Question

18. Equilateral triangles ABD and ACE are drawn on the sides AB
and AC of AABC as shown in the figure. Prove that :
(i) _DAC = EAB (ii) DC = BE.​

Answer 1

Question:

Equilateral triangles ABD and ACE are drawn on the sides AB

and AC of AABC as shown in the figure. Prove that :

(i) _DAC = EAB (ii) DC = BE.

Step-by-step explanation:

Given:

Triangle ABD and triangle ACE are equilateral triangles on sides AB and AC, respectively.

To prove:

(i) angleDAC = angleEAB

(ii) DC = BE.

Proof:

(i) We have triangle ABD ~ triangle ACE (all equilateral triangles are similar)

Therefore, angle DAB = angle EAC by CPST (Corresponding parts of similar triangle)

=> angle DAB + angle BAC = angle EAC + angle BAC

=> angleDAC = angleEAB

Hence, angleDAC = angleEAB proved.

(ii) In triangle DAC and triangle EAB, we have

DA/AE = AC/BA,

angleDAC = angleEAB

Therefore, triangle DAC ~ triangle EAB

Now, DA/AE = AC/BA = DC/BE

=> DA/AC = AC/DA = DC/BE

=> 1 = DC/BE

=> BE = DC

Hence, DC = BE proved.

Concept used > Similarly of triangles.

HOPE IT WAS HELPFUL