Class 9 Mathematics Notes Chapter 7 (Chapter 7) – Examplar Problem (Englisha) Book

Alright class, let's focus on Chapter 7: Triangles, specifically gearing up for competitive government exams using the NCERT Exemplar as our base. This chapter is fundamental for geometry, testing your understanding of shapes, sizes, and logical proof.

Chapter 7: Triangles - Detailed Notes for Government Exam Preparation (Based on NCERT Class 9 Exemplar)

1. Introduction & Basic Concepts

• Triangle: A closed figure formed by three intersecting lines (or three non-collinear points). It has three sides, three angles, and three vertices.
• Classification:
  • By Sides:
    • Scalene: All sides are of different lengths.
    • Isosceles: Any two sides are of equal length.
    • Equilateral: All three sides are of equal length.
  • By Angles:
    • Acute-angled: All angles are less than 90°.
    • Right-angled: One angle is exactly 90°.
    • Obtuse-angled: One angle is greater than 90°.
• Angle Sum Property: The sum of the three interior angles of any triangle is always 180°.
  • Application: If two angles are known, the third can be found.
• Exterior Angle Property: If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
  • Application: Relates an exterior angle to the interior angles not adjacent to it.

2. Congruence of Triangles

• Concept: Two triangles are congruent if they have the same shape and the same size. This means all corresponding sides are equal, and all corresponding angles are equal.
• Notation: Congruence is denoted by the symbol '≅'. If ∆ABC is congruent to ∆PQR, we write ∆ABC ≅ ∆PQR.
• Correspondence: The order of vertices matters. ∆ABC ≅ ∆PQR implies:
  • A ↔ P, B ↔ Q, C ↔ R (Vertices correspond)
  • AB = PQ, BC = QR, AC = PR (Corresponding sides are equal)
  • ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R (Corresponding angles are equal)
• CPCTC: Corresponding Parts of Congruent Triangles are Equal. This is the reason why we prove triangles congruent – to deduce equality of other parts.

3. Criteria for Congruence of Triangles

These are the rules used to prove if two triangles are congruent without checking all six corresponding parts.

• (i) SAS (Side-Angle-Side) Congruence Rule:
  • Statement: Two triangles are congruent if two sides and the included angle (the angle formed by those two sides) of one triangle are equal to the two sides and the included angle of the other triangle.
  • Key: The angle must be between the two specified sides. SSA or ASS is not a congruence rule.
• (ii) ASA (Angle-Side-Angle) Congruence Rule:
  • Statement: Two triangles are congruent if two angles and the included side (the side connecting the vertices of those two angles) of one triangle are equal to two angles and the included side of the other triangle.
• (iii) AAS (Angle-Angle-Side) Congruence Rule:
  • Statement: Two triangles are congruent if any two pairs of angles and one pair of non-included corresponding sides are equal.
  • Derivation: This follows from ASA because if two angles are equal, the third angle must also be equal (Angle Sum Property). So, AAS effectively becomes ASA.
• (iv) SSS (Side-Side-Side) Congruence Rule:
  • Statement: Two triangles are congruent if the three sides of one triangle are equal to the three corresponding sides of the other triangle.
• (v) RHS (Right angle-Hypotenuse-Side) Congruence Rule:
  • Statement: Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one corresponding side of the other triangle.
  • Key: Applicable only to right-angled triangles.

4. Properties of Isosceles Triangles

• Theorem 7.2: Angles opposite to equal sides of an isosceles triangle are equal.
  • If in ∆ABC, AB = AC, then ∠C = ∠B.
• Theorem 7.3 (Converse of Theorem 7.2): The sides opposite to equal angles of a triangle are equal.
  • If in ∆ABC, ∠B = ∠C, then AC = AB.
• Corollary: An equilateral triangle is also equiangular (all angles 60°), and vice-versa.

5. Inequalities in a Triangle

These rules relate sides and angles when they are not equal.

• Theorem 7.6: If two sides of a triangle are unequal, the angle opposite to the longer side is larger (greater).
  • If in ∆ABC, AC > AB, then ∠B > ∠C.
• Theorem 7.7 (Converse of Theorem 7.6): In any triangle, the side opposite to the larger (greater) angle is longer.
  • If in ∆ABC, ∠B > ∠C, then AC > AB.
• Theorem 7.8 (Triangle Inequality): The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
  • AB + BC > AC
  • BC + AC > AB
  • AC + AB > BC
  • Application: Used to determine if given side lengths can form a triangle. The difference between any two sides is less than the third side.

