Answer:
∠E = 38°
Explanation:
You want the measure of angle E in the triangle shown.
Linear pair
Angle BFA and the one marked 118° form a linear pair, so are supplementary.
∠BFA = 180° -118° = 62°
Alternate interior angles
Transversal BF lies between parallel lines BD and AE, creating alternate interior angles BFA and FBD. These are congruent, so ...
∠FBD = ∠BFA = 62°
Angle bisector
BF bisects angle ABD, so angle ABD is twice the measure of angle FBD:
∠ABD = 2×∠FBD = 2×62° = 124°
Exterior angle
Angle ABD is an exterior angle to triangle BCD, so is equal to the sum of the remote interior angles:
∠ABD = ∠BCD +∠CDB
Then the acute angle at D is ...
∠CDB = ∠ABD -∠BCD = 124° -86° = 38°
Corresponding angles
Transversal CE crosses parallel lines BD and AE, so the corresponding angles it creates are congruent. This means ...
∠E = ∠CDB = 38°
The measure of angle E is 38°.
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Additional comment
You can also find angle A = 56° using the sum of angles in triangle ABF, then angle E using the sum of angles in triangle ACE. The result is the same. The number of math operations is slightly more. The number of geometric relations you need to invoke for that is probably smaller.
In short, the more familiar you are with the geometric relationships, the less work you have to do.
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