Question

in the diagram below, $overline{ad}paralleloverline{eg}$, $overline{bh}perpoverline{fc}$, and $mangle abh = 149^{circ}$. find $mangle efh$. you may assume lines that appear straight are straight, but the figure is not otherwise drawn

Explanation:

Step1: Find ∠HBC

Since ∠ABH = 149° and ∠ABC is a straight - line angle (180°), then ∠HBC=180° - ∠ABH.
∠HBC = 180°-149° = 31°.

Step2: Find ∠BCH

Because BH⊥FC, ∠BHC = 90°. In triangle BCH, using the angle - sum property of a triangle (the sum of interior angles of a triangle is 180°), let ∠BCH=x. Then x + ∠HBC+∠BHC = 180°. Substituting the known values, we get x+31° + 90°=180°, so x = 180°-(90° + 31°)=59°.

Step3: Find ∠EFH

Since AD∥EG, ∠EFH and ∠BCH are corresponding angles. Corresponding angles formed by a transversal intersecting two parallel lines are equal. So ∠EFH = ∠BCH.

Answer:

59°