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CBSE Class 9 Mathematics Question 8 of 8

Triangles — Question 5

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Question 5

ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C.

Answer

Given :

Δ ABC is an isosceles with AB = AC.

Draw AP ⊥ BC,

∴ ∠APB = ∠APC = 90°

ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C. NCERT Class 9 Mathematics CBSE Solutions.

In Δ APB and Δ APC,

⇒ ∠APB = ∠APC (Each equal to 90°)

⇒ AB = AC (Since ABC is an isosceles triangle)

⇒ AP = AP (Common)

∴ Δ APB ≅ Δ APC (By R.H.S. congruence rule)

We know that,

Corresponding parts of congruent triangle are equal.

∴ ∠B = ∠C (By C.P.C.T.)

Hence, proved that ∠B = ∠C.

Triangles - Interactive Study Notes | Bright Tutorials
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Triangles — Interactive Study Guide

Congruence Criteria at a Glance

CriterionYou NeedRemember
SAS2 sides + included angleAngle MUST be between the 2 sides
ASA2 angles + included sideSide MUST be between the 2 angles
AAS2 angles + any sideSide can be anywhere
SSS3 sidesAll three sides match
RHSRight angle + hypotenuse + sideONLY for right triangles
NEVER use AAA (only proves similarity) or SSA (ambiguous case)!

Proof Writing Pattern

Every congruence proof follows this pattern:

  1. Identify the two triangles to compare.
  2. List three pairs of equal elements (sides/angles) with reasons.
  3. State the criterion (SAS/ASA/AAS/SSS/RHS).
  4. Conclude congruence.
  5. Use CPCT for any further deduction.

Quick Self-Check

  1. In ΔABC, AB = AC. ∠B = 55°. Find ∠A. (∠C = 55°, ∠A = 70°)
  2. Can a triangle have sides 2, 3, 6? (No: 2+3=5 < 6, violates triangle inequality)
  3. In a triangle, the longest side is opposite to which angle? (The largest angle)

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