Answer: ∠ A = ∠B= 48° and ∠ C = 84°
Step-by-step explanation:
Here, m∠ADE: m∠ADB = 2:9
Let m∠ADE = 2x and m∠ADB = 9x
Where x is any value.
By joining the points D and E (construction)
Since, Here DE ║ AB.
⇒ ∠DAB = 2 x
⇒ ∠ A = 4x ( because AD is the angle bisector so, ∠DAB=∠DAE = 2x )
Now, Let O is the intersection point of angle bisectors AD and BE.
Then, By the property of angle bisctor.
O is the circumcenter of the triangle ABC.
Therefore, OA = OB
⇒ ∠DAB = ∠EBA = 2 x
But BE is the angle bisceor,
Therefore, ∠EBA = ∠EBC=2 x
But, ∠B = ∠EBA + ∠EBC
⇒ ∠B = 4x
Now, since BD is the same transversal on the parallel lines AB and ED,
⇒ ∠B = ∠ EDC
⇒ ∠ EDC = 4x
Since, ∠ADB + ∠ADE + ∠EDC = 180°
⇒ 9x + 2x + 4x = 180°
⇒ 15x = 180°
⇒ x = 12°
Thus, the measures of the angles of ΔABC are,
∠ A = 4x =4×12 = 48°
∠ B = 4x =4×12= 48°
⇒ ∠ C = 180° - 48°- 48°= 84°
