Question

In triangle ABC BE and CF are altitudes on the side AC and AB respectively such that BE=CF. Using RHS congruence rule prove that AB = AC

Answer 1

Congruence of triangles:
Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:
There are 4 criteria for congruence of triangles.

Here we use ASA Congruence.

ASA(angle side angle):
Two Triangles are congruent if two angles and the included side of One triangle are equal to two angles & the included side of the other trian gle.

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Given:
ΔABC in which BE perpendicular to AC & CF perpendicular to AB, such that BE=CF.

To Prove:
i) ΔABE ≅ ΔACF
ii) AB=AC

Proof:
(i) In ΔABE & ΔACF,
∠A = ∠A (Common)
∠AEB = ∠AFC (each 90°)
BE = CF (Given)

Therefore, ΔABE ≅ ΔACF (by ASA congruence rule)

(ii) since ΔABE ≅ ΔACF
Thus, AB = AC (by CPCT)


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Answer 2

Hello mate ^_^

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Solution:

In ∆BEC and ∆CFB

BE=CF                (Given)

∠BEC=∠CFB              (Each given equal to 90°)

BC=CB                (Common)

Therefore, by RHS rule, ∆BEC≅∆CFB

It means that ∠C=∠B        (Corresponding parts of congruent triangles are equal)

⇒AB=AC                (In a triangle, sides opposite to equal angles are equal)

Therefore, ∆ABC is isosceles.

hope, this will help you.

Thank you______❤

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