Question

in an isosceles triangle abc , ab=ac , d and e are points on vc such that be=cd and ad=ae

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Answer 1

Answer:

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Answer 2

As we have an isosceles triangle ABC with AB = AC, where D and E are points on BC such that BE = CD and we need to prove that the sides AD and AE are equal i.e. AD = AE.

So, we have

AB = AC……………..(i)

BE = CD…………….(ii)

Now, we know the property of triangles that opposite angles of the opposite sides are equal if sides are equal. It means ∠B∠B and ∠C∠C will be equal as opposite sides of ∠B∠B and ∠C∠C i.e. AC and AB, are equal. Hence, we get

∠B=∠C...............(iii)

And from equation (ii), we (iii) have BE = CD

Now, subtract DE from both sides of terms of the above equation. So, we get

BE – DE = CD – DE

Now, we can observe that the diagram is replaced by side BD and CD – DE by side CE. Hence, we get above equation as

BD = CE………………….(iv)

Now, in ΔABDΔABDand ΔAEC , we have

AB = AC (from equation (i))

∠B=∠C (from equation (iii))

BD = CE (from equation (iv))

Hence, ΔABDis congruent to ΔAECby SAS criteria of congruence. So, we get

ΔABD≅ΔAEC

So, now all the corresponding sides and angles of triangles ABD and ACE are equal by the C.P.C.T property of congruent triangles.

Hence, we get

AD = AE ( C.P.C.T)

So, it proved that AD and AE are equal.

Note: Another approach for proving AD = AE, we can prove the triangles ABE and ADE as congruent triangle in the following way:

AB = AC

BE = DC

∠B=∠C

By SAS criteria ΔABE≅ΔADC.So, it can be another approach. Getting the equation ∠B=∠C∠B=∠C is the key point for proving the triangles ABD and ACE to congruent problems and need to use property for getting it.

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