Exercise 7.2 — Properties of Triangles
Some more properties of triangles.
The Isosceles Triangle Theorem — Foundation of Exercise 7.2
Before diving into the exercise questions, it is essential to understand the core theorem that drives every single problem in Exercise 7.2. This theorem appears in Theorem 7.2 of your Class 9 NCERT / Telangana / Andhra Pradesh textbook.
Angles opposite to equal sides of an isosceles triangle are equal.
Conversely: If two angles of a triangle are equal, the sides opposite to them are also equal.
How the Theorem is Proved
In isosceles △ABC where AB = AC, the goal is to prove that ∠B = ∠C. The construction step is key: draw the angle bisector AD of ∠A so that it meets BC at D. This creates two smaller triangles — △ABD and △ACD — which can be compared.
Congruence Rules Used in This Exercise
All five questions in Exercise 7.2 rely on either the SAS rule or the AAS rule to prove triangle congruence. Understanding which rule to apply is the key skill tested in board exams.
| Rule | Full Name | What Must Match | Used In |
|---|---|---|---|
| SAS | Side–Angle–Side | Two sides & the included angle between them | Theorem proof, Q1, Q2 |
| AAS | Angle–Angle–Side | Two angles & a non-included side | Q3, Q4 |
| CPCT | Corresponding Parts of Congruent Triangles | Used after proving congruence to conclude equal sides/angles | All questions |
Question 1 — Angle Bisectors of an Isosceles Triangle
Since AB = AC, the base angles are equal: ∠B = ∠C. The bisectors of ∠B and ∠C each cut these equal angles in half, creating two new equal half-angles. Point O is where these bisectors meet.
Question 2 — Perpendicular Bisector Proves Isosceles Triangle
Here the direction is reversed — instead of starting with an isosceles triangle, we are given a perpendicular bisector and asked to prove the triangle is isosceles. The perpendicular bisector gives us two facts: AD ⊥ BC (right angles) and BD = DC (D is the midpoint of BC).
Question 3 — Equal Altitudes Prove Equal Sides
An altitude is a perpendicular from a vertex to the opposite side. Since AB = AC, we already know ∠B = ∠C. The altitudes are drawn to the equal sides, not to the base — so we compare triangles that share BC as a common side.
Question 4 — Equal Altitudes Prove the Triangle is Isosceles
This is the reverse of Question 3 — we are now given that the two altitudes are equal and asked to conclude the triangle is isosceles. Notice both triangles share the common angle ∠A.
Question 5 — Two Isosceles Triangles on the Same Base
Both triangles sit on base BC, with A above BC and D below (or both on the same side). Since each is isosceles, we get two pairs of equal base angles. The trick is to add them together.
Quick Reference — All 5 Questions at a Glance
| Question | Given | To Prove / Show | Method | Key Step |
|---|---|---|---|---|
| Q1(i) | AB = AC; bisectors of ∠B, ∠C meet at O | OB = OC | Isosceles theorem + converse | ∠OBC = ∠OCB → OB = OC |
| Q1(ii) | Above + OB = OC | AO bisects ∠A | SAS → CPCT | △OAB ≅ △OAC → ∠OAB = ∠OAC |
| Q2 | AD ⊥ BC; BD = DC | AB = AC | SAS → CPCT | △ADB ≅ △ADC → AB = AC |
| Q3 | AB = AC; BD ⊥ AC; CE ⊥ AB | BD = CE | AAS → CPCT | △BEC ≅ △CDB → CE = BD |
| Q4 | BD ⊥ AC; CE ⊥ AB; BD = CE | AB = AC | AAS → CPCT | △ABD ≅ △ACE → AB = AC |
| Q5 | AB = AC; DB = DC; same base BC | ∠ABD = ∠ACD | Adding angle equations | ∠ABC + ∠DBC = ∠ACB + ∠DCB |
Common Mistakes to Avoid in Exercise 7.2
- Not stating the congruence rule: Writing "△ABD ≅ △ACD" without specifying "(by SAS)" is incomplete and loses marks in board exams. Always name the rule — SAS, AAS, SSS, ASA, or RHS.
- Confusing SAS and AAS: SAS requires the angle to be between the two sides. If the equal side is not between the two angles, it is AAS — not SAS. Getting this wrong changes the validity of the proof.
- Forgetting to justify the "common side": In every proof where AD = AD or BC = CB, write the reason "(common side)" explicitly. Examiners check for this.
- Applying the isosceles theorem without mentioning which sides are equal: Always clearly state "AB = AC (given)" before writing "∠B = ∠C". The theorem requires the given condition to be written first.
- Skipping the construction in Q1: The join AO must be stated before using it in the proof. Never use a line in a proof that has not been constructed or given.
What Exercise 7.2 Prepares You For
The isosceles triangle properties proved in Exercise 7.2 are foundational for several topics in Class 9 and beyond. In Exercise 7.3, you use angle-side relationships to compare sides and angles of unequal triangles — a direct extension of the isosceles triangle ideas here.
For CBSE, Telangana, and Andhra Pradesh board examinations, questions from Exercise 7.2 regularly appear as 3-mark or 4-mark proofs. The most commonly tested problems are Q1, Q3, and Q5. Mastering the step-by-step format — Given → To Prove → Construction → Proof → Conclusion — is as important as knowing the content itself.
In Class 10, these congruence techniques reappear in Chapter 6 (Similar Triangles) and in coordinate geometry proofs. The ability to identify which congruence rule applies — a skill sharpened by Exercise 7.2 — is tested throughout Classes 9 and 10.
→ Exercise 7.1 — Congruence of Triangles
→ Exercise 7.3 — Inequalities in Triangles
→ Introduction to Triangles — Chapter Overview