Key Takeaways for Exams:

1. Master the 5 congruence rules (SAS, ASA, AAS, SSS, RHS) – know their conditions precisely.
2. Understand CPCTC and how to apply it after proving congruence.
3. Remember the properties of isosceles triangles (equal sides ↔ equal opposite angles).
4. Know the triangle inequality rules (sum of two sides > third side; larger angle ↔ longer opposite side).
5. Be comfortable applying Angle Sum Property and Exterior Angle Property.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, similar to what you might encounter:

1. In ∆PQR, if ∠R > ∠Q, then which of the following is true?
   (A) QR > PR
   (B) PQ > PR
   (C) PQ < PR
   (D) QR < PR

2. It is given that ∆ABC ≅ ∆FDE, where AB = 5 cm, ∠B = 40°, ∠A = 80°, and FD = 5 cm. What is the value of ∠E?
   (A) 40°
   (B) 50°
   (C) 60°
   (D) 80°

3. In triangles ABC and PQR, AB = AC, ∠C = ∠P, and ∠B = ∠Q. The two triangles are:
   (A) Isosceles but not congruent
   (B) Isosceles and congruent
   (C) Congruent but not isosceles
   (D) Neither congruent nor isosceles

4. Which of the following is NOT a criterion for congruence of triangles?
   (A) SAS
   (B) ASA
   (C) SSA
   (D) SSS

5. In ∆ABC, side AB is produced to D such that BD = BC. If ∠B = 60° and ∠A = 70°, then:
   (A) AD > CD
   (B) AD = CD
   (C) AD < CD
   (D) Cannot be determined

6. If ∆ABC ≅ ∆PQR, then which of the following is not necessarily true?
   (A) BC = QR
   (B) AC = PR
   (C) AB = PQ
   (D) AB = QR

7. In right-angled ∆ABC, ∠B = 90°. Which is the longest side?
   (A) AB
   (B) BC
   (C) AC
   (D) Depends on other angles

8. In ∆ABC, AB = 4 cm, AC = 4 cm, and ∠A = 70°. In ∆PQR, PQ = 4 cm, PR = 4 cm, and ∠R = 70°. By which congruence rule can we potentially prove ∆ABC ≅ ∆PQR?
   (A) SAS
   (B) SSS
   (C) RHS
   (D) The triangles cannot be proven congruent with the given information.

9. The lengths 6 cm, 4 cm, and 1.5 cm are given. Can these lengths form a triangle?
   (A) Yes
   (B) No
   (C) Only if it's a right-angled triangle
   (D) Only if it's an isosceles triangle

10. In ∆XYZ, XY = XZ. Point M is on YZ such that XM bisects ∠YXZ. Which congruence criterion can be used to prove ∆XYM ≅ ∆XZM?
    (A) SSS
    (B) ASA
    (C) SAS
    (D) RHS

Answers to MCQs:

1. (B) PQ > PR (Side opposite the larger angle is longer. Side opposite ∠R is PQ, side opposite ∠Q is PR).
2. (C) 60° (In ∆ABC, ∠C = 180° - 80° - 40° = 60°. Since ∆ABC ≅ ∆FDE, A↔F, B↔D, C↔E. So ∠E = ∠C = 60°).
3. (A) Isosceles but not congruent (∆ABC is isosceles as AB=AC implies ∠C=∠B. Given ∠C=∠P and ∠B=∠Q, so ∠P=∠Q, making ∆PQR isosceles. But we don't have side equality between triangles, only within them. We have ASA similarity, but not necessarily congruence).
4. (C) SSA (Side-Side-Angle is not a congruence criterion unless the angle is 90° - RHS case).
5. (A) AD > CD (In ∆ABC, ∠C = 180-70-60 = 50°. In ∆BCD, BC=BD, so it's isosceles. ∠CBD = 180-60 = 120°. ∠BDC = ∠BCD = (180-120)/2 = 30°. In ∆ACD, ∠ACD = ∠ACB + ∠BCD = 50+30 = 80°. ∠ADC = 30°. ∠CAD = 70°. Since ∠ACD (80°) > ∠CAD (70°) > ∠ADC (30°), the sides opposite are AD > CD > AC).
6. (D) AB = QR (Correspondence is A↔P, B↔Q, C↔R. So AB=PQ, BC=QR, AC=PR. AB=QR is not necessarily true).
7. (C) AC (In a right-angled triangle, the hypotenuse, which is opposite the 90° angle, is the longest side).
8. (D) The triangles cannot be proven congruent (In ∆ABC, the angle 70° is included between equal sides AB and AC. In ∆PQR, the angle 70° is ∠R, which is not included between the equal sides PQ and PR. So SAS doesn't apply, and we lack information for other criteria).
9. (B) No (Check Triangle Inequality: 4 + 1.5 = 5.5, which is NOT greater than 6. The sum of the two smaller sides must be strictly greater than the largest side).
10. (C) SAS (We have XY = XZ (Given Side). XM = XM (Common Side). ∠YXM = ∠ZXM (Angle, as XM bisects ∠YXZ). The angle is included between the sides. So, SAS applies).

Study these notes and practice applying the concepts, especially the congruence criteria and inequalities. Good luck!