I don’t understand what I should do next

9514 1404 393

  ∠E = ∠A = 34°; ∠AFB = ∠DFE = 107°-34° = 73°

Step-by-step explanation:

Here's where I'd go next.

73° (∠EDA) is exterior to ΔDAC, so is the sum of remote interior angles A and C. That is, ...

  ∠C + ∠A = 73°

  ∠C = 73° -∠A = 73° -39° = 34°

Triangle CBE is given as isosceles, so its base angles C and E are congruent, both 34°.

Exterior angle CDF is the sum of remote interior angles DEF and DFE, so we have ...

  107° = 34° +∠DFE

  ∠DFE = 107° -34° = 73°

The angle we just found (∠DFE) is a vertical angle with ∠AFB, so has the same measure.

  ∠AFB = 73° . . . . vertical angle with ∠DFE

In summary, it looks like we have added some steps to the derivation:

  4. ∠A = 34° . . . . exterior angle theorem (∠A +39° = 73°)

  5. ∠E = ∠A = 34° . . . . definition of isosceles triangle

  6. ∠DFE = 73° . . . . exterior angle theorem (∠E +34° = 107°)

  7. ∠AFB = 73° . . . . vertical angles

Additional comment

Any sort of math problem solving has you work from what you know to what you need to know by making use of previously learned relationships between the bits of information you have. Here, you really only need to know (a) angles of a linear pair total 180°; (b) angles in a triangle total 180°; (c) angles opposite congruent sides in a triangle are congruent. Everything we claim above can be concluded using these facts. The various theorems we used are simply shortcuts that allow you to bypass intermediate steps in the problem-solving process.

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