Question

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$\Large{\underline{\underline{\mathfrak{Question :}}}}$
In the adjoining figure,△ABC is an isosceles triangle in which AB = AC. If E and F be the midpoints of AC and AB respectively, prove that BE = CF.​

Answer 1

Congruency Of Triangles

We will solve this question by using the concept of congruency of triangle. Two triangles are said to be congruent when they are identical to each other or are similar.

Rules of congruency :-

• When 3 sides are equal ( S S S )
• Two angles and one side is same ( A S A )
• Two sides and the angle between them is equal ( S A S )
• Two angles and one side in the order ( A A S )

Remember that these rules must be in order.

Solution :-

We are given that the triangle is isosceles i.e. two sides and two angles are same.

AB = AC

angle ABC = angle ACB . . . (1.)

We are given that E and F are mid points, i.e. BF and CE are also equal, ( half of AB and half of AC ) . . . (2.)

BC is the common base for triangle EBC and ECB.

In triangle FBC and triangle ECB :-

BF = EC ( Proved above )

angle B = angle C ( By property of isosceles ∆ )

BC = BC ( Common )

Hence ∆ FBC and ∆ ECB are congruent by ( S A S ) criteria of congruency.

Corresponding part of congruent triangles are equal to each other,

so BE = CF ( By CPCT )

• Hence proved . . . !!

Answer 2

Use congruency rule to prove both triangles as congruent and by CPCT both are equal

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