common Mistakes
1. Forgetting "strictly greater than" in triangle inequality — The sum of two sides must be greater than (not equal to) the third side. 4 + 5 = 9 does NOT form a triangle with side 9.
2. Confusing isosceles with equilateral — An equilateral triangle has all three sides equal; an isosceles triangle has exactly two equal sides. Every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.
3. Wrong angle in isosceles triangles — In an isosceles triangle, the equal angles are opposite the equal sides, not adjacent to them. If AB = BC, then ∠C = ∠A (angles opposite the equal sides).
4. Adding angles incorrectly — Always double-check your addition when using the Angle Sum Property. 75° + 35° = 110°, not 100°. A small arithmetic error gives the wrong third angle.
5. Classifying by only sides or only angles — A complete classification requires both. A triangle can be "Isosceles and Acute-angled" — don't give just one category when the question asks for both.
6. Assuming equal angles mean equal sides for all triangles — Equal angles imply equal opposite sides only when you can confirm the triangle is equilateral. In a general triangle, knowing one angle tells you nothing about side lengths unless additional information is given.
7. Confusing perpendicular and parallel — Perpendicular lines meet at 90°. Parallel lines never meet. Do not mix up these terms in geometry answers.
Exam Checklist
Before submitting your geometry answers, verify each item:
- [ ] Triangle inequality: Did I check all three pairs (a+b>c, b+c>a, a+c>b)?
- [ ] Angle sum: Do my three angles add up to exactly 180°?
- [ ] Classification: Did I classify by both sides AND angles when asked?
- [ ] Isosceles angles: Did I pair equal angles with equal opposite sides?
- [ ] Triangle inequality: Did I use strictly greater than (not ≥)?
- [ ] Perpendicular: Did I mark the right angle symbol (⊥ or 90°) in diagrams?
- [ ] Units: Did I include ° for angles and cm/m for lengths?
- [ ] MCQ strategy: Did I eliminate clearly wrong options before picking an answer?
- [ ] Re-read the question: Does it ask for classification by sides, angles, or both?