Question : Let D and E be two points on the side BC of $\triangle ABC$ such that AD = AE and $\angle BAD = \angle EAC$. If AB=(3x+1) cm, BD = 9 cm, AC=34 cm and EC = (y + 1) cm, then the value of (x + y) is:

Option 1: 19

Option 2: 16

Option 3: 17

Option 4: 20

Correct Answer: 19

Solution :
If two sides of a triangle are equal, then the angles opposite to these sides are equal.
An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Angle - Angle (AA) property:-
In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.
$\angle$ BAD = $\angle$ EAC (Given in the question)     ----(1)
In $\triangle$ ADE
⇒ AD = AE (Given in the question)
⇒ $\angle$ ADE = $\angle$ AED    (angles of an isosceles triangles)  ----(2)
So $\angle$ ADB = $\angle$ AEC (Supplementary angles of $\angle$ ADE and $\angle$ AED)
Considering $\triangle$ABD and $\triangle$AEC
⇒ $\angle$ ADB = $\angle$ AEC (Equal angles)
$\angle$ BAD = $\angle$ EAC (Given in the question)
AD = AE (Given in the question)
By the Property of triangles ASA, we can conclude
$\triangle$ABD ~ $\triangle$ACE
So,
$\frac{\text{AB}}{\text{AC}} = \frac{\text{BD}}{\text{CE}} = \frac{\text{AD}}{\text{AE}}$
$\frac{3\text{x}+1}{34} = \frac{9}{\text{y}+1} = 1$ ---(3)
Solving the above equation we get
3x + 1 = 34
⇒ x = 11
Similarly,
y + 1 = 9
⇒ y = 8
The value of (x + y) = 11 + 8 = 19
$\therefore$ The value of (x + y) is 19 cm.
Hence, the correct answer is 19 cm.