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.

Let's assume that m∠CAB = x°.

Since AD is the angle bisector of ∠A, we can use the angle bisector theorem to find that
m∠DAB : m∠BAD = BD : AD.

Similarly, since BE is the angle bisector of ∠B, we can use the angle bisector theorem to find that
m∠DBE : m∠BED = DE : BE.

Since DE || AB, ∠DBE and ∠DAB are corresponding angles, so they are congruent. This means that m∠DBE = m∠DAB.

Now, let's consider ∠ADE. We are given that m∠ADE is 34° smaller than m∠CAB. So, m∠ADE = (x - 34)°.

Also, since ∠ADE and ∠BAD are adjacent angles, we can find that
m∠BAD + m∠ADE + m∠DBE = 180°.

Substituting the values we already know, we have:
m∠BAD + (x - 34) + m∠DAB = 180.

Since m∠DAB = m∠DBE, we can rewrite the equation as:
m∠BAD + (x - 34) + m∠DAB = 180.

Using the angle bisector theorem, we know that:
m∠DAB : m∠BAD = BD : AD.

So, we can write BD/AD = m∠DAB/m∠BAD = m∠DBE/m∠BED.

Substituting the known values, we have:
BD/AD = m∠DAB/m∠BAD = m∠DBE/m∠BED.
1/1 = m∠DAB/m∠BAD = m∠DBE/m∠BED. (Since we know that ∠DAB and ∠DBE are congruent.)

Now, we have two equations:
m∠BAD + (x - 34) + m∠DAB = 180
1/1 = m∠DAB/m∠BAD = m∠DBE/m∠BED.

Let's solve these two equations:

m∠BAD + (x - 34) + m∠DAB = 180 -> equation 1
1/1 = m∠DAB/m∠BAD = m∠DBE/m∠BED -> equation 2

From equation 2, we can conclude that m∠DAB = m∠BAD = 1/2.

Substituting this value in equation 1, we have:
1/2 + 1/2 + (x - 34) = 180
1 + (x - 34) = 180
x - 33 = 180
x = 180 + 33
x = 213

So, m∠CAB = 213°.

Now, we can find m∠DAB:
m∠DAB = 1/2 * m∠CAB
m∠DAB = 1/2 * 213
m∠DAB = 106.5°

Since m∠DAB = m∠DBE, we can conclude that m∠DBE = 106.5°.

Finally, we can find m∠ADE:
m∠ADE = m∠CAB - 34
m∠ADE = 213 - 34
m∠ADE = 179°.

So, the measures of the angles of ΔADE are:
m∠ADE = 179°
m∠DAB = 106.5°
m∠DBE = 106.5°.

To solve this problem, we can begin by using the angle bisector theorem to find the relationship between the lengths of segments AD, DB, AE, and EC.

According to the angle bisector theorem, we can write the following proportions:

1. AD/DB = AC/CB (since AD is an angle bisector of ∠A)
2. AE/EC = AB/BC (since AE is an angle bisector of ∠B)

We also know that DE is parallel to AB, which means we have the following similar triangles: ΔADE ~ ΔABC.

Using these proportions, we can find a relationship between AD and AE:

AD/DB = AE/EC (from the angle bisector theorem)

Since DE is parallel to AB, we can substitute AB/BC for AE/EC:

AD/DB = AB/BC

Now, let's use this information to solve the problem.

Given that m∠ADE is 34° smaller than m∠CAB, we can express this as an equation:

m∠ADE + 34° = m∠CAB

Let's denote m∠ADE as x. We can rewrite the equation as:

x + 34° = m∠CAB

We also know that m∠ADE + m∠ABC = 180° (since ΔABC is a triangle), so we have:

x + m∠ABC = 180°

Since AD is an angle bisector of ∠A, m∠ABC = m∠CAB. We can substitute this into the equation:

x + m∠CAB = 180°

Now substitute x + 34° for m∠CAB:

x + (x + 34°) = 180°

2x + 34° = 180°

Solving for x:

2x = 180° - 34°

2x = 146°

x = 73°

Therefore, m∠ADE is 73°.

Since ∠A and ∠B are each bisected by AD and BE, respectively, we know that ∠ADE and ∠BED are equal to half of their corresponding angles ∠A and ∠B.

Thus, m∠ADE = m∠BED = 73°.

To find the measure of ∠D, we can use the fact that the sum of the angles in a triangle is 180°:

∠D + ∠ADE + ∠BED = 180°

∠D + 73° + 73° = 180°

∠D + 146° = 180°

∠D = 180° - 146°

∠D = 34°

Therefore, the measure of ∠D is 34°.

To find the measure of ∠E, we can use the fact that the sum of the angles in a triangle is 180°:

∠E + ∠ADE + ∠BED = 180°

∠E + 73° + 73° = 180°

∠E + 146° = 180°

∠E = 180° - 146°

∠E = 34°

Therefore, the measure of ∠E is 34°.

The measures of the angles of ΔADE are:
∠D = 34°
∠A = ∠E = 73°

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.

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