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CBSE Class 9 Mathematics Question 7 of 8

Triangles — Question 4

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Question 4

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Answer

Triangle ABC with BE and CF as equal altitudes is shown in the figure below:

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles. NCERT Class 9 Mathematics CBSE Solutions.

Given :

BE is a altitude.

∴ ∠AEB = ∠CEB = 90°

CF is a altitude.

∴ ∠AFC = ∠BFC = 90°

Also, BE = CF.

In Δ BEC and Δ CFB,

⇒ ∠BEC = ∠CFB (Each equal to 90°)

⇒ BC = CB (Common)

⇒ BE = CF (Given)

⇒ Δ BEC ≅ Δ CFB (By R.H.S. congruence rule)

We know that,

Corresponding parts of congruent triangle are equal.

⇒ ∠BCE = ∠CBF (By C.P.C.T.)

As,

Sides opposite to equal angles of a triangle are equal.

∴ AB = AC.

Hence, proved that Δ ABC is an isosceles triangle.

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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