Mathematics
BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.
Triangles
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Answer
Triangle ABC with BE and CF as equal altitudes is shown in the figure below:
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.
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