In ΔABC, AD and BE are the angle bisectors of ∠A and ∠B and DE║ AB. Find the measures of the angles of ΔABC, if m∠ADE: m∠ADB = 2:9.

Answer: ∠A=48°,∠B=48°,∠C=84°.

Step by-step explanation:

Given: AD and BE are the angle bisectors of ∠A and ∠B

i. e ∠6=∠7 (∵ Angles formed after AD bisected ∠A)

∠4=∠5 (∵ Angles formed after BE bisected ∠B)

Also, DE║AB

⇒ ∠2=∠7 (∵ Alternate interior angles)

∠3=∠6 (∵ Alternate interior angles)

And ∠ADE : ∠ADB = ∠2:∠3 = 2:9 = 2x : 9x ... (1)

To Find: ∠A,∠B,∠C.

Solution: ∠2=∠7 (∵ Given) ... (2)

∠2=∠4 (∵ angles on the same segment) ... (3)

∠4=∠5 = ∠B/2 (∵ Given) ... (4)

∴ In Δ ABD

∠3+∠4+∠5+∠7 = 180 (∵ Sum of interior angles of a triangle)

From equation 2,3,4,5, Put values

9x+2x+2x+2x = 180°

⇒15x = 180°

⇒x=12°

Putting values in equation (4) ⇒ ∠ B = 2 * (2*12) = 48°

Also, ∠B=∠A=48°

Now, in Δ ABC

∠C+∠B+∠A = 180°

⇒48°+48°+∠C = 180°

⇒∠C=84°