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We are given the following information:

- \( \angle BAC = 120^\circ \).
- \( AD \) bisects \( \angle BAF \).
- \( AE \) bisects \( \angle CAF \).

We are asked to find the measure of \( \angle DAE \).

### Steps to Solve:

1. First, since \( AD \) bisects \( \angle BAF \), this implies that:
   \[
   \angle BAD = \angle DAF = \frac{1}{2} \angle BAF
   \]
2. Similarly, since \( AE \) bisects \( \angle CAF \), we have:
   \[
   \angle CAE = \angle EAF = \frac{1}{2} \angle CAF
   \]

3. Now, \( \angle BAC = \angle BAF + \angle CAF = 120^\circ \). Hence, we can express \( \angle BAF \) and \( \angle CAF \) in terms of this total angle:
   \[
   \angle BAF + \angle CAF = 120^\circ
   \]
   Let \( \angle BAF = x \), then \( \angle CAF = 120^\circ - x \).

4. The measure of \( \angle DAE \) is the sum of \( \angle DAF \) and \( \angle EAF \):
   \[
   \angle DAE = \angle DAF + \angle EAF = \frac{1}{2} x + \frac{1}{2} (120^\circ - x)
   \]

5. Simplifying the expression for \( \angle DAE \):
   \[
   \angle DAE = \frac{x}{2} + \frac{120^\circ - x}{2} = \frac{120^\circ}{2} = 60^\circ
   \]

Therefore, the measure of \( \angle DAE \) is \( 60^\circ \).

Would you like further details or clarifications on this solution?

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Here are 5 related questions to explore further:

1. What is the significance of angle bisectors in triangle geometry?
2. How would the solution change if \( \angle BAC \) was different?
3. Can you generalize this approach for other cases of bisected angles?
4. What are other ways to calculate angle measures using geometric properties?
5. How can angle bisectors help in constructing perpendicular bisectors?

**Tip:** Always check whether given angles are supplementary or complementary when dealing with intersecting lines. This can help simplify the problem.

∠BAC = 120°, and points D, E, and F are in the interior of ∠BAC as shown. AD bisects ∠BAF. AE bisects ∠CAF. Find m∠DAE.

This problem involves finding m∠DAE in a 120° angle where two angle bisectors are involved. It uses geometric concepts such as angle bisectors and measurements. Step-by-step solution included.

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