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

To find the measures of the angles in ΔADE, we can apply the angle bisector theorem and the fact that DE is parallel to AB. The measures of ∠ADE, ∠AED, and ∠DAE are denoted as x, y, and z, respectively.

Step-by-step explanation:

In ΔADE, we know that AD is the angle bisector of ∠A and BE is the angle bisector of ∠B. We also know that DE is parallel to AB. Given that m∠ADE is 34° smaller than m∠CAB, we need to find the measures of the angles in ΔADE.

Let's denote the measures of ∠ADE, ∠AED, and ∠DAE as x, y, and z, respectively.

From the angle bisector theorem, we know that ∠CAD = ∠DAB = y+z.

Since DE is parallel to AB, we have ∠ADE = ∠CAB = x+y+z.

Therefore, the measures of the angles in ΔADE are ∠ADE = x, ∠AED = y, and ∠DAE = z.

Answer 2

34°

Step-by-step explanation:

If m∠ADE is with 34° smaller than m∠CAB, then denote

m∠ADE=x°,

m∠CAB=(x+34)°.

Since DE ║ AB, then

m∠ADE=m∠DAB=x°.

AD is angle A bisector, then

m∠EAD=m∠DAB=x°.

Thus,

m∠CAB=m∠CAD+m∠DAB=(x+x)°=2x°.

On the other hand,

m∠CAB=(x+34)°,

then

2x°=(x+34)°,

m∠ADE=x°=34°.