Question

ABC and DBC are two isosceles triangles on the same side of BC. Prove that:
(i) DA (or AD) produced bisects BC at right angle.
(ii) angle BDA = angle CDA.​

Answer 1

Diagram drawn.

To prove:-

DA bisects BC

And

angle BDA= Angle CDA

To prove that DA bisects BC , we have to prove that ∆ABD is congruent to ∆ACD.

In ∆ABD & ∆ACD:

AD = AD (common)

Angle BDA = Angle CAD (parallel angles)

Angle BAD= Angle CAD(parallel angles)

(i)Since ,∆ABD congruent to ∆ACD by ASA , AD bisects BC at right angle.

(ii)Since,∆ABD congruent to ∆ACD , angle bda = angle cda

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Answer 2

Answer:

Solution:

Given: ABC and DBC are isosceles triangles

To Prove: ∠ABD = ∠ACD

Let's join point A and point B.

ABC and DBC are two isosceles triangles on the same base BC (see Fig. 7.33). Show that ∠ABD = ∠ACD

In △DAB and △DAC,

AB = AC (Given)

BD = CD (Given)

AD = AD (Common side)

∴ △ ABD ≅ △ ACD (By SSS congruence rule)

∴ ∠ABD = ∠ACD (By CPCT)