Question

In ΔABC, AD and BE are the angle bisectors of ∠A and ∠B and DE║AB. If m∠ADE is with 34° smaller than m∠CAB, find the measures of the angles of ΔADE.

Answer 1

Answer: The measures of angles of Δ ADE are, m∠ ADE = m∠DAE= 34° and m∠DEA = 112°

Explanation:

Here, AD and BE are the angle bisectors of ∠A and ∠B

Therefore, m∠BAD = m∠DAE = m∠ CAB/2 -------(1)

And, m∠ABE = m∠ EBD

Now, DE║AB

Therefore, m∠BAD = m∠ ADE ( By the alternative interior angle theorem)

⇒ m∠ CAB/2 =m ∠ ADE -----------(2)

But given, m∠ CAB - m∠ ADE = 34°

⇒ m∠ CAB - m∠ CAB/2 = 34°

⇒m∠ CAB/2 = 34⇒ m∠CAB = 68°

⇒ m∠ ADE = 68°/2 = 34° ( By equation (2) )

And, m∠DAE = 34°

Since, in Δ ADE,

m∠ ADE + m∠DAE + m∠DEA = 180°

⇒ m∠DEA = 112°